id	sid	tid	token	lemma	pos
bcpe-1790	1	1	bcp	bcp	PROPN
bcpe-1790	1	2	education	education	PROPN
bcpe-1790	1	3	&	&	CCONJ
bcpe-1790	1	4	psychology	psychology	PROPN
bcpe-1790	1	5	erhss	erhss	PROPN
bcpe-1790	1	6	2022	2022	NUM
bcpe-1790	1	7	volume	volume	NOUN
bcpe-1790	1	8	6	6	NUM
bcpe-1790	1	9	(	(	PUNCT
bcpe-1790	1	10	2022	2022	NUM
bcpe-1790	1	11	)	)	PUNCT
bcpe-1790	1	12	200	200	NUM
bcpe-1790	1	13	the	the	DET
bcpe-1790	1	14	innovative	innovative	ADJ
bcpe-1790	1	15	application	application	NOUN
bcpe-1790	1	16	of	of	ADP
bcpe-1790	1	17	polya	polya	NOUN
bcpe-1790	1	18	's	's	PART
bcpe-1790	1	19	problem	problem	NOUN
bcpe-1790	1	20	-	-	PUNCT
bcpe-1790	1	21	solving	solve	VERB
bcpe-1790	1	22	thought	thought	NOUN
bcpe-1790	1	23	in	in	ADP
bcpe-1790	1	24	the	the	DET
bcpe-1790	1	25	teaching	teaching	NOUN
bcpe-1790	1	26	of	of	ADP
bcpe-1790	1	27	mathematical	mathematical	ADJ
bcpe-1790	1	28	problem	problem	NOUN
bcpe-1790	1	29	-	-	PUNCT
bcpe-1790	1	30	solving	solving	NOUN
bcpe-1790	1	31	--	--	PUNCT
bcpe-1790	1	32	take	take	VERB
bcpe-1790	1	33	a	a	DET
bcpe-1790	1	34	conic	conic	ADJ
bcpe-1790	1	35	section	section	NOUN
bcpe-1790	1	36	of	of	ADP
bcpe-1790	1	37	the	the	DET
bcpe-1790	1	38	college	college	NOUN
bcpe-1790	1	39	entrance	entrance	NOUN
bcpe-1790	1	40	examination	examination	NOUN
bcpe-1790	1	41	as	as	ADP
bcpe-1790	1	42	an	an	DET
bcpe-1790	1	43	example	example	NOUN
bcpe-1790	1	44	yuwei	yuwei	PROPN
bcpe-1790	1	45	wu	wu	PROPN
bcpe-1790	1	46	*	*	PROPN
bcpe-1790	1	47	guangzhou	guangzhou	PROPN
bcpe-1790	1	48	university	university	PROPN
bcpe-1790	1	49	,	,	PUNCT
bcpe-1790	1	50	school	school	NOUN
bcpe-1790	1	51	of	of	ADP
bcpe-1790	1	52	mathematics	mathematic	NOUN
bcpe-1790	1	53	and	and	CCONJ
bcpe-1790	1	54	information	information	NOUN
bcpe-1790	1	55	science	science	NOUN
bcpe-1790	1	56	,	,	PUNCT
bcpe-1790	1	57	guangdong	guangdong	PROPN
bcpe-1790	1	58	,	,	PUNCT
bcpe-1790	1	59	china	china	PROPN
bcpe-1790	1	60	*	*	PUNCT
bcpe-1790	1	61	corresponding	correspond	VERB
bcpe-1790	1	62	author	author	NOUN
bcpe-1790	1	63	:	:	PUNCT
bcpe-1790	1	64	vj162774@163.com	vj162774@163.com	X
bcpe-1790	1	65	abstract	abstract	PROPN
bcpe-1790	1	66	.	.	PUNCT
bcpe-1790	2	1	in	in	ADP
bcpe-1790	2	2	view	view	NOUN
bcpe-1790	2	3	of	of	ADP
bcpe-1790	2	4	the	the	DET
bcpe-1790	2	5	current	current	ADJ
bcpe-1790	2	6	situation	situation	NOUN
bcpe-1790	2	7	of	of	ADP
bcpe-1790	2	8	mathematics	mathematics	NOUN
bcpe-1790	2	9	teaching	teaching	NOUN
bcpe-1790	2	10	thinking	thinking	NOUN
bcpe-1790	2	11	and	and	CCONJ
bcpe-1790	2	12	the	the	DET
bcpe-1790	2	13	problems	problem	NOUN
bcpe-1790	2	14	,	,	PUNCT
bcpe-1790	2	15	which	which	PRON
bcpe-1790	2	16	are	be	AUX
bcpe-1790	2	17	easily	easily	ADV
bcpe-1790	2	18	overlooked	overlook	VERB
bcpe-1790	2	19	in	in	ADP
bcpe-1790	2	20	high	high	ADJ
bcpe-1790	2	21	school	school	NOUN
bcpe-1790	2	22	classrooms	classroom	NOUN
bcpe-1790	2	23	using	use	VERB
bcpe-1790	2	24	polya	polya	PROPN
bcpe-1790	2	25	's	's	PART
bcpe-1790	2	26	problem	problem	NOUN
bcpe-1790	2	27	-	-	PUNCT
bcpe-1790	2	28	solving	solve	VERB
bcpe-1790	2	29	ideas	idea	NOUN
bcpe-1790	2	30	.	.	PUNCT
bcpe-1790	3	1	this	this	DET
bcpe-1790	3	2	paper	paper	NOUN
bcpe-1790	3	3	innovates	innovate	VERB
bcpe-1790	3	4	the	the	DET
bcpe-1790	3	5	"	"	PUNCT
bcpe-1790	3	6	'	'	PUNCT
bcpe-1790	3	7	how	how	SCONJ
bcpe-1790	3	8	to	to	PART
bcpe-1790	3	9	solve	solve	VERB
bcpe-1790	3	10	it	it	PRON
bcpe-1790	3	11	'	'	PART
bcpe-1790	3	12	list	list	NOUN
bcpe-1790	3	13	"	"	PUNCT
bcpe-1790	3	14	in	in	ADP
bcpe-1790	3	15	combination	combination	NOUN
bcpe-1790	3	16	with	with	ADP
bcpe-1790	3	17	the	the	DET
bcpe-1790	3	18	characteristics	characteristic	NOUN
bcpe-1790	3	19	of	of	ADP
bcpe-1790	3	20	the	the	DET
bcpe-1790	3	21	current	current	ADJ
bcpe-1790	3	22	era	era	NOUN
bcpe-1790	3	23	,	,	PUNCT
bcpe-1790	3	24	which	which	PRON
bcpe-1790	3	25	extends	extend	VERB
bcpe-1790	3	26	the	the	DET
bcpe-1790	3	27	"	"	PUNCT
bcpe-1790	3	28	four	four	NUM
bcpe-1790	3	29	steps	step	NOUN
bcpe-1790	3	30	to	to	PART
bcpe-1790	3	31	solve	solve	VERB
bcpe-1790	3	32	problems	problem	NOUN
bcpe-1790	3	33	"	"	PUNCT
bcpe-1790	3	34	into	into	ADP
bcpe-1790	3	35	five	five	NUM
bcpe-1790	3	36	steps	step	NOUN
bcpe-1790	3	37	.	.	PUNCT
bcpe-1790	4	1	to	to	PART
bcpe-1790	4	2	take	take	VERB
bcpe-1790	4	3	a	a	DET
bcpe-1790	4	4	conic	conic	ADJ
bcpe-1790	4	5	section	section	NOUN
bcpe-1790	4	6	of	of	ADP
bcpe-1790	4	7	the	the	DET
bcpe-1790	4	8	college	college	NOUN
bcpe-1790	4	9	entrance	entrance	NOUN
bcpe-1790	4	10	examination	examination	NOUN
bcpe-1790	4	11	as	as	ADP
bcpe-1790	4	12	the	the	DET
bcpe-1790	4	13	carrier	carrier	NOUN
bcpe-1790	4	14	,	,	PUNCT
bcpe-1790	4	15	the	the	DET
bcpe-1790	4	16	five	five	NUM
bcpe-1790	4	17	steps	step	NOUN
bcpe-1790	4	18	are	be	AUX
bcpe-1790	4	19	applied	apply	VERB
bcpe-1790	4	20	in	in	ADP
bcpe-1790	4	21	teaching	teach	VERB
bcpe-1790	4	22	mathematical	mathematical	ADJ
bcpe-1790	4	23	problemsolving	problemsolving	NOUN
bcpe-1790	4	24	.	.	PUNCT
bcpe-1790	5	1	it	it	PRON
bcpe-1790	5	2	is	be	AUX
bcpe-1790	5	3	expounded	expound	VERB
bcpe-1790	5	4	in	in	ADP
bcpe-1790	5	5	detail	detail	NOUN
bcpe-1790	5	6	to	to	PART
bcpe-1790	5	7	provide	provide	VERB
bcpe-1790	5	8	a	a	DET
bcpe-1790	5	9	method	method	NOUN
bcpe-1790	5	10	reference	reference	NOUN
bcpe-1790	5	11	for	for	ADP
bcpe-1790	5	12	optimizing	optimize	VERB
bcpe-1790	5	13	the	the	DET
bcpe-1790	5	14	teaching	teaching	NOUN
bcpe-1790	5	15	thinking	thinking	NOUN
bcpe-1790	5	16	of	of	ADP
bcpe-1790	5	17	high	high	ADJ
bcpe-1790	5	18	school	school	NOUN
bcpe-1790	5	19	mathematics	mathematic	NOUN
bcpe-1790	5	20	teachers	teacher	NOUN
bcpe-1790	5	21	.	.	PUNCT
bcpe-1790	6	1	keywords	keyword	NOUN
bcpe-1790	6	2	:	:	PUNCT
bcpe-1790	6	3	problem	problem	NOUN
bcpe-1790	6	4	-	-	PUNCT
bcpe-1790	6	5	solving	solve	VERB
bcpe-1790	6	6	thinking	thinking	NOUN
bcpe-1790	6	7	;	;	PUNCT
bcpe-1790	6	8	teaching	teach	VERB
bcpe-1790	6	9	thinking	thinking	NOUN
bcpe-1790	6	10	;	;	PUNCT
bcpe-1790	6	11	conic	conic	ADJ
bcpe-1790	6	12	section	section	NOUN
bcpe-1790	6	13	.	.	PUNCT
bcpe-1790	7	1	1	1	X
bcpe-1790	7	2	.	.	X
bcpe-1790	7	3	the	the	DET
bcpe-1790	7	4	question	question	NOUN
bcpe-1790	7	5	is	be	AUX
bcpe-1790	7	6	raised	raise	VERB
bcpe-1790	7	7	1.1	1.1	NUM
bcpe-1790	7	8	mathematical	mathematical	ADJ
bcpe-1790	7	9	teaching	teaching	NOUN
bcpe-1790	7	10	thinking	thinking	NOUN
bcpe-1790	7	11	and	and	CCONJ
bcpe-1790	7	12	its	its	PRON
bcpe-1790	7	13	present	present	ADJ
bcpe-1790	7	14	situation	situation	NOUN
bcpe-1790	7	15	mathematics	mathematic	NOUN
bcpe-1790	7	16	teaching	teaching	NOUN
bcpe-1790	7	17	thinking	thinking	NOUN
bcpe-1790	7	18	is	be	AUX
bcpe-1790	7	19	the	the	DET
bcpe-1790	7	20	rational	rational	ADJ
bcpe-1790	7	21	understanding	understanding	NOUN
bcpe-1790	7	22	of	of	ADP
bcpe-1790	7	23	mathematics	mathematic	NOUN
bcpe-1790	7	24	teaching	teaching	NOUN
bcpe-1790	7	25	activities	activity	NOUN
bcpe-1790	7	26	(	(	PUNCT
bcpe-1790	7	27	processes	process	NOUN
bcpe-1790	7	28	)	)	PUNCT
bcpe-1790	7	29	by	by	ADP
bcpe-1790	7	30	mathematics	mathematics	NOUN
bcpe-1790	7	31	teachers	teacher	NOUN
bcpe-1790	7	32	,	,	PUNCT
bcpe-1790	7	33	the	the	DET
bcpe-1790	7	34	internal	internal	ADJ
bcpe-1790	7	35	rational	rational	ADJ
bcpe-1790	7	36	activities	activity	NOUN
bcpe-1790	7	37	of	of	ADP
bcpe-1790	7	38	the	the	DET
bcpe-1790	7	39	interaction	interaction	NOUN
bcpe-1790	7	40	between	between	ADP
bcpe-1790	7	41	mathematics	mathematics	NOUN
bcpe-1790	7	42	teachers	teacher	NOUN
bcpe-1790	7	43	and	and	CCONJ
bcpe-1790	7	44	mathematics	mathematic	NOUN
bcpe-1790	7	45	teaching	teaching	NOUN
bcpe-1790	7	46	objects	object	NOUN
bcpe-1790	7	47	(	(	PUNCT
bcpe-1790	7	48	textbooks	textbook	NOUN
bcpe-1790	7	49	,	,	PUNCT
bcpe-1790	7	50	students	student	NOUN
bcpe-1790	7	51	,	,	PUNCT
bcpe-1790	7	52	etc	etc	X
bcpe-1790	7	53	.	.	X
bcpe-1790	7	54	)	)	PUNCT
bcpe-1790	7	55	ability	ability	NOUN
bcpe-1790	7	56	.	.	PUNCT
bcpe-1790	8	1	mathematics	mathematic	NOUN
bcpe-1790	8	2	teaching	teaching	NOUN
bcpe-1790	8	3	thinking	think	VERB
bcpe-1790	8	4	"	"	PUNCT
bcpe-1790	8	5	focuses	focus	VERB
bcpe-1790	8	6	on	on	ADP
bcpe-1790	8	7	teaching	teach	VERB
bcpe-1790	8	8	thinking	thinking	NOUN
bcpe-1790	8	9	that	that	PRON
bcpe-1790	8	10	meets	meet	VERB
bcpe-1790	8	11	the	the	DET
bcpe-1790	8	12	needs	need	NOUN
bcpe-1790	8	13	of	of	ADP
bcpe-1790	8	14	students	student	NOUN
bcpe-1790	8	15	,	,	PUNCT
bcpe-1790	8	16	rather	rather	ADV
bcpe-1790	8	17	than	than	ADP
bcpe-1790	8	18	thinking	think	VERB
bcpe-1790	8	19	only	only	ADV
bcpe-1790	8	20	for	for	ADP
bcpe-1790	8	21	teachers	teacher	NOUN
bcpe-1790	8	22	themselves	themselves	PRON
bcpe-1790	8	23	or	or	CCONJ
bcpe-1790	8	24	the	the	DET
bcpe-1790	8	25	content	content	NOUN
bcpe-1790	8	26	of	of	ADP
bcpe-1790	8	27	the	the	DET
bcpe-1790	8	28	subject	subject	NOUN
bcpe-1790	8	29	.	.	PUNCT
bcpe-1790	9	1	"[1	"[1	X
bcpe-1790	9	2	-	-	PUNCT
bcpe-1790	9	3	3	3	X
bcpe-1790	9	4	]	]	PUNCT
bcpe-1790	9	5	the	the	DET
bcpe-1790	9	6	teaching	teaching	NOUN
bcpe-1790	9	7	thinking	thinking	NOUN
bcpe-1790	9	8	of	of	ADP
bcpe-1790	9	9	mathematics	mathematics	NOUN
bcpe-1790	9	10	teachers	teacher	NOUN
bcpe-1790	9	11	has	have	AUX
bcpe-1790	9	12	always	always	ADV
bcpe-1790	9	13	been	be	AUX
bcpe-1790	9	14	an	an	DET
bcpe-1790	9	15	essential	essential	ADJ
bcpe-1790	9	16	part	part	NOUN
bcpe-1790	9	17	of	of	ADP
bcpe-1790	9	18	mathematics	mathematic	NOUN
bcpe-1790	9	19	teaching	teaching	NOUN
bcpe-1790	9	20	,	,	PUNCT
bcpe-1790	9	21	and	and	CCONJ
bcpe-1790	9	22	its	its	PRON
bcpe-1790	9	23	differences	difference	NOUN
bcpe-1790	9	24	will	will	AUX
bcpe-1790	9	25	directly	directly	ADV
bcpe-1790	9	26	affect	affect	VERB
bcpe-1790	9	27	the	the	DET
bcpe-1790	9	28	quality	quality	NOUN
bcpe-1790	9	29	of	of	ADP
bcpe-1790	9	30	teaching	teaching	NOUN
bcpe-1790	9	31	.	.	PUNCT
bcpe-1790	10	1	while	while	SCONJ
bcpe-1790	10	2	it	it	PRON
bcpe-1790	10	3	is	be	AUX
bcpe-1790	10	4	widely	widely	ADV
bcpe-1790	10	5	believed	believe	VERB
bcpe-1790	10	6	that	that	SCONJ
bcpe-1790	10	7	mathematics	mathematic	NOUN
bcpe-1790	10	8	is	be	AUX
bcpe-1790	10	9	the	the	DET
bcpe-1790	10	10	gymnastics	gymnastic	NOUN
bcpe-1790	10	11	of	of	ADP
bcpe-1790	10	12	the	the	DET
bcpe-1790	10	13	mind	mind	NOUN
bcpe-1790	10	14	,	,	PUNCT
bcpe-1790	10	15	its	its	PRON
bcpe-1790	10	16	function	function	NOUN
bcpe-1790	10	17	as	as	ADP
bcpe-1790	10	18	"	"	PUNCT
bcpe-1790	10	19	mind	mind	NOUN
bcpe-1790	10	20	gymnastics	gymnastic	NOUN
bcpe-1790	10	21	"	"	PUNCT
bcpe-1790	10	22	has	have	AUX
bcpe-1790	10	23	not	not	PART
bcpe-1790	10	24	been	be	AUX
bcpe-1790	10	25	fully	fully	ADV
bcpe-1790	10	26	utilized	utilize	VERB
bcpe-1790	10	27	.	.	PUNCT
bcpe-1790	11	1	it	it	PRON
bcpe-1790	11	2	is	be	AUX
bcpe-1790	11	3	mainly	mainly	ADV
bcpe-1790	11	4	due	due	ADJ
bcpe-1790	11	5	to	to	ADP
bcpe-1790	11	6	the	the	DET
bcpe-1790	11	7	following	follow	VERB
bcpe-1790	11	8	three	three	NUM
bcpe-1790	11	9	deficiencies	deficiency	NOUN
bcpe-1790	11	10	in	in	ADP
bcpe-1790	11	11	mathematics	mathematic	NOUN
bcpe-1790	11	12	teaching	teaching	NOUN
bcpe-1790	11	13	.	.	PUNCT
bcpe-1790	12	1	firstly	firstly	ADV
bcpe-1790	12	2	,	,	PUNCT
bcpe-1790	12	3	it	it	PRON
bcpe-1790	12	4	focuses	focus	VERB
bcpe-1790	12	5	on	on	ADP
bcpe-1790	12	6	knowledge	knowledge	NOUN
bcpe-1790	12	7	and	and	CCONJ
bcpe-1790	12	8	skills	skill	NOUN
bcpe-1790	12	9	while	while	SCONJ
bcpe-1790	12	10	ignoring	ignore	VERB
bcpe-1790	12	11	mathematical	mathematical	ADJ
bcpe-1790	12	12	thinking	thinking	NOUN
bcpe-1790	12	13	.	.	PUNCT
bcpe-1790	13	1	secondly	secondly	ADV
bcpe-1790	13	2	,	,	PUNCT
bcpe-1790	13	3	it	it	PRON
bcpe-1790	13	4	focuses	focus	VERB
bcpe-1790	13	5	on	on	ADP
bcpe-1790	13	6	thinking	think	VERB
bcpe-1790	13	7	results	result	NOUN
bcpe-1790	13	8	and	and	CCONJ
bcpe-1790	13	9	neglects	neglect	VERB
bcpe-1790	13	10	the	the	DET
bcpe-1790	13	11	thinking	thinking	NOUN
bcpe-1790	13	12	process	process	NOUN
bcpe-1790	13	13	[	[	X
bcpe-1790	13	14	4	4	NUM
bcpe-1790	13	15	-	-	SYM
bcpe-1790	13	16	5	5	NUM
bcpe-1790	13	17	]	]	PUNCT
bcpe-1790	13	18	.	.	PUNCT
bcpe-1790	14	1	1.2	1.2	NUM
bcpe-1790	14	2	polya	polya	NOUN
bcpe-1790	14	3	's	's	PART
bcpe-1790	14	4	problem	problem	NOUN
bcpe-1790	14	5	-	-	PUNCT
bcpe-1790	14	6	solving	solve	VERB
bcpe-1790	14	7	ideas	idea	NOUN
bcpe-1790	14	8	and	and	CCONJ
bcpe-1790	14	9	new	new	ADJ
bcpe-1790	14	10	mathematical	mathematical	ADJ
bcpe-1790	14	11	problems	problem	NOUN
bcpe-1790	14	12	in	in	ADP
bcpe-1790	14	13	the	the	DET
bcpe-1790	14	14	book	book	NOUN
bcpe-1790	14	15	"	"	PUNCT
bcpe-1790	14	16	how	how	SCONJ
bcpe-1790	14	17	to	to	PART
bcpe-1790	14	18	solve	solve	VERB
bcpe-1790	14	19	it	it	PRON
bcpe-1790	14	20	,	,	PUNCT
bcpe-1790	14	21	"	"	PUNCT
bcpe-1790	14	22	george	george	PROPN
bcpe-1790	14	23	polya	polya	PROPN
bcpe-1790	14	24	proposed	propose	VERB
bcpe-1790	14	25	four	four	NUM
bcpe-1790	14	26	steps	step	NOUN
bcpe-1790	14	27	for	for	ADP
bcpe-1790	14	28	solving	solve	VERB
bcpe-1790	14	29	problems	problem	NOUN
bcpe-1790	14	30	in	in	ADP
bcpe-1790	14	31	the	the	DET
bcpe-1790	14	32	process	process	NOUN
bcpe-1790	14	33	of	of	ADP
bcpe-1790	14	34	thinking	thinking	NOUN
bcpe-1790	14	35	in	in	ADP
bcpe-1790	14	36	mathematics	mathematic	NOUN
bcpe-1790	14	37	,	,	PUNCT
bcpe-1790	14	38	including	include	VERB
bcpe-1790	14	39	"	"	PUNCT
bcpe-1790	14	40	understanding	understand	VERB
bcpe-1790	14	41	the	the	DET
bcpe-1790	14	42	problem	problem	NOUN
bcpe-1790	14	43	,	,	PUNCT
bcpe-1790	14	44	"	"	PUNCT
bcpe-1790	14	45	"	"	PUNCT
bcpe-1790	14	46	devising	devise	VERB
bcpe-1790	14	47	a	a	DET
bcpe-1790	14	48	plan	plan	NOUN
bcpe-1790	14	49	,	,	PUNCT
bcpe-1790	14	50	"	"	PUNCT
bcpe-1790	14	51	"	"	PUNCT
bcpe-1790	14	52	carrying	carry	VERB
bcpe-1790	14	53	out	out	ADP
bcpe-1790	14	54	the	the	DET
bcpe-1790	14	55	plan	plan	NOUN
bcpe-1790	14	56	,	,	PUNCT
bcpe-1790	14	57	"	"	PUNCT
bcpe-1790	14	58	and	and	CCONJ
bcpe-1790	14	59	"	"	PUNCT
bcpe-1790	14	60	looking	look	VERB
bcpe-1790	14	61	back	back	ADV
bcpe-1790	14	62	"	"	PUNCT
bcpe-1790	14	63	four	four	NUM
bcpe-1790	14	64	steps	step	NOUN
bcpe-1790	15	1	[	[	X
bcpe-1790	15	2	6]."understanding	6]."understande	VERB
bcpe-1790	15	3	the	the	DET
bcpe-1790	15	4	problem	problem	NOUN
bcpe-1790	15	5	"	"	PUNCT
bcpe-1790	15	6	is	be	AUX
bcpe-1790	15	7	to	to	PART
bcpe-1790	15	8	let	let	VERB
bcpe-1790	15	9	students	student	NOUN
bcpe-1790	15	10	understand	understand	VERB
bcpe-1790	15	11	the	the	DET
bcpe-1790	15	12	meaning	meaning	NOUN
bcpe-1790	15	13	of	of	ADP
bcpe-1790	15	14	the	the	DET
bcpe-1790	15	15	problem	problem	NOUN
bcpe-1790	15	16	,	,	PUNCT
bcpe-1790	15	17	and	and	CCONJ
bcpe-1790	15	18	to	to	PART
bcpe-1790	15	19	understand	understand	VERB
bcpe-1790	15	20	the	the	DET
bcpe-1790	15	21	known	know	VERB
bcpe-1790	15	22	and	and	CCONJ
bcpe-1790	15	23	unknown	unknown	ADJ
bcpe-1790	15	24	quantities	quantity	NOUN
bcpe-1790	15	25	of	of	ADP
bcpe-1790	15	26	the	the	DET
bcpe-1790	15	27	problem	problem	NOUN
bcpe-1790	15	28	,	,	PUNCT
bcpe-1790	15	29	which	which	PRON
bcpe-1790	15	30	is	be	AUX
bcpe-1790	15	31	the	the	DET
bcpe-1790	15	32	basis	basis	NOUN
bcpe-1790	15	33	for	for	ADP
bcpe-1790	15	34	solving	solve	VERB
bcpe-1790	15	35	the	the	DET
bcpe-1790	15	36	problem	problem	NOUN
bcpe-1790	15	37	.	.	PUNCT
bcpe-1790	16	1	"	"	PUNCT
bcpe-1790	16	2	devising	devise	VERB
bcpe-1790	16	3	a	a	DET
bcpe-1790	16	4	plan	plan	NOUN
bcpe-1790	16	5	"	"	PUNCT
bcpe-1790	16	6	is	be	AUX
bcpe-1790	16	7	the	the	DET
bcpe-1790	16	8	core	core	NOUN
bcpe-1790	16	9	of	of	ADP
bcpe-1790	16	10	problem	problem	NOUN
bcpe-1790	16	11	-	-	PUNCT
bcpe-1790	16	12	solving	solving	NOUN
bcpe-1790	16	13	.	.	PUNCT
bcpe-1790	17	1	"	"	PUNCT
bcpe-1790	17	2	carrying	carry	VERB
bcpe-1790	17	3	out	out	ADP
bcpe-1790	17	4	the	the	DET
bcpe-1790	17	5	plan	plan	NOUN
bcpe-1790	17	6	"	"	PUNCT
bcpe-1790	17	7	is	be	AUX
bcpe-1790	17	8	mainly	mainly	ADV
bcpe-1790	17	9	to	to	PART
bcpe-1790	17	10	comprehensively	comprehensively	ADV
bcpe-1790	17	11	express	express	VERB
bcpe-1790	17	12	the	the	DET
bcpe-1790	17	13	previous	previous	ADJ
bcpe-1790	17	14	step	step	NOUN
bcpe-1790	17	15	to	to	PART
bcpe-1790	17	16	ensure	ensure	VERB
bcpe-1790	17	17	the	the	DET
bcpe-1790	17	18	accuracy	accuracy	NOUN
bcpe-1790	17	19	of	of	ADP
bcpe-1790	17	20	the	the	DET
bcpe-1790	17	21	results	result	NOUN
bcpe-1790	17	22	,	,	PUNCT
bcpe-1790	17	23	which	which	PRON
bcpe-1790	17	24	is	be	AUX
bcpe-1790	17	25	the	the	DET
bcpe-1790	17	26	key	key	NOUN
bcpe-1790	17	27	to	to	ADP
bcpe-1790	17	28	problem	problem	NOUN
bcpe-1790	17	29	-	-	PUNCT
bcpe-1790	17	30	solving	solving	NOUN
bcpe-1790	17	31	.	.	PUNCT
bcpe-1790	17	32	"	"	PUNCT
bcpe-1790	18	1	looking	look	VERB
bcpe-1790	18	2	back	back	ADV
bcpe-1790	18	3	"	"	PUNCT
bcpe-1790	18	4	is	be	AUX
bcpe-1790	18	5	a	a	DET
bcpe-1790	18	6	"	"	PUNCT
bcpe-1790	18	7	reunderstanding	reunderstanding	NOUN
bcpe-1790	18	8	"	"	PUNCT
bcpe-1790	18	9	of	of	ADP
bcpe-1790	18	10	problem	problem	NOUN
bcpe-1790	18	11	-	-	PUNCT
bcpe-1790	18	12	solving	solve	VERB
bcpe-1790	18	13	activities	activity	NOUN
bcpe-1790	18	14	and	and	CCONJ
bcpe-1790	18	15	trying	try	VERB
bcpe-1790	18	16	to	to	PART
bcpe-1790	18	17	draw	draw	VERB
bcpe-1790	18	18	inferences	inference	NOUN
bcpe-1790	18	19	about	about	ADP
bcpe-1790	18	20	the	the	DET
bcpe-1790	18	21	problem	problem	NOUN
bcpe-1790	18	22	from	from	ADP
bcpe-1790	18	23	one	one	NUM
bcpe-1790	18	24	case	case	NOUN
bcpe-1790	18	25	to	to	ADP
bcpe-1790	18	26	another	another	PRON
bcpe-1790	18	27	,	,	PUNCT
bcpe-1790	18	28	and	and	CCONJ
bcpe-1790	18	29	then	then	ADV
bcpe-1790	18	30	make	make	VERB
bcpe-1790	18	31	logical	logical	ADJ
bcpe-1790	18	32	presuppositions	presupposition	NOUN
bcpe-1790	18	33	for	for	ADP
bcpe-1790	18	34	solving	solve	VERB
bcpe-1790	18	35	this	this	DET
bcpe-1790	18	36	type	type	NOUN
bcpe-1790	18	37	of	of	ADP
bcpe-1790	18	38	problem	problem	NOUN
bcpe-1790	18	39	,	,	PUNCT
bcpe-1790	18	40	which	which	PRON
bcpe-1790	18	41	is	be	AUX
bcpe-1790	18	42	the	the	DET
bcpe-1790	18	43	sublimation	sublimation	NOUN
bcpe-1790	18	44	of	of	ADP
bcpe-1790	18	45	problem	problem	NOUN
bcpe-1790	18	46	-	-	PUNCT
bcpe-1790	18	47	solving	solving	NOUN
bcpe-1790	18	48	.	.	PUNCT
bcpe-1790	19	1	because	because	SCONJ
bcpe-1790	19	2	most	most	ADJ
bcpe-1790	19	3	of	of	ADP
bcpe-1790	19	4	the	the	DET
bcpe-1790	19	5	current	current	ADJ
bcpe-1790	19	6	problem	problem	NOUN
bcpe-1790	19	7	-	-	PUNCT
bcpe-1790	19	8	solving	solve	VERB
bcpe-1790	19	9	process	process	NOUN
bcpe-1790	19	10	and	and	CCONJ
bcpe-1790	19	11	teaching	teaching	NOUN
bcpe-1790	19	12	focus	focus	NOUN
bcpe-1790	19	13	on	on	ADP
bcpe-1790	19	14	how	how	SCONJ
bcpe-1790	19	15	to	to	PART
bcpe-1790	19	16	find	find	VERB
bcpe-1790	19	17	the	the	DET
bcpe-1790	19	18	ideas	idea	NOUN
bcpe-1790	19	19	and	and	CCONJ
bcpe-1790	19	20	methods	method	NOUN
bcpe-1790	19	21	to	to	PART
bcpe-1790	19	22	solve	solve	VERB
bcpe-1790	19	23	the	the	DET
bcpe-1790	19	24	problem	problem	NOUN
bcpe-1790	19	25	,	,	PUNCT
bcpe-1790	19	26	teachers	teacher	NOUN
bcpe-1790	19	27	often	often	ADV
bcpe-1790	19	28	ignore	ignore	VERB
bcpe-1790	19	29	the	the	DET
bcpe-1790	19	30	"	"	PUNCT
bcpe-1790	19	31	looking	look	VERB
bcpe-1790	19	32	back	back	ADV
bcpe-1790	19	33	"	"	PUNCT
bcpe-1790	19	34	step	step	NOUN
bcpe-1790	19	35	in	in	ADP
bcpe-1790	19	36	the	the	DET
bcpe-1790	19	37	teaching	teaching	NOUN
bcpe-1790	19	38	process	process	NOUN
bcpe-1790	19	39	.	.	PUNCT
bcpe-1790	20	1	students	student	NOUN
bcpe-1790	20	2	only	only	ADV
bcpe-1790	20	3	know	know	VERB
bcpe-1790	20	4	the	the	DET
bcpe-1790	20	5	problem	problem	NOUN
bcpe-1790	20	6	-	-	PUNCT
bcpe-1790	20	7	solving	solve	VERB
bcpe-1790	20	8	tactics	tactic	NOUN
bcpe-1790	20	9	and	and	CCONJ
bcpe-1790	20	10	feel	feel	VERB
bcpe-1790	20	11	that	that	SCONJ
bcpe-1790	20	12	the	the	DET
bcpe-1790	20	13	problem	problem	NOUN
bcpe-1790	20	14	-	-	PUNCT
bcpe-1790	20	15	solving	solve	VERB
bcpe-1790	20	16	process	process	NOUN
bcpe-1790	20	17	is	be	AUX
bcpe-1790	20	18	rushed	rush	VERB
bcpe-1790	20	19	.	.	PUNCT
bcpe-1790	21	1	they	they	PRON
bcpe-1790	21	2	do	do	AUX
bcpe-1790	21	3	not	not	PART
bcpe-1790	21	4	know	know	VERB
bcpe-1790	21	5	how	how	SCONJ
bcpe-1790	21	6	the	the	DET
bcpe-1790	21	7	solution	solution	NOUN
bcpe-1790	21	8	was	be	AUX
bcpe-1790	21	9	discovered	discover	VERB
bcpe-1790	21	10	,	,	PUNCT
bcpe-1790	21	11	nor	nor	CCONJ
bcpe-1790	21	12	the	the	DET
bcpe-1790	21	13	extensive	extensive	ADJ
bcpe-1790	21	14	connection	connection	NOUN
bcpe-1790	21	15	between	between	ADP
bcpe-1790	21	16	the	the	DET
bcpe-1790	21	17	problem	problem	NOUN
bcpe-1790	21	18	and	and	CCONJ
bcpe-1790	21	19	other	other	ADJ
bcpe-1790	21	20	concepts	concept	NOUN
bcpe-1790	21	21	,	,	PUNCT
bcpe-1790	21	22	theories	theory	NOUN
bcpe-1790	21	23	,	,	PUNCT
bcpe-1790	21	24	methods	method	NOUN
bcpe-1790	21	25	,	,	PUNCT
bcpe-1790	21	26	and	and	CCONJ
bcpe-1790	21	27	problems	problem	NOUN
bcpe-1790	21	28	,	,	PUNCT
bcpe-1790	21	29	let	let	VERB
bcpe-1790	21	30	alone	alone	ADV
bcpe-1790	21	31	how	how	SCONJ
bcpe-1790	21	32	the	the	DET
bcpe-1790	21	33	problem	problem	NOUN
bcpe-1790	21	34	should	should	AUX
bcpe-1790	21	35	be	be	AUX
bcpe-1790	21	36	solved	solve	VERB
bcpe-1790	21	37	and	and	CCONJ
bcpe-1790	21	38	how	how	SCONJ
bcpe-1790	21	39	the	the	DET
bcpe-1790	21	40	topic	topic	NOUN
bcpe-1790	21	41	will	will	AUX
bcpe-1790	21	42	develop	develop	VERB
bcpe-1790	21	43	.	.	PUNCT
bcpe-1790	22	1	even	even	ADV
bcpe-1790	22	2	if	if	SCONJ
bcpe-1790	22	3	there	there	PRON
bcpe-1790	22	4	is	be	VERB
bcpe-1790	22	5	a	a	DET
bcpe-1790	22	6	"	"	PUNCT
bcpe-1790	22	7	looking	look	VERB
bcpe-1790	22	8	back	back	ADV
bcpe-1790	22	9	"	"	PUNCT
bcpe-1790	22	10	step	step	NOUN
bcpe-1790	22	11	in	in	ADP
bcpe-1790	22	12	the	the	DET
bcpe-1790	22	13	class	class	NOUN
bcpe-1790	22	14	,	,	PUNCT
bcpe-1790	22	15	it	it	PRON
bcpe-1790	22	16	is	be	AUX
bcpe-1790	22	17	bcp	bcp	PROPN
bcpe-1790	22	18	education	education	PROPN
bcpe-1790	22	19	&	&	CCONJ
bcpe-1790	22	20	psychology	psychology	PROPN
bcpe-1790	22	21	erhss	erhss	PROPN
bcpe-1790	22	22	2022	2022	NUM
bcpe-1790	22	23	volume	volume	NOUN
bcpe-1790	22	24	6	6	NUM
bcpe-1790	22	25	(	(	PUNCT
bcpe-1790	22	26	2022	2022	NUM
bcpe-1790	22	27	)	)	PUNCT
bcpe-1790	22	28	201	201	NUM
bcpe-1790	22	29	limited	limit	VERB
bcpe-1790	22	30	to	to	ADP
bcpe-1790	22	31	answering	answer	VERB
bcpe-1790	22	32	the	the	DET
bcpe-1790	22	33	question	question	NOUN
bcpe-1790	22	34	itself	itself	PRON
bcpe-1790	22	35	and	and	CCONJ
bcpe-1790	22	36	to	to	ADP
bcpe-1790	22	37	the	the	DET
bcpe-1790	22	38	process	process	NOUN
bcpe-1790	22	39	of	of	ADP
bcpe-1790	22	40	"	"	PUNCT
bcpe-1790	22	41	checking	check	VERB
bcpe-1790	22	42	whether	whether	SCONJ
bcpe-1790	22	43	the	the	DET
bcpe-1790	22	44	answer	answer	NOUN
bcpe-1790	22	45	is	be	AUX
bcpe-1790	22	46	correct	correct	ADJ
bcpe-1790	22	47	"	"	PUNCT
bcpe-1790	22	48	or	or	CCONJ
bcpe-1790	22	49	recalling	recall	VERB
bcpe-1790	22	50	the	the	DET
bcpe-1790	22	51	question	question	NOUN
bcpe-1790	22	52	answered	answer	VERB
bcpe-1790	22	53	in	in	ADP
bcpe-1790	22	54	memory	memory	NOUN
bcpe-1790	22	55	.	.	PUNCT
bcpe-1790	23	1	such	such	DET
bcpe-1790	23	2	a	a	DET
bcpe-1790	23	3	"	"	PUNCT
bcpe-1790	23	4	looking	look	VERB
bcpe-1790	23	5	back	back	ADV
bcpe-1790	23	6	"	"	PUNCT
bcpe-1790	23	7	is	be	AUX
bcpe-1790	23	8	too	too	ADV
bcpe-1790	23	9	formal	formal	ADJ
bcpe-1790	23	10	and	and	CCONJ
bcpe-1790	23	11	low	low	ADJ
bcpe-1790	23	12	-	-	PUNCT
bcpe-1790	23	13	level	level	NOUN
bcpe-1790	23	14	,	,	PUNCT
bcpe-1790	23	15	which	which	PRON
bcpe-1790	23	16	can	can	AUX
bcpe-1790	23	17	not	not	PART
bcpe-1790	23	18	achieve	achieve	VERB
bcpe-1790	23	19	the	the	DET
bcpe-1790	23	20	purpose	purpose	NOUN
bcpe-1790	23	21	of	of	ADP
bcpe-1790	23	22	cultivating	cultivate	VERB
bcpe-1790	23	23	students	student	NOUN
bcpe-1790	23	24	'	'	PART
bcpe-1790	23	25	divergent	divergent	ADJ
bcpe-1790	23	26	thinking	thinking	NOUN
bcpe-1790	23	27	and	and	CCONJ
bcpe-1790	23	28	critical	critical	ADJ
bcpe-1790	23	29	thinking	thinking	NOUN
bcpe-1790	23	30	,	,	PUNCT
bcpe-1790	23	31	and	and	CCONJ
bcpe-1790	23	32	does	do	AUX
bcpe-1790	23	33	not	not	PART
bcpe-1790	23	34	meet	meet	VERB
bcpe-1790	23	35	the	the	DET
bcpe-1790	23	36	true	true	ADJ
bcpe-1790	23	37	meaning	meaning	NOUN
bcpe-1790	23	38	of	of	ADP
bcpe-1790	23	39	"	"	PUNCT
bcpe-1790	23	40	looking	look	VERB
bcpe-1790	23	41	back	back	ADV
bcpe-1790	23	42	"	"	PUNCT
bcpe-1790	23	43	originally	originally	ADV
bcpe-1790	23	44	referred	refer	VERB
bcpe-1790	23	45	to	to	ADP
bcpe-1790	23	46	by	by	ADP
bcpe-1790	23	47	polya	polya	PRON
bcpe-1790	23	48	[	[	X
bcpe-1790	23	49	7	7	NUM
bcpe-1790	23	50	]	]	PUNCT
bcpe-1790	23	51	.	.	PUNCT
bcpe-1790	24	1	the	the	DET
bcpe-1790	24	2	reason	reason	NOUN
bcpe-1790	24	3	is	be	AUX
bcpe-1790	24	4	that	that	SCONJ
bcpe-1790	24	5	teachers	teacher	NOUN
bcpe-1790	24	6	'	'	PART
bcpe-1790	24	7	thinking	thinking	NOUN
bcpe-1790	24	8	in	in	ADP
bcpe-1790	24	9	the	the	DET
bcpe-1790	24	10	teaching	teaching	NOUN
bcpe-1790	24	11	process	process	NOUN
bcpe-1790	24	12	is	be	AUX
bcpe-1790	24	13	too	too	ADV
bcpe-1790	24	14	limited	limited	ADJ
bcpe-1790	24	15	,	,	PUNCT
bcpe-1790	24	16	and	and	CCONJ
bcpe-1790	24	17	they	they	PRON
bcpe-1790	24	18	blindly	blindly	ADV
bcpe-1790	24	19	pursue	pursue	VERB
bcpe-1790	24	20	how	how	SCONJ
bcpe-1790	24	21	to	to	PART
bcpe-1790	24	22	"	"	PUNCT
bcpe-1790	24	23	instill	instill	VERB
bcpe-1790	24	24	"	"	PUNCT
bcpe-1790	24	25	success	success	NOUN
bcpe-1790	24	26	without	without	ADP
bcpe-1790	24	27	showing	show	VERB
bcpe-1790	24	28	a	a	DET
bcpe-1790	24	29	complete	complete	ADJ
bcpe-1790	24	30	thinking	thinking	NOUN
bcpe-1790	24	31	process	process	NOUN
bcpe-1790	24	32	,	,	PUNCT
bcpe-1790	24	33	and	and	CCONJ
bcpe-1790	24	34	they	they	PRON
bcpe-1790	24	35	lack	lack	VERB
bcpe-1790	24	36	a	a	DET
bcpe-1790	24	37	deep	deep	ADJ
bcpe-1790	24	38	understanding	understanding	NOUN
bcpe-1790	24	39	of	of	ADP
bcpe-1790	24	40	the	the	DET
bcpe-1790	24	41	meaning	meaning	NOUN
bcpe-1790	24	42	of	of	ADP
bcpe-1790	24	43	review	review	NOUN
bcpe-1790	24	44	after	after	ADP
bcpe-1790	24	45	solving	solve	VERB
bcpe-1790	24	46	problems	problem	NOUN
bcpe-1790	24	47	.	.	PUNCT
bcpe-1790	25	1	therefore	therefore	ADV
bcpe-1790	25	2	,	,	PUNCT
bcpe-1790	25	3	it	it	PRON
bcpe-1790	25	4	is	be	AUX
bcpe-1790	25	5	necessary	necessary	ADJ
bcpe-1790	25	6	to	to	PART
bcpe-1790	25	7	discuss	discuss	VERB
bcpe-1790	25	8	further	far	ADV
bcpe-1790	25	9	the	the	DET
bcpe-1790	25	10	step	step	NOUN
bcpe-1790	25	11	of	of	ADP
bcpe-1790	25	12	"	"	PUNCT
bcpe-1790	25	13	looking	look	VERB
bcpe-1790	25	14	back	back	ADV
bcpe-1790	25	15	"	"	PUNCT
bcpe-1790	25	16	to	to	PART
bcpe-1790	25	17	innovate	innovate	VERB
bcpe-1790	25	18	the	the	DET
bcpe-1790	25	19	problem	problem	NOUN
bcpe-1790	25	20	-	-	PUNCT
bcpe-1790	25	21	solving	solve	VERB
bcpe-1790	25	22	thought	thought	NOUN
bcpe-1790	25	23	of	of	ADP
bcpe-1790	25	24	polya	polya	NOUN
bcpe-1790	25	25	.	.	PUNCT
bcpe-1790	26	1	therefore	therefore	ADV
bcpe-1790	26	2	,	,	PUNCT
bcpe-1790	26	3	the	the	DET
bcpe-1790	26	4	teaching	teaching	NOUN
bcpe-1790	26	5	thinking	thinking	NOUN
bcpe-1790	26	6	of	of	ADP
bcpe-1790	26	7	mathematics	mathematics	NOUN
bcpe-1790	26	8	teachers	teacher	NOUN
bcpe-1790	26	9	can	can	AUX
bcpe-1790	26	10	be	be	AUX
bcpe-1790	26	11	effectively	effectively	ADV
bcpe-1790	26	12	improved	improve	VERB
bcpe-1790	26	13	,	,	PUNCT
bcpe-1790	26	14	and	and	CCONJ
bcpe-1790	26	15	the	the	DET
bcpe-1790	26	16	effect	effect	NOUN
bcpe-1790	26	17	of	of	ADP
bcpe-1790	26	18	mathematics	mathematics	NOUN
bcpe-1790	26	19	teaching	teaching	NOUN
bcpe-1790	26	20	can	can	AUX
bcpe-1790	26	21	be	be	AUX
bcpe-1790	26	22	improved	improve	VERB
bcpe-1790	26	23	.	.	PUNCT
bcpe-1790	27	1	2	2	X
bcpe-1790	27	2	.	.	X
bcpe-1790	27	3	the	the	DET
bcpe-1790	27	4	innovative	innovative	ADJ
bcpe-1790	27	5	thinking	thinking	NOUN
bcpe-1790	27	6	of	of	ADP
bcpe-1790	27	7	polya	polya	PROPN
bcpe-1790	27	8	's	's	PART
bcpe-1790	27	9	problem	problem	NOUN
bcpe-1790	27	10	-	-	PUNCT
bcpe-1790	27	11	solving	solve	VERB
bcpe-1790	27	12	ideas	idea	NOUN
bcpe-1790	27	13	in	in	ADP
bcpe-1790	27	14	view	view	NOUN
bcpe-1790	27	15	of	of	ADP
bcpe-1790	27	16	the	the	DET
bcpe-1790	27	17	existing	exist	VERB
bcpe-1790	27	18	problems	problem	NOUN
bcpe-1790	27	19	and	and	CCONJ
bcpe-1790	27	20	defects	defect	NOUN
bcpe-1790	27	21	,	,	PUNCT
bcpe-1790	27	22	this	this	DET
bcpe-1790	27	23	paper	paper	NOUN
bcpe-1790	27	24	attempts	attempt	VERB
bcpe-1790	27	25	to	to	PART
bcpe-1790	27	26	innovate	innovate	VERB
bcpe-1790	27	27	polya	polya	PROPN
bcpe-1790	27	28	's	's	PART
bcpe-1790	27	29	four	four	NUM
bcpe-1790	27	30	steps	step	NOUN
bcpe-1790	27	31	to	to	PART
bcpe-1790	27	32	solve	solve	VERB
bcpe-1790	27	33	problems	problem	NOUN
bcpe-1790	27	34	.	.	PUNCT
bcpe-1790	28	1	it	it	PRON
bcpe-1790	28	2	proposes	propose	VERB
bcpe-1790	28	3	to	to	PART
bcpe-1790	28	4	divide	divide	VERB
bcpe-1790	28	5	them	they	PRON
bcpe-1790	28	6	into	into	ADP
bcpe-1790	28	7	five	five	NUM
bcpe-1790	28	8	steps	step	NOUN
bcpe-1790	28	9	:	:	PUNCT
bcpe-1790	28	10	"	"	PUNCT
bcpe-1790	28	11	observing	observe	VERB
bcpe-1790	28	12	and	and	CCONJ
bcpe-1790	28	13	doubting	doubt	VERB
bcpe-1790	28	14	,	,	PUNCT
bcpe-1790	28	15	"	"	PUNCT
bcpe-1790	28	16	"	"	PUNCT
bcpe-1790	28	17	multivariate	multivariate	NOUN
bcpe-1790	28	18	analyzing	analyzing	NOUN
bcpe-1790	28	19	doubts	doubt	NOUN
bcpe-1790	28	20	,	,	PUNCT
bcpe-1790	28	21	"	"	PUNCT
bcpe-1790	28	22	"	"	PUNCT
bcpe-1790	28	23	accurately	accurately	ADV
bcpe-1790	28	24	solving	solve	VERB
bcpe-1790	28	25	doubts	doubt	NOUN
bcpe-1790	28	26	,	,	PUNCT
bcpe-1790	28	27	"	"	PUNCT
bcpe-1790	28	28	"	"	PUNCT
bcpe-1790	28	29	reviewing	review	VERB
bcpe-1790	28	30	and	and	CCONJ
bcpe-1790	28	31	doubting	doubt	VERB
bcpe-1790	28	32	,	,	PUNCT
bcpe-1790	28	33	"	"	PUNCT
bcpe-1790	28	34	and	and	CCONJ
bcpe-1790	28	35	"	"	PUNCT
bcpe-1790	28	36	reflecting	reflect	VERB
bcpe-1790	28	37	and	and	CCONJ
bcpe-1790	28	38	doubting	doubt	VERB
bcpe-1790	28	39	.	.	PUNCT
bcpe-1790	28	40	"	"	PUNCT
bcpe-1790	29	1	"	"	PUNCT
bcpe-1790	29	2	looking	look	VERB
bcpe-1790	29	3	back	back	ADV
bcpe-1790	29	4	"	"	PUNCT
bcpe-1790	29	5	is	be	AUX
bcpe-1790	29	6	refined	refine	VERB
bcpe-1790	29	7	into	into	ADP
bcpe-1790	29	8	two	two	NUM
bcpe-1790	29	9	steps	step	NOUN
bcpe-1790	29	10	"	"	PUNCT
bcpe-1790	29	11	reviewing	review	VERB
bcpe-1790	29	12	and	and	CCONJ
bcpe-1790	29	13	doubting	doubt	VERB
bcpe-1790	29	14	"	"	PUNCT
bcpe-1790	29	15	and	and	CCONJ
bcpe-1790	29	16	"	"	PUNCT
bcpe-1790	29	17	reflecting	reflect	VERB
bcpe-1790	29	18	and	and	CCONJ
bcpe-1790	29	19	doubting	doubt	VERB
bcpe-1790	29	20	.	.	PUNCT
bcpe-1790	29	21	"	"	PUNCT
bcpe-1790	30	1	2.1	2.1	NUM
bcpe-1790	30	2	observing	observe	VERB
bcpe-1790	30	3	and	and	CCONJ
bcpe-1790	30	4	doubting	doubt	VERB
bcpe-1790	30	5	finding	find	VERB
bcpe-1790	30	6	the	the	DET
bcpe-1790	30	7	crux	crux	NOUN
bcpe-1790	30	8	of	of	ADP
bcpe-1790	30	9	the	the	DET
bcpe-1790	30	10	problem	problem	NOUN
bcpe-1790	30	11	from	from	ADP
bcpe-1790	30	12	observing	observe	VERB
bcpe-1790	30	13	and	and	CCONJ
bcpe-1790	30	14	doubting	doubt	VERB
bcpe-1790	30	15	,	,	PUNCT
bcpe-1790	30	16	based	base	VERB
bcpe-1790	30	17	on	on	ADP
bcpe-1790	30	18	existing	exist	VERB
bcpe-1790	30	19	knowledge	knowledge	NOUN
bcpe-1790	30	20	,	,	PUNCT
bcpe-1790	30	21	it	it	PRON
bcpe-1790	30	22	is	be	AUX
bcpe-1790	30	23	conducted	conduct	VERB
bcpe-1790	30	24	an	an	DET
bcpe-1790	30	25	in	in	ADP
bcpe-1790	30	26	-	-	PUNCT
bcpe-1790	30	27	depth	depth	NOUN
bcpe-1790	30	28	analysis	analysis	NOUN
bcpe-1790	30	29	of	of	ADP
bcpe-1790	30	30	doubts	doubt	NOUN
bcpe-1790	30	31	(	(	PUNCT
bcpe-1790	30	32	topics	topic	NOUN
bcpe-1790	30	33	)	)	PUNCT
bcpe-1790	30	34	,	,	PUNCT
bcpe-1790	30	35	which	which	PRON
bcpe-1790	30	36	clarifies	clarify	VERB
bcpe-1790	30	37	the	the	DET
bcpe-1790	30	38	mathematical	mathematical	ADJ
bcpe-1790	30	39	problems	problem	NOUN
bcpe-1790	30	40	that	that	PRON
bcpe-1790	30	41	need	need	VERB
bcpe-1790	30	42	to	to	PART
bcpe-1790	30	43	be	be	AUX
bcpe-1790	30	44	solved	solve	VERB
bcpe-1790	30	45	in	in	ADP
bcpe-1790	30	46	the	the	DET
bcpe-1790	30	47	doubts	doubt	NOUN
bcpe-1790	30	48	.	.	PUNCT
bcpe-1790	31	1	it	it	PRON
bcpe-1790	31	2	is	be	AUX
bcpe-1790	31	3	known	know	VERB
bcpe-1790	31	4	that	that	SCONJ
bcpe-1790	31	5	knowledge	knowledge	NOUN
bcpe-1790	31	6	points	point	NOUN
bcpe-1790	31	7	are	be	AUX
bcpe-1790	31	8	related	relate	VERB
bcpe-1790	31	9	to	to	ADP
bcpe-1790	31	10	the	the	DET
bcpe-1790	31	11	problems	problem	NOUN
bcpe-1790	31	12	and	and	CCONJ
bcpe-1790	31	13	understanding	understand	VERB
bcpe-1790	31	14	the	the	DET
bcpe-1790	31	15	known	know	VERB
bcpe-1790	31	16	conditions	condition	NOUN
bcpe-1790	31	17	and	and	CCONJ
bcpe-1790	31	18	unknowns	unknown	NOUN
bcpe-1790	31	19	of	of	ADP
bcpe-1790	31	20	the	the	DET
bcpe-1790	31	21	topic	topic	NOUN
bcpe-1790	31	22	.	.	PUNCT
bcpe-1790	32	1	moreover	moreover	ADV
bcpe-1790	32	2	,	,	PUNCT
bcpe-1790	32	3	it	it	PRON
bcpe-1790	32	4	looks	look	VERB
bcpe-1790	32	5	for	for	ADP
bcpe-1790	32	6	the	the	DET
bcpe-1790	32	7	obstacles	obstacle	NOUN
bcpe-1790	32	8	that	that	PRON
bcpe-1790	32	9	need	need	VERB
bcpe-1790	32	10	to	to	PART
bcpe-1790	32	11	be	be	AUX
bcpe-1790	32	12	overcome	overcome	VERB
bcpe-1790	32	13	from	from	ADP
bcpe-1790	32	14	the	the	DET
bcpe-1790	32	15	condition	condition	NOUN
bcpe-1790	32	16	to	to	ADP
bcpe-1790	32	17	the	the	DET
bcpe-1790	32	18	conclusion	conclusion	NOUN
bcpe-1790	32	19	to	to	PART
bcpe-1790	32	20	find	find	VERB
bcpe-1790	32	21	the	the	DET
bcpe-1790	32	22	crux	crux	NOUN
bcpe-1790	32	23	of	of	ADP
bcpe-1790	32	24	the	the	DET
bcpe-1790	32	25	problem	problem	NOUN
bcpe-1790	32	26	.	.	PUNCT
bcpe-1790	33	1	2.2	2.2	NUM
bcpe-1790	33	2	multivariate	multivariate	NOUN
bcpe-1790	33	3	analyzing	analyzing	NOUN
bcpe-1790	33	4	doubts	doubt	NOUN
bcpe-1790	33	5	exploring	explore	VERB
bcpe-1790	33	6	the	the	DET
bcpe-1790	33	7	path	path	NOUN
bcpe-1790	33	8	of	of	ADP
bcpe-1790	33	9	solving	solve	VERB
bcpe-1790	33	10	the	the	DET
bcpe-1790	33	11	problem	problem	NOUN
bcpe-1790	33	12	multivariate	multivariate	NOUN
bcpe-1790	33	13	analyzing	analyze	VERB
bcpe-1790	33	14	doubts	doubt	NOUN
bcpe-1790	33	15	,	,	PUNCT
bcpe-1790	33	16	that	that	ADV
bcpe-1790	33	17	is	is	ADV
bcpe-1790	33	18	,	,	PUNCT
bcpe-1790	33	19	multi	multi	ADJ
bcpe-1790	33	20	-	-	ADJ
bcpe-1790	33	21	level	level	ADJ
bcpe-1790	33	22	and	and	CCONJ
bcpe-1790	33	23	multi	multi	ADJ
bcpe-1790	33	24	-	-	ADJ
bcpe-1790	33	25	angle	angle	NOUN
bcpe-1790	33	26	analysis	analysis	NOUN
bcpe-1790	33	27	of	of	ADP
bcpe-1790	33	28	doubts	doubt	NOUN
bcpe-1790	33	29	,	,	PUNCT
bcpe-1790	33	30	aims	aim	VERB
bcpe-1790	33	31	to	to	PART
bcpe-1790	33	32	resolve	resolve	VERB
bcpe-1790	33	33	the	the	DET
bcpe-1790	33	34	crux	crux	NOUN
bcpe-1790	33	35	of	of	ADP
bcpe-1790	33	36	the	the	DET
bcpe-1790	33	37	problem	problem	NOUN
bcpe-1790	33	38	and	and	CCONJ
bcpe-1790	33	39	find	find	VERB
bcpe-1790	33	40	an	an	DET
bcpe-1790	33	41	effective	effective	ADJ
bcpe-1790	33	42	solution	solution	NOUN
bcpe-1790	33	43	.	.	PUNCT
bcpe-1790	34	1	recall	recall	VERB
bcpe-1790	34	2	whether	whether	SCONJ
bcpe-1790	34	3	there	there	PRON
bcpe-1790	34	4	is	be	VERB
bcpe-1790	34	5	a	a	DET
bcpe-1790	34	6	connection	connection	NOUN
bcpe-1790	34	7	between	between	ADP
bcpe-1790	34	8	the	the	DET
bcpe-1790	34	9	current	current	ADJ
bcpe-1790	34	10	problem	problem	NOUN
bcpe-1790	34	11	and	and	CCONJ
bcpe-1790	34	12	similar	similar	ADJ
bcpe-1790	34	13	problems	problem	NOUN
bcpe-1790	34	14	or	or	CCONJ
bcpe-1790	34	15	similar	similar	ADJ
bcpe-1790	34	16	models	model	NOUN
bcpe-1790	34	17	.	.	PUNCT
bcpe-1790	35	1	if	if	SCONJ
bcpe-1790	35	2	so	so	ADV
bcpe-1790	35	3	,	,	PUNCT
bcpe-1790	35	4	try	try	VERB
bcpe-1790	35	5	to	to	PART
bcpe-1790	35	6	solve	solve	VERB
bcpe-1790	35	7	the	the	DET
bcpe-1790	35	8	problem	problem	NOUN
bcpe-1790	35	9	in	in	ADP
bcpe-1790	35	10	the	the	DET
bcpe-1790	35	11	same	same	ADJ
bcpe-1790	35	12	way	way	NOUN
bcpe-1790	35	13	.	.	PUNCT
bcpe-1790	36	1	if	if	SCONJ
bcpe-1790	36	2	not	not	PART
bcpe-1790	36	3	,	,	PUNCT
bcpe-1790	36	4	it	it	PRON
bcpe-1790	36	5	analyzes	analyze	VERB
bcpe-1790	36	6	the	the	DET
bcpe-1790	36	7	relationship	relationship	NOUN
bcpe-1790	36	8	between	between	ADP
bcpe-1790	36	9	the	the	DET
bcpe-1790	36	10	known	know	VERB
bcpe-1790	36	11	conditions	condition	NOUN
bcpe-1790	36	12	and	and	CCONJ
bcpe-1790	36	13	the	the	DET
bcpe-1790	36	14	problem	problem	NOUN
bcpe-1790	36	15	and	and	CCONJ
bcpe-1790	36	16	thinks	think	VERB
bcpe-1790	36	17	about	about	ADP
bcpe-1790	36	18	what	what	DET
bcpe-1790	36	19	information	information	NOUN
bcpe-1790	36	20	can	can	AUX
bcpe-1790	36	21	be	be	AUX
bcpe-1790	36	22	obtained	obtain	VERB
bcpe-1790	36	23	from	from	ADP
bcpe-1790	36	24	the	the	DET
bcpe-1790	36	25	known	know	VERB
bcpe-1790	36	26	and	and	CCONJ
bcpe-1790	36	27	which	which	DET
bcpe-1790	36	28	information	information	NOUN
bcpe-1790	36	29	is	be	AUX
bcpe-1790	36	30	related	relate	VERB
bcpe-1790	36	31	to	to	ADP
bcpe-1790	36	32	the	the	DET
bcpe-1790	36	33	problem	problem	NOUN
bcpe-1790	36	34	,	,	PUNCT
bcpe-1790	36	35	or	or	CCONJ
bcpe-1790	36	36	what	what	DET
bcpe-1790	36	37	conditions	condition	NOUN
bcpe-1790	36	38	can	can	AUX
bcpe-1790	36	39	be	be	AUX
bcpe-1790	36	40	inversely	inversely	ADV
bcpe-1790	36	41	deduced	deduce	VERB
bcpe-1790	36	42	from	from	ADP
bcpe-1790	36	43	the	the	DET
bcpe-1790	36	44	problem	problem	NOUN
bcpe-1790	36	45	,	,	PUNCT
bcpe-1790	36	46	and	and	CCONJ
bcpe-1790	36	47	how	how	SCONJ
bcpe-1790	36	48	the	the	DET
bcpe-1790	36	49	inversely	inversely	ADV
bcpe-1790	36	50	deduced	deduce	VERB
bcpe-1790	36	51	conditions	condition	NOUN
bcpe-1790	36	52	are	be	AUX
bcpe-1790	36	53	related	relate	VERB
bcpe-1790	36	54	to	to	ADP
bcpe-1790	36	55	the	the	DET
bcpe-1790	36	56	known	know	VERB
bcpe-1790	36	57	conditions	condition	NOUN
bcpe-1790	36	58	.	.	PUNCT
bcpe-1790	37	1	it	it	PRON
bcpe-1790	37	2	is	be	AUX
bcpe-1790	37	3	used	use	VERB
bcpe-1790	37	4	to	to	PART
bcpe-1790	37	5	build	build	VERB
bcpe-1790	37	6	a	a	DET
bcpe-1790	37	7	"	"	PUNCT
bcpe-1790	37	8	bridge	bridge	NOUN
bcpe-1790	37	9	"	"	PUNCT
bcpe-1790	37	10	between	between	ADP
bcpe-1790	37	11	the	the	DET
bcpe-1790	37	12	known	know	VERB
bcpe-1790	37	13	conditions	condition	NOUN
bcpe-1790	37	14	and	and	CCONJ
bcpe-1790	37	15	the	the	DET
bcpe-1790	37	16	problem	problem	NOUN
bcpe-1790	37	17	and	and	CCONJ
bcpe-1790	37	18	find	find	VERB
bcpe-1790	37	19	the	the	DET
bcpe-1790	37	20	optimal	optimal	ADJ
bcpe-1790	37	21	solution	solution	NOUN
bcpe-1790	37	22	path	path	NOUN
bcpe-1790	37	23	.	.	PUNCT
bcpe-1790	38	1	2.3	2.3	NUM
bcpe-1790	38	2	accurately	accurately	ADV
bcpe-1790	38	3	solving	solve	VERB
bcpe-1790	38	4	doubts	doubt	NOUN
bcpe-1790	38	5	accurately	accurately	ADV
bcpe-1790	38	6	implementing	implement	VERB
bcpe-1790	38	7	the	the	DET
bcpe-1790	38	8	plan	plan	NOUN
bcpe-1790	38	9	accurately	accurately	ADV
bcpe-1790	38	10	solving	solve	VERB
bcpe-1790	38	11	doubts	doubt	NOUN
bcpe-1790	38	12	means	mean	VERB
bcpe-1790	38	13	implementing	implement	VERB
bcpe-1790	38	14	the	the	DET
bcpe-1790	38	15	plan	plan	NOUN
bcpe-1790	38	16	strictly	strictly	ADV
bcpe-1790	38	17	and	and	CCONJ
bcpe-1790	38	18	accurately	accurately	ADV
bcpe-1790	38	19	to	to	PART
bcpe-1790	38	20	solve	solve	VERB
bcpe-1790	38	21	doubts	doubt	NOUN
bcpe-1790	38	22	,	,	PUNCT
bcpe-1790	38	23	ensuring	ensure	VERB
bcpe-1790	38	24	the	the	DET
bcpe-1790	38	25	accuracy	accuracy	NOUN
bcpe-1790	38	26	of	of	ADP
bcpe-1790	38	27	the	the	DET
bcpe-1790	38	28	problem	problem	NOUN
bcpe-1790	38	29	-	-	PUNCT
bcpe-1790	38	30	solving	solve	VERB
bcpe-1790	38	31	results	result	NOUN
bcpe-1790	38	32	,	,	PUNCT
bcpe-1790	38	33	and	and	CCONJ
bcpe-1790	38	34	aiming	aim	VERB
bcpe-1790	38	35	to	to	PART
bcpe-1790	38	36	test	test	VERB
bcpe-1790	38	37	whether	whether	SCONJ
bcpe-1790	38	38	the	the	DET
bcpe-1790	38	39	formulated	formulate	VERB
bcpe-1790	38	40	plan	plan	NOUN
bcpe-1790	38	41	is	be	AUX
bcpe-1790	38	42	feasible	feasible	ADJ
bcpe-1790	38	43	to	to	PART
bcpe-1790	38	44	achieve	achieve	VERB
bcpe-1790	38	45	the	the	DET
bcpe-1790	38	46	purpose	purpose	NOUN
bcpe-1790	38	47	of	of	ADP
bcpe-1790	38	48	problem	problem	NOUN
bcpe-1790	38	49	-	-	PUNCT
bcpe-1790	38	50	solving	solving	NOUN
bcpe-1790	38	51	.	.	PUNCT
bcpe-1790	39	1	"	"	PUNCT
bcpe-1790	39	2	accurately	accurately	ADV
bcpe-1790	39	3	solving	solve	VERB
bcpe-1790	39	4	doubts	doubt	NOUN
bcpe-1790	39	5	"	"	PUNCT
bcpe-1790	39	6	is	be	AUX
bcpe-1790	39	7	much	much	ADV
bcpe-1790	39	8	easier	easy	ADJ
bcpe-1790	39	9	than	than	ADP
bcpe-1790	39	10	"	"	PUNCT
bcpe-1790	39	11	multivariate	multivariate	NOUN
bcpe-1790	39	12	analyzing	analyzing	NOUN
bcpe-1790	39	13	doubts	doubt	NOUN
bcpe-1790	39	14	.	.	PUNCT
bcpe-1790	39	15	"	"	PUNCT
bcpe-1790	40	1	it	it	PRON
bcpe-1790	40	2	mainly	mainly	ADV
bcpe-1790	40	3	implements	implement	VERB
bcpe-1790	40	4	the	the	DET
bcpe-1790	40	5	already	already	ADV
bcpe-1790	40	6	formed	form	VERB
bcpe-1790	40	7	problem	problem	NOUN
bcpe-1790	40	8	-	-	PUNCT
bcpe-1790	40	9	solving	solve	VERB
bcpe-1790	40	10	ideas	idea	NOUN
bcpe-1790	40	11	on	on	ADP
bcpe-1790	40	12	the	the	DET
bcpe-1790	40	13	test	test	NOUN
bcpe-1790	40	14	paper	paper	NOUN
bcpe-1790	40	15	.	.	PUNCT
bcpe-1790	41	1	the	the	DET
bcpe-1790	41	2	most	most	ADV
bcpe-1790	41	3	important	important	ADJ
bcpe-1790	41	4	thing	thing	NOUN
bcpe-1790	41	5	at	at	ADP
bcpe-1790	41	6	this	this	DET
bcpe-1790	41	7	stage	stage	NOUN
bcpe-1790	41	8	is	be	AUX
bcpe-1790	41	9	to	to	PART
bcpe-1790	41	10	be	be	AUX
bcpe-1790	41	11	patient	patient	ADJ
bcpe-1790	41	12	and	and	CCONJ
bcpe-1790	41	13	careful	careful	ADJ
bcpe-1790	41	14	.	.	PUNCT
bcpe-1790	42	1	in	in	ADP
bcpe-1790	42	2	order	order	NOUN
bcpe-1790	42	3	to	to	PART
bcpe-1790	42	4	ensure	ensure	VERB
bcpe-1790	42	5	that	that	SCONJ
bcpe-1790	42	6	every	every	DET
bcpe-1790	42	7	detail	detail	NOUN
bcpe-1790	42	8	is	be	AUX
bcpe-1790	42	9	accurate	accurate	ADJ
bcpe-1790	42	10	,	,	PUNCT
bcpe-1790	42	11	every	every	DET
bcpe-1790	42	12	step	step	NOUN
bcpe-1790	42	13	must	must	AUX
bcpe-1790	42	14	be	be	AUX
bcpe-1790	42	15	carefully	carefully	ADV
bcpe-1790	42	16	checked	check	VERB
bcpe-1790	42	17	.	.	PUNCT
bcpe-1790	43	1	every	every	DET
bcpe-1790	43	2	point	point	NOUN
bcpe-1790	43	3	should	should	AUX
bcpe-1790	43	4	be	be	AUX
bcpe-1790	43	5	clarified	clarify	VERB
bcpe-1790	43	6	so	so	SCONJ
bcpe-1790	43	7	that	that	SCONJ
bcpe-1790	43	8	the	the	DET
bcpe-1790	43	9	entire	entire	ADJ
bcpe-1790	43	10	problem	problem	NOUN
bcpe-1790	43	11	-	-	PUNCT
bcpe-1790	43	12	solving	solve	VERB
bcpe-1790	43	13	process	process	NOUN
bcpe-1790	43	14	is	be	AUX
bcpe-1790	43	15	free	free	ADJ
bcpe-1790	43	16	from	from	ADP
bcpe-1790	43	17	the	the	DET
bcpe-1790	43	18	ambiguity	ambiguity	NOUN
bcpe-1790	43	19	that	that	PRON
bcpe-1790	43	20	may	may	AUX
bcpe-1790	43	21	hide	hide	VERB
bcpe-1790	43	22	errors	error	NOUN
bcpe-1790	43	23	.	.	PUNCT
bcpe-1790	44	1	2.4	2.4	NUM
bcpe-1790	44	2	reviewing	reviewing	NOUN
bcpe-1790	44	3	and	and	CCONJ
bcpe-1790	44	4	doubting	doubt	VERB
bcpe-1790	44	5	perceive	perceive	NOUN
bcpe-1790	44	6	the	the	DET
bcpe-1790	44	7	problem	problem	NOUN
bcpe-1790	44	8	-	-	PUNCT
bcpe-1790	44	9	solving	solve	VERB
bcpe-1790	44	10	process	process	NOUN
bcpe-1790	44	11	reviewing	reviewing	NOUN
bcpe-1790	44	12	and	and	CCONJ
bcpe-1790	44	13	doubting	doubt	VERB
bcpe-1790	44	14	,	,	PUNCT
bcpe-1790	44	15	that	that	ADV
bcpe-1790	44	16	is	is	ADV
bcpe-1790	44	17	,	,	PUNCT
bcpe-1790	44	18	review	review	VERB
bcpe-1790	44	19	and	and	CCONJ
bcpe-1790	44	20	sort	sort	VERB
bcpe-1790	44	21	out	out	ADP
bcpe-1790	44	22	the	the	DET
bcpe-1790	44	23	entire	entire	ADJ
bcpe-1790	44	24	problem	problem	NOUN
bcpe-1790	44	25	-	-	PUNCT
bcpe-1790	44	26	solving	solve	VERB
bcpe-1790	44	27	process	process	NOUN
bcpe-1790	44	28	,	,	PUNCT
bcpe-1790	44	29	reflect	reflect	VERB
bcpe-1790	44	30	on	on	ADP
bcpe-1790	44	31	whether	whether	SCONJ
bcpe-1790	44	32	the	the	DET
bcpe-1790	44	33	problem	problem	NOUN
bcpe-1790	44	34	-	-	PUNCT
bcpe-1790	44	35	solving	solve	VERB
bcpe-1790	44	36	process	process	NOUN
bcpe-1790	44	37	has	have	AUX
bcpe-1790	44	38	found	find	VERB
bcpe-1790	44	39	the	the	DET
bcpe-1790	44	40	right	right	ADJ
bcpe-1790	44	41	direction	direction	NOUN
bcpe-1790	44	42	,	,	PUNCT
bcpe-1790	44	43	whether	whether	SCONJ
bcpe-1790	44	44	there	there	PRON
bcpe-1790	44	45	are	be	VERB
bcpe-1790	44	46	loopholes	loophole	NOUN
bcpe-1790	44	47	in	in	ADP
bcpe-1790	44	48	one	one	NUM
bcpe-1790	44	49	's	's	PART
bcpe-1790	44	50	subject	subject	ADJ
bcpe-1790	44	51	knowledge	knowledge	NOUN
bcpe-1790	44	52	,	,	PUNCT
bcpe-1790	44	53	and	and	CCONJ
bcpe-1790	44	54	perceive	perceive	VERB
bcpe-1790	44	55	the	the	DET
bcpe-1790	44	56	mathematical	mathematical	ADJ
bcpe-1790	44	57	knowledge	knowledge	NOUN
bcpe-1790	44	58	,	,	PUNCT
bcpe-1790	44	59	mathematical	mathematical	ADJ
bcpe-1790	44	60	methods	method	NOUN
bcpe-1790	44	61	,	,	PUNCT
bcpe-1790	44	62	and	and	CCONJ
bcpe-1790	44	63	mathematical	mathematical	ADJ
bcpe-1790	44	64	ideas	idea	NOUN
bcpe-1790	44	65	used	use	VERB
bcpe-1790	44	66	in	in	ADP
bcpe-1790	44	67	solving	solve	VERB
bcpe-1790	44	68	problems	problem	NOUN
bcpe-1790	44	69	,	,	PUNCT
bcpe-1790	44	70	and	and	CCONJ
bcpe-1790	44	71	think	think	VERB
bcpe-1790	44	72	about	about	ADP
bcpe-1790	44	73	it	it	PRON
bcpe-1790	44	74	.	.	PUNCT
bcpe-1790	45	1	it	it	PRON
bcpe-1790	45	2	can	can	AUX
bcpe-1790	45	3	use	use	VERB
bcpe-1790	45	4	different	different	ADJ
bcpe-1790	45	5	methods	method	NOUN
bcpe-1790	45	6	to	to	PART
bcpe-1790	45	7	solve	solve	VERB
bcpe-1790	45	8	bcp	bcp	PROPN
bcpe-1790	45	9	education	education	PROPN
bcpe-1790	45	10	&	&	CCONJ
bcpe-1790	45	11	psychology	psychology	PROPN
bcpe-1790	45	12	erhss	erhss	PROPN
bcpe-1790	45	13	2022	2022	NUM
bcpe-1790	45	14	volume	volume	NOUN
bcpe-1790	45	15	6	6	NUM
bcpe-1790	45	16	(	(	PUNCT
bcpe-1790	45	17	2022	2022	NUM
bcpe-1790	45	18	)	)	PUNCT
bcpe-1790	45	19	202	202	NUM
bcpe-1790	45	20	the	the	DET
bcpe-1790	45	21	problem	problem	NOUN
bcpe-1790	45	22	and	and	CCONJ
bcpe-1790	45	23	consider	consider	VERB
bcpe-1790	45	24	whether	whether	SCONJ
bcpe-1790	45	25	the	the	DET
bcpe-1790	45	26	problem	problem	NOUN
bcpe-1790	45	27	can	can	AUX
bcpe-1790	45	28	be	be	AUX
bcpe-1790	45	29	summarized	summarize	VERB
bcpe-1790	45	30	and	and	CCONJ
bcpe-1790	45	31	classified	classify	VERB
bcpe-1790	45	32	to	to	PART
bcpe-1790	45	33	achieve	achieve	VERB
bcpe-1790	45	34	the	the	DET
bcpe-1790	45	35	problem	problem	NOUN
bcpe-1790	45	36	-	-	PUNCT
bcpe-1790	45	37	solving	solve	VERB
bcpe-1790	45	38	effect	effect	NOUN
bcpe-1790	45	39	of	of	ADP
bcpe-1790	45	40	"	"	PUNCT
bcpe-1790	45	41	doing	do	VERB
bcpe-1790	45	42	one	one	NUM
bcpe-1790	45	43	,	,	PUNCT
bcpe-1790	45	44	knowing	know	VERB
bcpe-1790	45	45	one	one	NUM
bcpe-1790	45	46	class	class	NOUN
bcpe-1790	45	47	.	.	PUNCT
bcpe-1790	45	48	"	"	PUNCT
bcpe-1790	46	1	2.5	2.5	NUM
bcpe-1790	46	2	reflecting	reflect	VERB
bcpe-1790	46	3	and	and	CCONJ
bcpe-1790	46	4	doubting	doubt	VERB
bcpe-1790	46	5	the	the	DET
bcpe-1790	46	6	problem	problem	NOUN
bcpe-1790	46	7	of	of	ADP
bcpe-1790	46	8	sublimation	sublimation	NOUN
bcpe-1790	46	9	and	and	CCONJ
bcpe-1790	46	10	expansion	expansion	NOUN
bcpe-1790	46	11	reflecting	reflect	VERB
bcpe-1790	46	12	and	and	CCONJ
bcpe-1790	46	13	doubting	doubt	VERB
bcpe-1790	46	14	mean	mean	VERB
bcpe-1790	46	15	necessary	necessary	ADJ
bcpe-1790	46	16	extension	extension	NOUN
bcpe-1790	46	17	and	and	CCONJ
bcpe-1790	46	18	expansion	expansion	NOUN
bcpe-1790	46	19	of	of	ADP
bcpe-1790	46	20	the	the	DET
bcpe-1790	46	21	original	original	ADJ
bcpe-1790	46	22	topic	topic	NOUN
bcpe-1790	46	23	,	,	PUNCT
bcpe-1790	46	24	reflecting	reflect	VERB
bcpe-1790	46	25	on	on	ADP
bcpe-1790	46	26	whether	whether	SCONJ
bcpe-1790	46	27	the	the	DET
bcpe-1790	46	28	problem	problem	NOUN
bcpe-1790	46	29	-	-	PUNCT
bcpe-1790	46	30	solving	solve	VERB
bcpe-1790	46	31	process	process	NOUN
bcpe-1790	46	32	can	can	AUX
bcpe-1790	46	33	withstand	withstand	VERB
bcpe-1790	46	34	scrutiny	scrutiny	NOUN
bcpe-1790	46	35	,	,	PUNCT
bcpe-1790	46	36	and	and	CCONJ
bcpe-1790	46	37	then	then	ADV
bcpe-1790	46	38	discovering	discover	VERB
bcpe-1790	46	39	the	the	DET
bcpe-1790	46	40	rules	rule	NOUN
bcpe-1790	46	41	.	.	PUNCT
bcpe-1790	47	1	reflecting	reflect	VERB
bcpe-1790	47	2	and	and	CCONJ
bcpe-1790	47	3	doubting	doubt	VERB
bcpe-1790	47	4	are	be	AUX
bcpe-1790	47	5	different	different	ADJ
bcpe-1790	47	6	from	from	ADP
bcpe-1790	47	7	reviewing	review	VERB
bcpe-1790	47	8	and	and	CCONJ
bcpe-1790	47	9	doubting	doubt	VERB
bcpe-1790	47	10	.	.	PUNCT
bcpe-1790	48	1	it	it	PRON
bcpe-1790	48	2	is	be	AUX
bcpe-1790	48	3	the	the	DET
bcpe-1790	48	4	growth	growth	NOUN
bcpe-1790	48	5	of	of	ADP
bcpe-1790	48	6	problemsolving	problemsolve	VERB
bcpe-1790	48	7	thinking	thinking	NOUN
bcpe-1790	48	8	based	base	VERB
bcpe-1790	48	9	on	on	ADP
bcpe-1790	48	10	reviewing	review	VERB
bcpe-1790	48	11	and	and	CCONJ
bcpe-1790	48	12	doubting	doubt	VERB
bcpe-1790	48	13	.	.	PUNCT
bcpe-1790	49	1	as	as	SCONJ
bcpe-1790	49	2	polya	polya	PROPN
bcpe-1790	49	3	said	say	VERB
bcpe-1790	49	4	,	,	PUNCT
bcpe-1790	49	5	"	"	PUNCT
bcpe-1790	49	6	mathematics	mathematic	NOUN
bcpe-1790	49	7	is	be	AUX
bcpe-1790	49	8	not	not	PART
bcpe-1790	49	9	a	a	DET
bcpe-1790	49	10	pile	pile	NOUN
bcpe-1790	49	11	of	of	ADP
bcpe-1790	49	12	symbols	symbol	NOUN
bcpe-1790	49	13	and	and	CCONJ
bcpe-1790	49	14	formulas	formula	NOUN
bcpe-1790	49	15	,	,	PUNCT
bcpe-1790	49	16	but	but	CCONJ
bcpe-1790	49	17	a	a	DET
bcpe-1790	49	18	lively	lively	ADJ
bcpe-1790	49	19	intellectual	intellectual	ADJ
bcpe-1790	49	20	activity	activity	NOUN
bcpe-1790	49	21	.	.	PUNCT
bcpe-1790	49	22	"	"	PUNCT
bcpe-1790	50	1	[	[	X
bcpe-1790	50	2	8	8	X
bcpe-1790	50	3	]	]	PUNCT
bcpe-1790	50	4	this	this	DET
bcpe-1790	50	5	developmental	developmental	ADJ
bcpe-1790	50	6	teaching	teaching	NOUN
bcpe-1790	50	7	link	link	NOUN
bcpe-1790	50	8	stimulates	stimulate	VERB
bcpe-1790	50	9	students	student	NOUN
bcpe-1790	50	10	'	'	PART
bcpe-1790	50	11	subjective	subjective	ADJ
bcpe-1790	50	12	enthusiasm	enthusiasm	NOUN
bcpe-1790	50	13	for	for	ADP
bcpe-1790	50	14	learning	learn	VERB
bcpe-1790	50	15	and	and	CCONJ
bcpe-1790	50	16	improves	improve	VERB
bcpe-1790	50	17	their	their	PRON
bcpe-1790	50	18	understanding	understanding	NOUN
bcpe-1790	50	19	.	.	PUNCT
bcpe-1790	51	1	it	it	PRON
bcpe-1790	51	2	has	have	AUX
bcpe-1790	51	3	accumulated	accumulate	VERB
bcpe-1790	51	4	valuable	valuable	ADJ
bcpe-1790	51	5	experience	experience	NOUN
bcpe-1790	51	6	and	and	CCONJ
bcpe-1790	51	7	laid	lay	VERB
bcpe-1790	51	8	a	a	DET
bcpe-1790	51	9	solid	solid	ADJ
bcpe-1790	51	10	foundation	foundation	NOUN
bcpe-1790	51	11	for	for	ADP
bcpe-1790	51	12	solving	solve	VERB
bcpe-1790	51	13	more	more	ADJ
bcpe-1790	51	14	,	,	PUNCT
bcpe-1790	51	15	deeper	deep	ADJ
bcpe-1790	51	16	,	,	PUNCT
bcpe-1790	51	17	and	and	CCONJ
bcpe-1790	51	18	wider	wide	ADJ
bcpe-1790	51	19	mathematical	mathematical	ADJ
bcpe-1790	51	20	problems	problem	NOUN
bcpe-1790	51	21	in	in	ADP
bcpe-1790	51	22	the	the	DET
bcpe-1790	51	23	future	future	NOUN
bcpe-1790	51	24	.	.	PUNCT
bcpe-1790	52	1	3	3	X
bcpe-1790	52	2	.	.	NOUN
bcpe-1790	52	3	practical	practical	ADJ
bcpe-1790	52	4	applications	application	NOUN
bcpe-1790	52	5	the	the	DET
bcpe-1790	52	6	topic	topic	NOUN
bcpe-1790	52	7	(	(	PUNCT
bcpe-1790	52	8	2021	2021	NUM
bcpe-1790	52	9	national	national	ADJ
bcpe-1790	52	10	unified	unify	VERB
bcpe-1790	52	11	college	college	NOUN
bcpe-1790	52	12	entrance	entrance	NOUN
bcpe-1790	52	13	examination	examination	NOUN
bcpe-1790	52	14	paper	paper	NOUN
bcpe-1790	52	15	science	science	NOUN
bcpe-1790	52	16	volume	volume	NOUN
bcpe-1790	52	17	b	b	PROPN
bcpe-1790	52	18	question	question	NOUN
bcpe-1790	52	19	21	21	NUM
bcpe-1790	52	20	)	)	PUNCT
bcpe-1790	52	21	has	have	VERB
bcpe-1790	52	22	a	a	DET
bcpe-1790	52	23	known	know	VERB
bcpe-1790	52	24	focus	focus	NOUN
bcpe-1790	52	25	𝐹𝐹	𝐹𝐹	NOUN
bcpe-1790	52	26	on	on	ADP
bcpe-1790	52	27	the	the	DET
bcpe-1790	52	28	parabola	parabola	PROPN
bcpe-1790	52	29	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	52	30	:	:	PUNCT
bcpe-1790	52	31	𝑥𝑥2	𝑥𝑥2	X
bcpe-1790	52	32	=	=	SYM
bcpe-1790	52	33	2𝑝𝑝𝑝𝑝(𝑝𝑝	2𝑝𝑝𝑝𝑝(𝑝𝑝	NUM
bcpe-1790	52	34	>	>	X
bcpe-1790	52	35	0	0	NUM
bcpe-1790	52	36	)	)	PUNCT
bcpe-1790	52	37	,	,	PUNCT
bcpe-1790	52	38	and	and	CCONJ
bcpe-1790	52	39	the	the	DET
bcpe-1790	52	40	minimum	minimum	ADJ
bcpe-1790	52	41	distance	distance	NOUN
bcpe-1790	52	42	between	between	ADP
bcpe-1790	52	43	𝐹𝐹	𝐹𝐹	PROPN
bcpe-1790	52	44	and	and	CCONJ
bcpe-1790	52	45	the	the	DET
bcpe-1790	52	46	point	point	NOUN
bcpe-1790	52	47	on	on	ADP
bcpe-1790	52	48	the	the	DET
bcpe-1790	52	49	circle	circle	NOUN
bcpe-1790	52	50	𝑀𝑀	𝑀𝑀	PROPN
bcpe-1790	52	51	:	:	PUNCT
bcpe-1790	52	52	𝑥𝑥2	𝑥𝑥2	X
bcpe-1790	52	53	+	+	CCONJ
bcpe-1790	52	54	(	(	PUNCT
bcpe-1790	52	55	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	52	56	+	+	PROPN
bcpe-1790	52	57	4)2	4)2	NOUN
bcpe-1790	52	58	=	=	SYM
bcpe-1790	52	59	1	1	NUM
bcpe-1790	52	60	is	be	AUX
bcpe-1790	52	61	4	4	NUM
bcpe-1790	52	62	.	.	PUNCT
bcpe-1790	53	1	(	(	PUNCT
bcpe-1790	53	2	1	1	X
bcpe-1790	53	3	)	)	PUNCT
bcpe-1790	53	4	ask	ask	NOUN
bcpe-1790	53	5	𝑝𝑝.	𝑝𝑝.	PROPN
bcpe-1790	53	6	(	(	PUNCT
bcpe-1790	53	7	2	2	NUM
bcpe-1790	53	8	)	)	PUNCT
bcpe-1790	53	9	if	if	SCONJ
bcpe-1790	53	10	the	the	DET
bcpe-1790	53	11	point	point	NOUN
bcpe-1790	53	12	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	53	13	is	be	AUX
bcpe-1790	53	14	on	on	ADP
bcpe-1790	53	15	𝑀𝑀	𝑀𝑀	PROPN
bcpe-1790	53	16	,	,	PUNCT
bcpe-1790	53	17	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	53	18	and	and	CCONJ
bcpe-1790	53	19	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	53	20	are	be	AUX
bcpe-1790	53	21	the	the	DET
bcpe-1790	53	22	two	two	NUM
bcpe-1790	53	23	tangent	tangent	ADJ
bcpe-1790	53	24	lines	line	NOUN
bcpe-1790	53	25	of	of	ADP
bcpe-1790	53	26	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	53	27	,	,	PUNCT
bcpe-1790	53	28	𝑃𝑃	𝑃𝑃	PROPN
bcpe-1790	53	29	,	,	PUNCT
bcpe-1790	53	30	and	and	CCONJ
bcpe-1790	53	31	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	53	32	are	be	AUX
bcpe-1790	53	33	the	the	DET
bcpe-1790	53	34	tangent	tangent	ADJ
bcpe-1790	53	35	points	point	NOUN
bcpe-1790	53	36	,	,	PUNCT
bcpe-1790	53	37	find	find	VERB
bcpe-1790	53	38	the	the	DET
bcpe-1790	53	39	maximum	maximum	ADJ
bcpe-1790	53	40	value	value	NOUN
bcpe-1790	53	41	of	of	ADP
bcpe-1790	53	42	the	the	DET
bcpe-1790	53	43	area	area	NOUN
bcpe-1790	53	44	of	of	ADP
bcpe-1790	53	45	∆𝑃𝑃𝑃𝑃𝑃𝑃.	∆𝑃𝑃𝑃𝑃𝑃𝑃.	PROPN
bcpe-1790	53	46	in	in	ADP
bcpe-1790	53	47	high	high	ADJ
bcpe-1790	53	48	school	school	NOUN
bcpe-1790	53	49	teaching	teaching	NOUN
bcpe-1790	53	50	,	,	PUNCT
bcpe-1790	53	51	most	most	ADJ
bcpe-1790	53	52	students	student	NOUN
bcpe-1790	53	53	have	have	AUX
bcpe-1790	53	54	reached	reach	VERB
bcpe-1790	53	55	the	the	DET
bcpe-1790	53	56	level	level	NOUN
bcpe-1790	53	57	of	of	ADP
bcpe-1790	53	58	solving	solve	VERB
bcpe-1790	53	59	the	the	DET
bcpe-1790	53	60	first	first	ADJ
bcpe-1790	53	61	question	question	NOUN
bcpe-1790	53	62	,	,	PUNCT
bcpe-1790	53	63	so	so	CCONJ
bcpe-1790	53	64	only	only	ADV
bcpe-1790	53	65	the	the	DET
bcpe-1790	53	66	second	second	ADJ
bcpe-1790	53	67	question	question	NOUN
bcpe-1790	53	68	is	be	AUX
bcpe-1790	53	69	analyzed	analyze	VERB
bcpe-1790	53	70	to	to	PART
bcpe-1790	53	71	illustrate	illustrate	VERB
bcpe-1790	53	72	further	far	ADV
bcpe-1790	53	73	the	the	DET
bcpe-1790	53	74	innovative	innovative	ADJ
bcpe-1790	53	75	application	application	NOUN
bcpe-1790	53	76	value	value	NOUN
bcpe-1790	53	77	of	of	ADP
bcpe-1790	53	78	polya	polya	NOUN
bcpe-1790	53	79	's	's	PART
bcpe-1790	53	80	problem	problem	NOUN
bcpe-1790	53	81	-	-	PUNCT
bcpe-1790	53	82	solving	solve	VERB
bcpe-1790	53	83	thought	thought	NOUN
bcpe-1790	53	84	.	.	PUNCT
bcpe-1790	54	1	in	in	ADP
bcpe-1790	54	2	this	this	DET
bcpe-1790	54	3	paper	paper	NOUN
bcpe-1790	54	4	,	,	PUNCT
bcpe-1790	54	5	simulation	simulation	NOUN
bcpe-1790	54	6	teaching	teaching	NOUN
bcpe-1790	54	7	is	be	AUX
bcpe-1790	54	8	carried	carry	VERB
bcpe-1790	54	9	out	out	ADP
bcpe-1790	54	10	through	through	ADP
bcpe-1790	54	11	dialogue	dialogue	NOUN
bcpe-1790	54	12	.	.	PUNCT
bcpe-1790	55	1	the	the	DET
bcpe-1790	55	2	code	code	NOUN
bcpe-1790	55	3	name	name	NOUN
bcpe-1790	55	4	of	of	ADP
bcpe-1790	55	5	the	the	DET
bcpe-1790	55	6	teacher	teacher	NOUN
bcpe-1790	55	7	is	be	AUX
bcpe-1790	55	8	"	"	PUNCT
bcpe-1790	55	9	t	t	PROPN
bcpe-1790	55	10	,	,	PUNCT
bcpe-1790	55	11	"	"	PUNCT
bcpe-1790	55	12	and	and	CCONJ
bcpe-1790	55	13	the	the	DET
bcpe-1790	55	14	code	code	NOUN
bcpe-1790	55	15	name	name	NOUN
bcpe-1790	55	16	of	of	ADP
bcpe-1790	55	17	the	the	DET
bcpe-1790	55	18	student	student	NOUN
bcpe-1790	55	19	is	be	AUX
bcpe-1790	55	20	"	"	PUNCT
bcpe-1790	55	21	s.	s.	PROPN
bcpe-1790	55	22	"	"	PUNCT
bcpe-1790	56	1	[	[	X
bcpe-1790	56	2	9	9	NUM
bcpe-1790	56	3	-	-	SYM
bcpe-1790	56	4	10	10	NUM
bcpe-1790	56	5	]	]	PUNCT
bcpe-1790	56	6	analysis	analysis	NOUN
bcpe-1790	56	7	of	of	ADP
bcpe-1790	56	8	the	the	DET
bcpe-1790	56	9	second	second	ADJ
bcpe-1790	56	10	question	question	NOUN
bcpe-1790	56	11	:	:	PUNCT
bcpe-1790	56	12	(	(	PUNCT
bcpe-1790	56	13	1	1	X
bcpe-1790	56	14	)	)	PUNCT
bcpe-1790	56	15	observing	observe	VERB
bcpe-1790	56	16	and	and	CCONJ
bcpe-1790	56	17	doubting	doubt	VERB
bcpe-1790	56	18	t	t	PROPN
bcpe-1790	56	19	:	:	PUNCT
bcpe-1790	56	20	read	read	VERB
bcpe-1790	56	21	the	the	DET
bcpe-1790	56	22	second	second	ADJ
bcpe-1790	56	23	question	question	NOUN
bcpe-1790	56	24	carefully	carefully	ADV
bcpe-1790	56	25	and	and	CCONJ
bcpe-1790	56	26	talk	talk	VERB
bcpe-1790	56	27	about	about	ADP
bcpe-1790	56	28	what	what	DET
bcpe-1790	56	29	information	information	NOUN
bcpe-1790	56	30	you	you	PRON
bcpe-1790	56	31	have	have	AUX
bcpe-1790	56	32	obtained	obtain	VERB
bcpe-1790	56	33	?	?	PUNCT
bcpe-1790	57	1	moreover	moreover	ADV
bcpe-1790	57	2	,	,	PUNCT
bcpe-1790	57	3	what	what	PRON
bcpe-1790	57	4	is	be	AUX
bcpe-1790	57	5	the	the	DET
bcpe-1790	57	6	unknown	unknown	ADJ
bcpe-1790	57	7	quantity	quantity	NOUN
bcpe-1790	57	8	of	of	ADP
bcpe-1790	57	9	the	the	DET
bcpe-1790	57	10	title	title	NOUN
bcpe-1790	57	11	?	?	PUNCT
bcpe-1790	58	1	is	be	AUX
bcpe-1790	58	2	this	this	PRON
bcpe-1790	58	3	a	a	DET
bcpe-1790	58	4	question	question	NOUN
bcpe-1790	58	5	of	of	ADP
bcpe-1790	58	6	what	what	PRON
bcpe-1790	58	7	?	?	PUNCT
bcpe-1790	59	1	s:𝑃𝑃𝑃𝑃	s:𝑃𝑃𝑃𝑃	INTJ
bcpe-1790	59	2	and	and	CCONJ
bcpe-1790	59	3	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	59	4	are	be	AUX
bcpe-1790	59	5	the	the	DET
bcpe-1790	59	6	two	two	NUM
bcpe-1790	59	7	tangent	tangent	ADJ
bcpe-1790	59	8	lines	line	NOUN
bcpe-1790	59	9	of	of	ADP
bcpe-1790	59	10	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	59	11	,	,	PUNCT
bcpe-1790	59	12	𝑃𝑃	𝑃𝑃	PROPN
bcpe-1790	59	13	,	,	PUNCT
bcpe-1790	59	14	and	and	CCONJ
bcpe-1790	59	15	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	59	16	are	be	AUX
bcpe-1790	59	17	the	the	DET
bcpe-1790	59	18	tangent	tangent	ADJ
bcpe-1790	59	19	points	point	NOUN
bcpe-1790	59	20	.	.	PUNCT
bcpe-1790	60	1	the	the	DET
bcpe-1790	60	2	unknown	unknown	ADJ
bcpe-1790	60	3	quantity	quantity	NOUN
bcpe-1790	60	4	is	be	AUX
bcpe-1790	60	5	the	the	DET
bcpe-1790	60	6	maximum	maximum	ADJ
bcpe-1790	60	7	value	value	NOUN
bcpe-1790	60	8	of	of	ADP
bcpe-1790	60	9	the	the	DET
bcpe-1790	60	10	area	area	NOUN
bcpe-1790	60	11	of	of	ADP
bcpe-1790	60	12	∆𝑃𝑃𝑃𝑃𝑃𝑃.	∆𝑃𝑃𝑃𝑃𝑃𝑃.	PROPN
bcpe-1790	60	13	it	it	PRON
bcpe-1790	60	14	is	be	AUX
bcpe-1790	60	15	a	a	DET
bcpe-1790	60	16	question	question	NOUN
bcpe-1790	60	17	of	of	ADP
bcpe-1790	60	18	the	the	DET
bcpe-1790	60	19	best	good	ADJ
bcpe-1790	60	20	value	value	NOUN
bcpe-1790	60	21	.	.	PUNCT
bcpe-1790	61	1	t	t	PROPN
bcpe-1790	61	2	:	:	PUNCT
bcpe-1790	61	3	what	what	PRON
bcpe-1790	61	4	is	be	AUX
bcpe-1790	61	5	the	the	DET
bcpe-1790	61	6	most	most	ADV
bcpe-1790	61	7	common	common	ADJ
bcpe-1790	61	8	method	method	NOUN
bcpe-1790	61	9	among	among	ADP
bcpe-1790	61	10	the	the	DET
bcpe-1790	61	11	solutions	solution	NOUN
bcpe-1790	61	12	to	to	ADP
bcpe-1790	61	13	the	the	DET
bcpe-1790	61	14	conic	conic	ADJ
bcpe-1790	61	15	section	section	NOUN
bcpe-1790	61	16	maximum	maximum	ADJ
bcpe-1790	61	17	value	value	NOUN
bcpe-1790	61	18	problem	problem	NOUN
bcpe-1790	61	19	?	?	PUNCT
bcpe-1790	62	1	s	s	AUX
bcpe-1790	62	2	:	:	PUNCT
bcpe-1790	62	3	transform	transform	VERB
bcpe-1790	62	4	the	the	DET
bcpe-1790	62	5	conic	conic	ADJ
bcpe-1790	62	6	section	section	NOUN
bcpe-1790	62	7	maximum	maximum	ADJ
bcpe-1790	62	8	value	value	NOUN
bcpe-1790	62	9	problem	problem	NOUN
bcpe-1790	62	10	into	into	ADP
bcpe-1790	62	11	the	the	DET
bcpe-1790	62	12	maximum	maximum	ADJ
bcpe-1790	62	13	value	value	NOUN
bcpe-1790	62	14	problem	problem	NOUN
bcpe-1790	62	15	of	of	ADP
bcpe-1790	62	16	quadratic	quadratic	ADJ
bcpe-1790	62	17	function	function	NOUN
bcpe-1790	62	18	or	or	CCONJ
bcpe-1790	62	19	trigonometric	trigonometric	ADJ
bcpe-1790	62	20	function	function	NOUN
bcpe-1790	62	21	,	,	PUNCT
bcpe-1790	62	22	and	and	CCONJ
bcpe-1790	62	23	then	then	ADV
bcpe-1790	62	24	use	use	VERB
bcpe-1790	62	25	the	the	DET
bcpe-1790	62	26	basic	basic	ADJ
bcpe-1790	62	27	inequality	inequality	NOUN
bcpe-1790	62	28	method	method	NOUN
bcpe-1790	62	29	,	,	PUNCT
bcpe-1790	62	30	the	the	DET
bcpe-1790	62	31	matching	matching	NOUN
bcpe-1790	62	32	method	method	NOUN
bcpe-1790	62	33	,	,	PUNCT
bcpe-1790	62	34	element	element	NOUN
bcpe-1790	62	35	replacement	replacement	NOUN
bcpe-1790	62	36	method	method	NOUN
bcpe-1790	62	37	,	,	PUNCT
bcpe-1790	62	38	derivative	derivative	ADJ
bcpe-1790	62	39	method	method	NOUN
bcpe-1790	62	40	,	,	PUNCT
bcpe-1790	62	41	number	number	NOUN
bcpe-1790	62	42	-	-	PUNCT
bcpe-1790	62	43	shape	shape	NOUN
bcpe-1790	62	44	combination	combination	NOUN
bcpe-1790	62	45	method	method	NOUN
bcpe-1790	62	46	,	,	PUNCT
bcpe-1790	62	47	and	and	CCONJ
bcpe-1790	62	48	other	other	ADJ
bcpe-1790	62	49	methods	method	NOUN
bcpe-1790	62	50	to	to	PART
bcpe-1790	62	51	find	find	VERB
bcpe-1790	62	52	the	the	DET
bcpe-1790	62	53	maximum	maximum	ADJ
bcpe-1790	62	54	value	value	NOUN
bcpe-1790	62	55	.	.	PUNCT
bcpe-1790	63	1	t	t	NOUN
bcpe-1790	63	2	:	:	PUNCT
bcpe-1790	63	3	in	in	ADP
bcpe-1790	63	4	order	order	NOUN
bcpe-1790	63	5	to	to	PART
bcpe-1790	63	6	use	use	VERB
bcpe-1790	63	7	these	these	DET
bcpe-1790	63	8	methods	method	NOUN
bcpe-1790	63	9	,	,	PUNCT
bcpe-1790	63	10	what	what	PRON
bcpe-1790	63	11	are	be	AUX
bcpe-1790	63	12	the	the	DET
bcpe-1790	63	13	prerequisites	prerequisite	NOUN
bcpe-1790	63	14	?	?	PUNCT
bcpe-1790	64	1	s	s	X
bcpe-1790	64	2	:	:	PUNCT
bcpe-1790	64	3	convert	convert	VERB
bcpe-1790	64	4	∆𝑃𝑃𝑃𝑃𝑃𝑃	∆𝑃𝑃𝑃𝑃𝑃𝑃	PRON
bcpe-1790	64	5	area	area	NOUN
bcpe-1790	64	6	expressions	expression	NOUN
bcpe-1790	64	7	into	into	ADP
bcpe-1790	64	8	quadratic	quadratic	ADJ
bcpe-1790	64	9	or	or	CCONJ
bcpe-1790	64	10	trigonometric	trigonometric	ADJ
bcpe-1790	64	11	expressions	expression	NOUN
bcpe-1790	64	12	.	.	PUNCT
bcpe-1790	65	1	(	(	PUNCT
bcpe-1790	65	2	2	2	X
bcpe-1790	65	3	)	)	PUNCT
bcpe-1790	65	4	multivariate	multivariate	NOUN
bcpe-1790	65	5	analyzing	analyzing	NOUN
bcpe-1790	65	6	doubts	doubt	NOUN
bcpe-1790	65	7	at	at	ADP
bcpe-1790	65	8	this	this	DET
bcpe-1790	65	9	time	time	NOUN
bcpe-1790	65	10	,	,	PUNCT
bcpe-1790	65	11	students	student	NOUN
bcpe-1790	65	12	may	may	AUX
bcpe-1790	65	13	be	be	AUX
bcpe-1790	65	14	limited	limit	VERB
bcpe-1790	65	15	to	to	ADP
bcpe-1790	65	16	finding	find	VERB
bcpe-1790	65	17	connections	connection	NOUN
bcpe-1790	65	18	between	between	ADP
bcpe-1790	65	19	general	general	ADJ
bcpe-1790	65	20	conditions	condition	NOUN
bcpe-1790	65	21	and	and	CCONJ
bcpe-1790	65	22	∆𝑃𝑃𝑃𝑃𝑃𝑃	∆𝑃𝑃𝑃𝑃𝑃𝑃	NUM
bcpe-1790	65	23	area	area	NOUN
bcpe-1790	65	24	expressions	expression	NOUN
bcpe-1790	65	25	,	,	PUNCT
bcpe-1790	65	26	so	so	SCONJ
bcpe-1790	65	27	a	a	DET
bcpe-1790	65	28	teacher	teacher	NOUN
bcpe-1790	65	29	can	can	AUX
bcpe-1790	65	30	guide	guide	VERB
bcpe-1790	65	31	students	student	NOUN
bcpe-1790	65	32	to	to	PART
bcpe-1790	65	33	use	use	VERB
bcpe-1790	65	34	reverse	reverse	NOUN
bcpe-1790	65	35	thinking	think	VERB
bcpe-1790	65	36	to	to	PART
bcpe-1790	65	37	start	start	VERB
bcpe-1790	65	38	from	from	ADP
bcpe-1790	65	39	the	the	DET
bcpe-1790	65	40	problem	problem	NOUN
bcpe-1790	65	41	to	to	PART
bcpe-1790	65	42	find	find	VERB
bcpe-1790	65	43	the	the	DET
bcpe-1790	65	44	relationship	relationship	NOUN
bcpe-1790	65	45	between	between	ADP
bcpe-1790	65	46	known	know	VERB
bcpe-1790	65	47	conditions	condition	NOUN
bcpe-1790	65	48	and	and	CCONJ
bcpe-1790	65	49	known	know	VERB
bcpe-1790	65	50	conditions	condition	NOUN
bcpe-1790	65	51	.	.	PUNCT
bcpe-1790	66	1	t	t	NOUN
bcpe-1790	66	2	:	:	PUNCT
bcpe-1790	66	3	how	how	SCONJ
bcpe-1790	66	4	can	can	AUX
bcpe-1790	66	5	i	i	PRON
bcpe-1790	66	6	find	find	VERB
bcpe-1790	66	7	the	the	DET
bcpe-1790	66	8	area	area	NOUN
bcpe-1790	66	9	of	of	ADP
bcpe-1790	66	10	∆𝑃𝑃𝑃𝑃𝑃𝑃	∆𝑃𝑃𝑃𝑃𝑃𝑃	PROPN
bcpe-1790	66	11	?	?	PUNCT
bcpe-1790	67	1	have	have	AUX
bcpe-1790	67	2	you	you	PRON
bcpe-1790	67	3	seen	see	VERB
bcpe-1790	67	4	similar	similar	ADJ
bcpe-1790	67	5	problems	problem	NOUN
bcpe-1790	67	6	before	before	ADV
bcpe-1790	67	7	?	?	PUNCT
bcpe-1790	68	1	s	s	VERB
bcpe-1790	68	2	:	:	PUNCT
bcpe-1790	68	3	using	use	VERB
bcpe-1790	68	4	the	the	DET
bcpe-1790	68	5	formula	formula	NOUN
bcpe-1790	68	6	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	68	7	=	=	SYM
bcpe-1790	68	8	1	1	NUM
bcpe-1790	68	9	2	2	NUM
bcpe-1790	68	10	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	68	11	∙	∙	PROPN
bcpe-1790	68	12	𝑑𝑑	𝑑𝑑	PROPN
bcpe-1790	68	13	,	,	PUNCT
bcpe-1790	68	14	𝑑𝑑	𝑑𝑑	VERB
bcpe-1790	68	15	is	be	AUX
bcpe-1790	68	16	the	the	DET
bcpe-1790	68	17	distance	distance	NOUN
bcpe-1790	68	18	from	from	ADP
bcpe-1790	68	19	a	a	DET
bcpe-1790	68	20	point	point	NOUN
bcpe-1790	68	21	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	68	22	to	to	ADP
bcpe-1790	68	23	a	a	DET
bcpe-1790	68	24	straight	straight	ADJ
bcpe-1790	68	25	line	line	NOUN
bcpe-1790	68	26	𝑃𝑃𝑃𝑃.	𝑃𝑃𝑃𝑃.	VERB
bcpe-1790	68	27	the	the	DET
bcpe-1790	68	28	conic	conic	ADJ
bcpe-1790	68	29	section	section	NOUN
bcpe-1790	68	30	is	be	AUX
bcpe-1790	68	31	often	often	ADV
bcpe-1790	68	32	asked	ask	VERB
bcpe-1790	68	33	for	for	ADP
bcpe-1790	68	34	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	68	35	,	,	PUNCT
bcpe-1790	68	36	i	i	PRON
bcpe-1790	68	37	remember	remember	VERB
bcpe-1790	68	38	its	its	PRON
bcpe-1790	68	39	expression	expression	NOUN
bcpe-1790	68	40	is	be	AUX
bcpe-1790	68	41	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	68	42	=	=	SYM
bcpe-1790	68	43	√1	√1	PROPN
bcpe-1790	69	1	+	+	CCONJ
bcpe-1790	69	2	𝑘𝑘2|𝑥𝑥1	𝑘𝑘2|𝑥𝑥1	PRON
bcpe-1790	69	3	−	−	NOUN
bcpe-1790	69	4	𝑥𝑥2|	𝑥𝑥2|	PROPN
bcpe-1790	69	5	,	,	PUNCT
bcpe-1790	69	6	and	and	CCONJ
bcpe-1790	69	7	𝑑𝑑	𝑑𝑑	NOUN
bcpe-1790	69	8	can	can	AUX
bcpe-1790	69	9	be	be	AUX
bcpe-1790	69	10	expressed	express	VERB
bcpe-1790	69	11	by	by	ADP
bcpe-1790	69	12	directly	directly	ADV
bcpe-1790	69	13	applying	apply	VERB
bcpe-1790	69	14	the	the	DET
bcpe-1790	69	15	point	point	NOUN
bcpe-1790	69	16	-	-	PUNCT
bcpe-1790	69	17	to	to	ADP
bcpe-1790	69	18	-	-	PUNCT
bcpe-1790	69	19	distance	distance	NOUN
bcpe-1790	69	20	formula	formula	NOUN
bcpe-1790	69	21	.	.	PUNCT
bcpe-1790	70	1	t	t	NOUN
bcpe-1790	70	2	:	:	PUNCT
bcpe-1790	70	3	what	what	PRON
bcpe-1790	70	4	do	do	AUX
bcpe-1790	70	5	you	you	PRON
bcpe-1790	70	6	need	need	VERB
bcpe-1790	70	7	to	to	PART
bcpe-1790	70	8	know	know	VERB
bcpe-1790	70	9	to	to	PART
bcpe-1790	70	10	get	get	VERB
bcpe-1790	70	11	further	further	ADJ
bcpe-1790	70	12	expressions	expression	NOUN
bcpe-1790	70	13	?	?	PUNCT
bcpe-1790	71	1	can	can	AUX
bcpe-1790	71	2	these	these	DET
bcpe-1790	71	3	expressions	expression	NOUN
bcpe-1790	71	4	be	be	AUX
bcpe-1790	71	5	obtained	obtain	VERB
bcpe-1790	71	6	based	base	VERB
bcpe-1790	71	7	on	on	ADP
bcpe-1790	71	8	the	the	DET
bcpe-1790	71	9	existing	exist	VERB
bcpe-1790	71	10	conditions	condition	NOUN
bcpe-1790	71	11	?	?	PUNCT
bcpe-1790	72	1	bcp	bcp	NOUN
bcpe-1790	72	2	education	education	PROPN
bcpe-1790	72	3	&	&	CCONJ
bcpe-1790	72	4	psychology	psychology	PROPN
bcpe-1790	72	5	erhss	erhss	PROPN
bcpe-1790	72	6	2022	2022	NUM
bcpe-1790	72	7	volume	volume	NOUN
bcpe-1790	72	8	6	6	NUM
bcpe-1790	72	9	(	(	PUNCT
bcpe-1790	72	10	2022	2022	NUM
bcpe-1790	72	11	)	)	PUNCT
bcpe-1790	72	12	203	203	NUM
bcpe-1790	72	13	s	s	NOUN
bcpe-1790	72	14	:	:	PUNCT
bcpe-1790	72	15	we	we	PRON
bcpe-1790	72	16	should	should	AUX
bcpe-1790	72	17	know	know	VERB
bcpe-1790	72	18	the	the	DET
bcpe-1790	72	19	equation	equation	NOUN
bcpe-1790	72	20	for	for	ADP
bcpe-1790	72	21	straight	straight	ADJ
bcpe-1790	72	22	-	-	PUNCT
bcpe-1790	72	23	line	line	NOUN
bcpe-1790	72	24	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	72	25	,	,	PUNCT
bcpe-1790	72	26	and	and	CCONJ
bcpe-1790	72	27	expressions	expression	NOUN
bcpe-1790	72	28	for	for	ADP
bcpe-1790	72	29	𝑥𝑥1	𝑥𝑥1	NOUN
bcpe-1790	72	30	+	+	CCONJ
bcpe-1790	72	31	𝑥𝑥2	𝑥𝑥2	PRON
bcpe-1790	72	32	and	and	CCONJ
bcpe-1790	72	33	𝑥𝑥1𝑥𝑥2	𝑥𝑥1𝑥𝑥2	ADV
bcpe-1790	72	34	.	.	PUNCT
bcpe-1790	73	1	it	it	PRON
bcpe-1790	73	2	is	be	AUX
bcpe-1790	73	3	known	know	VERB
bcpe-1790	73	4	that	that	SCONJ
bcpe-1790	73	5	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	73	6	and	and	CCONJ
bcpe-1790	73	7	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	73	8	are	be	AUX
bcpe-1790	73	9	the	the	DET
bcpe-1790	73	10	two	two	NUM
bcpe-1790	73	11	tangents	tangent	NOUN
bcpe-1790	73	12	of	of	ADP
bcpe-1790	73	13	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	73	14	,	,	PUNCT
bcpe-1790	73	15	and	and	CCONJ
bcpe-1790	73	16	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	73	17	and	and	CCONJ
bcpe-1790	73	18	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	73	19	are	be	AUX
bcpe-1790	73	20	the	the	DET
bcpe-1790	73	21	tangent	tangent	ADJ
bcpe-1790	73	22	points	point	NOUN
bcpe-1790	73	23	.	.	PUNCT
bcpe-1790	74	1	suppose	suppose	VERB
bcpe-1790	74	2	the	the	DET
bcpe-1790	74	3	coordinates	coordinate	NOUN
bcpe-1790	74	4	of	of	ADP
bcpe-1790	74	5	points	point	NOUN
bcpe-1790	74	6	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	74	7	and	and	CCONJ
bcpe-1790	74	8	𝑃𝑃	𝑃𝑃	PROPN
bcpe-1790	74	9	are(𝑥𝑥1	are(𝑥𝑥1	PROPN
bcpe-1790	74	10	,	,	PUNCT
bcpe-1790	74	11	𝑝𝑝1	𝑝𝑝1	PROPN
bcpe-1790	74	12	)	)	PUNCT
bcpe-1790	74	13	,	,	PUNCT
bcpe-1790	74	14	(	(	PUNCT
bcpe-1790	74	15	𝑥𝑥2	𝑥𝑥2	ADV
bcpe-1790	74	16	,	,	PUNCT
bcpe-1790	74	17	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	74	18	)	)	PUNCT
bcpe-1790	74	19	.	.	PUNCT
bcpe-1790	75	1	according	accord	VERB
bcpe-1790	75	2	to	to	ADP
bcpe-1790	75	3	the	the	DET
bcpe-1790	75	4	knowledge	knowledge	NOUN
bcpe-1790	75	5	of	of	ADP
bcpe-1790	75	6	derivatives	derivative	NOUN
bcpe-1790	75	7	,	,	PUNCT
bcpe-1790	75	8	𝑘𝑘𝑃𝑃𝑃𝑃	𝑘𝑘𝑃𝑃𝑃𝑃	X
bcpe-1790	75	9	=	=	NOUN
bcpe-1790	75	10	1	1	NUM
bcpe-1790	75	11	2	2	NUM
bcpe-1790	75	12	𝑥𝑥1,𝑘𝑘𝑃𝑃𝑃𝑃	𝑥𝑥1,𝑘𝑘𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	75	13	=	=	SYM
bcpe-1790	75	14	1	1	NUM
bcpe-1790	75	15	2	2	NUM
bcpe-1790	75	16	𝑥𝑥2	𝑥𝑥2	NOUN
bcpe-1790	75	17	,	,	PUNCT
bcpe-1790	75	18	then	then	ADV
bcpe-1790	75	19	the	the	DET
bcpe-1790	75	20	equation	equation	NOUN
bcpe-1790	75	21	of	of	ADP
bcpe-1790	75	22	the	the	DET
bcpe-1790	75	23	straight	straight	ADJ
bcpe-1790	75	24	line	line	NOUN
bcpe-1790	75	25	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	75	26	is	be	AUX
bcpe-1790	75	27	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	75	28	−	−	PROPN
bcpe-1790	75	29	𝑝𝑝1	𝑝𝑝1	NOUN
bcpe-1790	75	30	=	=	SYM
bcpe-1790	75	31	1	1	NUM
bcpe-1790	75	32	2	2	NUM
bcpe-1790	75	33	𝑥𝑥1(𝑥𝑥	𝑥𝑥1(𝑥𝑥	NOUN
bcpe-1790	75	34	−	−	PROPN
bcpe-1790	75	35	𝑥𝑥1	𝑥𝑥1	NOUN
bcpe-1790	75	36	)	)	PUNCT
bcpe-1790	75	37	,	,	PUNCT
bcpe-1790	75	38	and	and	CCONJ
bcpe-1790	75	39	the	the	DET
bcpe-1790	75	40	equation	equation	NOUN
bcpe-1790	75	41	of	of	ADP
bcpe-1790	75	42	the	the	DET
bcpe-1790	75	43	straight	straight	ADJ
bcpe-1790	75	44	line	line	NOUN
bcpe-1790	75	45	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	75	46	is	be	AUX
bcpe-1790	75	47	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	75	48	−	−	PROPN
bcpe-1790	75	49	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	75	50	=	=	NOUN
bcpe-1790	75	51	1	1	NUM
bcpe-1790	75	52	2	2	NUM
bcpe-1790	75	53	𝑥𝑥2(𝑥𝑥	𝑥𝑥2(𝑥𝑥	NOUN
bcpe-1790	75	54	−	−	NOUN
bcpe-1790	75	55	𝑥𝑥2	𝑥𝑥2	NOUN
bcpe-1790	75	56	)	)	PUNCT
bcpe-1790	75	57	.	.	PUNCT
bcpe-1790	76	1	t	t	PROPN
bcpe-1790	76	2	:	:	PUNCT
bcpe-1790	76	3	can	can	AUX
bcpe-1790	76	4	you	you	PRON
bcpe-1790	76	5	simplify	simplify	VERB
bcpe-1790	76	6	the	the	DET
bcpe-1790	76	7	equation	equation	NOUN
bcpe-1790	76	8	?	?	PUNCT
bcpe-1790	77	1	which	which	PRON
bcpe-1790	77	2	hidden	hide	VERB
bcpe-1790	77	3	condition	condition	NOUN
bcpe-1790	77	4	is	be	AUX
bcpe-1790	77	5	not	not	PART
bcpe-1790	77	6	used	use	VERB
bcpe-1790	77	7	?	?	PUNCT
bcpe-1790	78	1	s	s	X
bcpe-1790	78	2	:	:	PUNCT
bcpe-1790	78	3	the	the	DET
bcpe-1790	78	4	two	two	NUM
bcpe-1790	78	5	points	point	NOUN
bcpe-1790	78	6	𝑃𝑃,𝑃𝑃	𝑃𝑃,𝑃𝑃	NOUN
bcpe-1790	78	7	are	be	AUX
bcpe-1790	78	8	on	on	ADP
bcpe-1790	78	9	the	the	DET
bcpe-1790	78	10	parabola	parabola	PROPN
bcpe-1790	78	11	𝐶𝐶:𝑝𝑝	𝐶𝐶:𝑝𝑝	PROPN
bcpe-1790	78	12	=	=	NOUN
bcpe-1790	78	13	1	1	NUM
bcpe-1790	78	14	4	4	NUM
bcpe-1790	78	15	𝑥𝑥2	𝑥𝑥2	NOUN
bcpe-1790	78	16	,	,	PUNCT
bcpe-1790	78	17	and	and	CCONJ
bcpe-1790	78	18	the	the	DET
bcpe-1790	78	19	two	two	NUM
bcpe-1790	78	20	straight	straight	ADJ
bcpe-1790	78	21	-	-	PUNCT
bcpe-1790	78	22	line	line	NOUN
bcpe-1790	78	23	equations	equation	NOUN
bcpe-1790	78	24	can	can	AUX
bcpe-1790	78	25	be	be	AUX
bcpe-1790	78	26	simplified	simplify	VERB
bcpe-1790	78	27	as	as	ADP
bcpe-1790	78	28	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	NOUN
bcpe-1790	78	29	=	=	SYM
bcpe-1790	79	1	1	1	NUM
bcpe-1790	79	2	2	2	NUM
bcpe-1790	79	3	𝑥𝑥1𝑥𝑥	𝑥𝑥1𝑥𝑥	NOUN
bcpe-1790	79	4	−	−	PROPN
bcpe-1790	79	5	𝑝𝑝1	𝑝𝑝1	PROPN
bcpe-1790	79	6	,	,	PUNCT
bcpe-1790	79	7	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	X
bcpe-1790	79	8	=	=	PUNCT
bcpe-1790	79	9	1	1	NUM
bcpe-1790	79	10	2	2	NUM
bcpe-1790	79	11	𝑥𝑥2𝑥𝑥	𝑥𝑥2𝑥𝑥	NOUN
bcpe-1790	79	12	−	−	PROPN
bcpe-1790	80	1	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	80	2	.	.	PUNCT
bcpe-1790	81	1	t	t	PROPN
bcpe-1790	81	2	:	:	PUNCT
bcpe-1790	81	3	can	can	AUX
bcpe-1790	81	4	you	you	PRON
bcpe-1790	81	5	find	find	VERB
bcpe-1790	81	6	the	the	DET
bcpe-1790	81	7	equation	equation	NOUN
bcpe-1790	81	8	of	of	ADP
bcpe-1790	81	9	straight	straight	ADJ
bcpe-1790	81	10	line	line	NOUN
bcpe-1790	81	11	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	81	12	if	if	SCONJ
bcpe-1790	81	13	you	you	PRON
bcpe-1790	81	14	know	know	VERB
bcpe-1790	81	15	the	the	DET
bcpe-1790	81	16	equation	equation	NOUN
bcpe-1790	81	17	of	of	ADP
bcpe-1790	81	18	straight	straight	ADJ
bcpe-1790	81	19	line	line	NOUN
bcpe-1790	81	20	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	81	21	and	and	CCONJ
bcpe-1790	81	22	straight	straight	ADJ
bcpe-1790	81	23	-	-	PUNCT
bcpe-1790	81	24	line	line	NOUN
bcpe-1790	81	25	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	81	26	?	?	PUNCT
bcpe-1790	82	1	s1	s1	NOUN
bcpe-1790	82	2	:	:	PUNCT
bcpe-1790	82	3	it	it	PRON
bcpe-1790	82	4	is	be	AUX
bcpe-1790	82	5	known	know	VERB
bcpe-1790	82	6	that	that	SCONJ
bcpe-1790	82	7	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	82	8	is	be	AUX
bcpe-1790	82	9	the	the	DET
bcpe-1790	82	10	intersection	intersection	NOUN
bcpe-1790	82	11	of	of	ADP
bcpe-1790	82	12	the	the	DET
bcpe-1790	82	13	straight	straight	ADJ
bcpe-1790	82	14	line	line	NOUN
bcpe-1790	82	15	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	82	16	and	and	CCONJ
bcpe-1790	82	17	the	the	DET
bcpe-1790	82	18	straight	straight	ADJ
bcpe-1790	82	19	line	line	NOUN
bcpe-1790	82	20	𝑃𝑃𝑃𝑃.	𝑃𝑃𝑃𝑃.	NOUN
bcpe-1790	82	21	let	let	VERB
bcpe-1790	82	22	the	the	DET
bcpe-1790	82	23	coordinates	coordinate	NOUN
bcpe-1790	82	24	of	of	ADP
bcpe-1790	82	25	the	the	DET
bcpe-1790	82	26	point	point	NOUN
bcpe-1790	82	27	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	82	28	be	be	AUX
bcpe-1790	82	29	(	(	PUNCT
bcpe-1790	82	30	𝑥𝑥0	𝑥𝑥0	INTJ
bcpe-1790	82	31	,	,	PUNCT
bcpe-1790	82	32	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	82	33	)	)	PUNCT
bcpe-1790	82	34	,	,	PUNCT
bcpe-1790	82	35	and	and	CCONJ
bcpe-1790	82	36	substitute	substitute	VERB
bcpe-1790	82	37	it	it	PRON
bcpe-1790	82	38	into	into	ADP
bcpe-1790	82	39	the	the	DET
bcpe-1790	82	40	equation	equation	NOUN
bcpe-1790	82	41	of	of	ADP
bcpe-1790	82	42	the	the	DET
bcpe-1790	82	43	straight	straight	ADJ
bcpe-1790	82	44	line	line	NOUN
bcpe-1790	82	45	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	82	46	and	and	CCONJ
bcpe-1790	82	47	the	the	DET
bcpe-1790	82	48	straight	straight	ADJ
bcpe-1790	82	49	line	line	NOUN
bcpe-1790	82	50	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	82	51	to	to	PART
bcpe-1790	82	52	get	get	VERB
bcpe-1790	82	53	�	�	PROPN
bcpe-1790	82	54	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	82	55	=	=	NOUN
bcpe-1790	82	56	1	1	NUM
bcpe-1790	82	57	2	2	NUM
bcpe-1790	82	58	𝑥𝑥1𝑥𝑥0	𝑥𝑥1𝑥𝑥0	NOUN
bcpe-1790	82	59	−	−	NOUN
bcpe-1790	83	1	𝑝𝑝1	𝑝𝑝1	PROPN
bcpe-1790	83	2	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	83	3	=	=	NOUN
bcpe-1790	83	4	1	1	NUM
bcpe-1790	83	5	2	2	NUM
bcpe-1790	83	6	𝑥𝑥2𝑥𝑥0	𝑥𝑥2𝑥𝑥0	NOUN
bcpe-1790	83	7	−	−	NOUN
bcpe-1790	83	8	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	83	9	,	,	PUNCT
bcpe-1790	83	10	so	so	CCONJ
bcpe-1790	83	11	the	the	DET
bcpe-1790	83	12	equation	equation	NOUN
bcpe-1790	83	13	of	of	ADP
bcpe-1790	83	14	straight	straight	ADJ
bcpe-1790	83	15	line	line	NOUN
bcpe-1790	83	16	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	83	17	is	be	AUX
bcpe-1790	83	18	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	83	19	=	=	SYM
bcpe-1790	83	20	1	1	NUM
bcpe-1790	83	21	2	2	NUM
bcpe-1790	83	22	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	83	23	−	−	NOUN
bcpe-1790	83	24	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	83	25	,	,	PUNCT
bcpe-1790	83	26	that	that	ADV
bcpe-1790	83	27	is	is	ADV
bcpe-1790	83	28	,	,	PUNCT
bcpe-1790	83	29	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	83	30	=	=	NOUN
bcpe-1790	83	31	1	1	NUM
bcpe-1790	83	32	2	2	NUM
bcpe-1790	83	33	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	83	34	−	−	PROPN
bcpe-1790	83	35	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	83	36	.	.	PROPN
bcpe-1790	84	1	t	t	PROPN
bcpe-1790	84	2	:	:	PUNCT
bcpe-1790	84	3	some	some	DET
bcpe-1790	84	4	students	student	NOUN
bcpe-1790	84	5	do	do	AUX
bcpe-1790	84	6	not	not	PART
bcpe-1790	84	7	understand	understand	VERB
bcpe-1790	84	8	why	why	SCONJ
bcpe-1790	84	9	they	they	PRON
bcpe-1790	84	10	can	can	AUX
bcpe-1790	84	11	directly	directly	ADV
bcpe-1790	84	12	write	write	VERB
bcpe-1790	84	13	the	the	DET
bcpe-1790	84	14	equation	equation	NOUN
bcpe-1790	84	15	of	of	ADP
bcpe-1790	84	16	a	a	DET
bcpe-1790	84	17	straight	straight	ADJ
bcpe-1790	84	18	line	line	NOUN
bcpe-1790	84	19	𝑃𝑃𝑃𝑃.	𝑃𝑃𝑃𝑃.	NOUN
bcpe-1790	84	20	please	please	INTJ
bcpe-1790	84	21	explain	explain	VERB
bcpe-1790	84	22	.	.	PUNCT
bcpe-1790	85	1	s1	s1	NOUN
bcpe-1790	85	2	:	:	PUNCT
bcpe-1790	85	3	according	accord	VERB
bcpe-1790	85	4	to	to	ADP
bcpe-1790	85	5	the	the	DET
bcpe-1790	85	6	knowledge	knowledge	NOUN
bcpe-1790	85	7	of	of	ADP
bcpe-1790	85	8	plane	plane	NOUN
bcpe-1790	85	9	geometry	geometry	NOUN
bcpe-1790	85	10	,	,	PUNCT
bcpe-1790	85	11	it	it	PRON
bcpe-1790	85	12	is	be	AUX
bcpe-1790	85	13	known	know	VERB
bcpe-1790	85	14	that	that	SCONJ
bcpe-1790	85	15	two	two	NUM
bcpe-1790	85	16	points	point	NOUN
bcpe-1790	85	17	have	have	VERB
bcpe-1790	85	18	only	only	ADV
bcpe-1790	85	19	one	one	NUM
bcpe-1790	85	20	straight	straight	ADJ
bcpe-1790	85	21	line	line	NOUN
bcpe-1790	85	22	.	.	PUNCT
bcpe-1790	86	1	that	that	PRON
bcpe-1790	86	2	is	be	AUX
bcpe-1790	86	3	,	,	PUNCT
bcpe-1790	86	4	two	two	NUM
bcpe-1790	86	5	points	point	NOUN
bcpe-1790	86	6	determine	determine	VERB
bcpe-1790	86	7	a	a	DET
bcpe-1790	86	8	straight	straight	ADJ
bcpe-1790	86	9	line	line	NOUN
bcpe-1790	86	10	.	.	PUNCT
bcpe-1790	87	1	because	because	SCONJ
bcpe-1790	87	2	𝑃𝑃(𝑥𝑥1,𝑝𝑝1),𝑃𝑃(𝑥𝑥2,𝑝𝑝2	𝑃𝑃(𝑥𝑥1,𝑝𝑝1),𝑃𝑃(𝑥𝑥2,𝑝𝑝2	ADV
bcpe-1790	87	3	)	)	PUNCT
bcpe-1790	87	4	are	be	AUX
bcpe-1790	87	5	two	two	NUM
bcpe-1790	87	6	different	different	ADJ
bcpe-1790	87	7	points	point	NOUN
bcpe-1790	87	8	,	,	PUNCT
bcpe-1790	87	9	and	and	CCONJ
bcpe-1790	87	10	their	their	PRON
bcpe-1790	87	11	coordinates	coordinate	NOUN
bcpe-1790	87	12	satisfy	satisfy	VERB
bcpe-1790	87	13	the	the	DET
bcpe-1790	87	14	equation	equation	NOUN
bcpe-1790	87	15	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	87	16	=	=	SYM
bcpe-1790	87	17	1	1	NUM
bcpe-1790	87	18	2	2	NUM
bcpe-1790	87	19	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	87	20	−	−	NOUN
bcpe-1790	87	21	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	87	22	,	,	PUNCT
bcpe-1790	87	23	that	that	ADV
bcpe-1790	87	24	is	is	ADV
bcpe-1790	87	25	,	,	PUNCT
bcpe-1790	87	26	the	the	DET
bcpe-1790	87	27	coordinates	coordinate	NOUN
bcpe-1790	87	28	of	of	ADP
bcpe-1790	87	29	𝑃𝑃	𝑃𝑃	ADJ
bcpe-1790	87	30	and	and	CCONJ
bcpe-1790	87	31	𝑃𝑃	𝑃𝑃	ADJ
bcpe-1790	87	32	points	point	NOUN
bcpe-1790	87	33	are	be	AUX
bcpe-1790	87	34	the	the	DET
bcpe-1790	87	35	solution	solution	NOUN
bcpe-1790	87	36	of	of	ADP
bcpe-1790	87	37	the	the	DET
bcpe-1790	87	38	equation	equation	NOUN
bcpe-1790	87	39	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	87	40	=	=	SYM
bcpe-1790	87	41	1	1	NUM
bcpe-1790	87	42	2	2	NUM
bcpe-1790	87	43	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	87	44	−	−	PROPN
bcpe-1790	87	45	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	87	46	,	,	PUNCT
bcpe-1790	87	47	the	the	DET
bcpe-1790	87	48	equation	equation	NOUN
bcpe-1790	87	49	of	of	ADP
bcpe-1790	87	50	line	line	NOUN
bcpe-1790	87	51	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	87	52	is	be	AUX
bcpe-1790	87	53	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	87	54	=	=	SYM
bcpe-1790	87	55	1	1	NUM
bcpe-1790	87	56	2	2	NUM
bcpe-1790	87	57	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	87	58	−	−	PROPN
bcpe-1790	87	59	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	87	60	.	.	PROPN
bcpe-1790	88	1	t	t	PROPN
bcpe-1790	88	2	:	:	PUNCT
bcpe-1790	88	3	how	how	SCONJ
bcpe-1790	88	4	to	to	PART
bcpe-1790	88	5	find	find	VERB
bcpe-1790	88	6	the	the	DET
bcpe-1790	88	7	expression	expression	NOUN
bcpe-1790	88	8	of	of	ADP
bcpe-1790	88	9	𝑥𝑥1	𝑥𝑥1	NOUN
bcpe-1790	88	10	+	+	CCONJ
bcpe-1790	88	11	𝑥𝑥2	𝑥𝑥2	PRON
bcpe-1790	88	12	and	and	CCONJ
bcpe-1790	88	13	𝑥𝑥1𝑥𝑥2	𝑥𝑥1𝑥𝑥2	ADV
bcpe-1790	88	14	?	?	PUNCT
bcpe-1790	89	1	s	s	VERB
bcpe-1790	89	2	:	:	PUNCT
bcpe-1790	89	3	the	the	DET
bcpe-1790	89	4	equation	equation	NOUN
bcpe-1790	89	5	of	of	ADP
bcpe-1790	89	6	the	the	DET
bcpe-1790	89	7	simultaneous	simultaneous	ADJ
bcpe-1790	89	8	line	line	NOUN
bcpe-1790	89	9	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	89	10	and	and	CCONJ
bcpe-1790	89	11	the	the	DET
bcpe-1790	89	12	parabola	parabola	NOUN
bcpe-1790	89	13	𝐶𝐶can	𝐶𝐶can	PROPN
bcpe-1790	89	14	be	be	AUX
bcpe-1790	89	15	obtained	obtain	VERB
bcpe-1790	89	16	according	accord	VERB
bcpe-1790	89	17	to	to	ADP
bcpe-1790	89	18	veda	veda	PROPN
bcpe-1790	89	19	's	's	PART
bcpe-1790	89	20	theorem	theorem	PROPN
bcpe-1790	89	21	.	.	PUNCT
bcpe-1790	90	1	(	(	PUNCT
bcpe-1790	90	2	3	3	X
bcpe-1790	90	3	)	)	PUNCT
bcpe-1790	90	4	accurately	accurately	ADV
bcpe-1790	90	5	solving	solve	VERB
bcpe-1790	90	6	doubts	doubt	NOUN
bcpe-1790	90	7	let	let	VERB
bcpe-1790	90	8	𝑃𝑃(𝑥𝑥1	𝑃𝑃(𝑥𝑥1	NOUN
bcpe-1790	90	9	,	,	PUNCT
bcpe-1790	90	10	𝑝𝑝1),𝑃𝑃(𝑥𝑥2,𝑝𝑝2),𝑃𝑃(𝑥𝑥0	𝑝𝑝1),𝑃𝑃(𝑥𝑥2,𝑝𝑝2),𝑃𝑃(𝑥𝑥0	PROPN
bcpe-1790	90	11	,	,	PUNCT
bcpe-1790	90	12	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	90	13	)	)	PUNCT
bcpe-1790	90	14	,	,	PUNCT
bcpe-1790	90	15	there	there	PRON
bcpe-1790	90	16	is	be	VERB
bcpe-1790	90	17	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	90	18	+	+	CCONJ
bcpe-1790	90	19	(	(	PUNCT
bcpe-1790	90	20	𝑝𝑝0	𝑝𝑝0	X
bcpe-1790	90	21	+	+	NOUN
bcpe-1790	90	22	4)2	4)2	NOUN
bcpe-1790	90	23	=	=	SYM
bcpe-1790	90	24	1	1	X
bcpe-1790	90	25	.	.	PUNCT
bcpe-1790	90	26	given	give	VERB
bcpe-1790	90	27	𝐶𝐶:𝑝𝑝	𝐶𝐶:𝑝𝑝	PROPN
bcpe-1790	90	28	=	=	SYM
bcpe-1790	90	29	1	1	NUM
bcpe-1790	90	30	4	4	NUM
bcpe-1790	90	31	𝑥𝑥2	𝑥𝑥2	NOUN
bcpe-1790	90	32	,	,	PUNCT
bcpe-1790	90	33	derivation	derivation	NOUN
bcpe-1790	90	34	of	of	ADP
bcpe-1790	90	35	the	the	DET
bcpe-1790	90	36	function	function	NOUN
bcpe-1790	90	37	gives	give	VERB
bcpe-1790	90	38	𝑝𝑝′	𝑝𝑝′	NOUN
bcpe-1790	90	39	=	=	SYM
bcpe-1790	90	40	1	1	NUM
bcpe-1790	90	41	2	2	NUM
bcpe-1790	90	42	𝑥𝑥.	𝑥𝑥.	NOUN
bcpe-1790	90	43	∴	∴	VERB
bcpe-1790	90	44	the	the	DET
bcpe-1790	90	45	equation	equation	NOUN
bcpe-1790	90	46	of	of	ADP
bcpe-1790	90	47	the	the	DET
bcpe-1790	90	48	line	line	NOUN
bcpe-1790	90	49	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	90	50	is	be	AUX
bcpe-1790	90	51	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	90	52	−	−	PROPN
bcpe-1790	90	53	𝑝𝑝1	𝑝𝑝1	NOUN
bcpe-1790	90	54	=	=	SYM
bcpe-1790	90	55	1	1	NUM
bcpe-1790	90	56	2	2	NUM
bcpe-1790	90	57	𝑥𝑥1(𝑥𝑥	𝑥𝑥1(𝑥𝑥	NOUN
bcpe-1790	90	58	−	−	PROPN
bcpe-1790	90	59	𝑥𝑥1	𝑥𝑥1	NOUN
bcpe-1790	90	60	)	)	PUNCT
bcpe-1790	90	61	,	,	PUNCT
bcpe-1790	90	62	that	that	ADV
bcpe-1790	90	63	is	is	ADV
bcpe-1790	90	64	,	,	PUNCT
bcpe-1790	90	65	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	90	66	=	=	NOUN
bcpe-1790	90	67	1	1	NUM
bcpe-1790	90	68	2	2	NUM
bcpe-1790	90	69	𝑥𝑥1𝑥𝑥	𝑥𝑥1𝑥𝑥	NOUN
bcpe-1790	90	70	−	−	PROPN
bcpe-1790	90	71	𝑝𝑝1	𝑝𝑝1	PROPN
bcpe-1790	90	72	.	.	PUNCT
bcpe-1790	91	1	similarly	similarly	ADV
bcpe-1790	91	2	,	,	PUNCT
bcpe-1790	91	3	the	the	DET
bcpe-1790	91	4	equation	equation	NOUN
bcpe-1790	91	5	of	of	ADP
bcpe-1790	91	6	the	the	DET
bcpe-1790	91	7	straight	straight	ADJ
bcpe-1790	91	8	line	line	NOUN
bcpe-1790	91	9	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	91	10	can	can	AUX
bcpe-1790	91	11	be	be	AUX
bcpe-1790	91	12	obtained	obtain	VERB
bcpe-1790	91	13	as	as	ADP
bcpe-1790	91	14	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	91	15	=	=	SYM
bcpe-1790	91	16	1	1	NUM
bcpe-1790	91	17	2	2	NUM
bcpe-1790	91	18	𝑥𝑥2𝑥𝑥	𝑥𝑥2𝑥𝑥	NOUN
bcpe-1790	91	19	−	−	PROPN
bcpe-1790	91	20	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	91	21	.	.	PUNCT
bcpe-1790	92	1	∵	∵	ADJ
bcpe-1790	92	2	point	point	NOUN
bcpe-1790	92	3	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	92	4	is	be	AUX
bcpe-1790	92	5	the	the	DET
bcpe-1790	92	6	common	common	ADJ
bcpe-1790	92	7	point	point	NOUN
bcpe-1790	92	8	of	of	ADP
bcpe-1790	92	9	the	the	DET
bcpe-1790	92	10	two	two	NUM
bcpe-1790	92	11	lines	line	NOUN
bcpe-1790	92	12	.	.	PUNCT
bcpe-1790	93	1	there	there	PRON
bcpe-1790	93	2	is	be	VERB
bcpe-1790	93	3	�	�	PROPN
bcpe-1790	93	4	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	93	5	=	=	NOUN
bcpe-1790	93	6	1	1	NUM
bcpe-1790	93	7	2	2	NUM
bcpe-1790	93	8	𝑥𝑥1𝑥𝑥0	𝑥𝑥1𝑥𝑥0	NOUN
bcpe-1790	93	9	−	−	NOUN
bcpe-1790	93	10	𝑝𝑝1	𝑝𝑝1	PROPN
bcpe-1790	93	11	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	94	1	=	=	NOUN
bcpe-1790	94	2	1	1	NUM
bcpe-1790	94	3	2	2	NUM
bcpe-1790	94	4	𝑥𝑥2𝑥𝑥0	𝑥𝑥2𝑥𝑥0	NOUN
bcpe-1790	94	5	−	−	NOUN
bcpe-1790	94	6	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	94	7	.	.	PUNCT
bcpe-1790	95	1	∴	∴	VERB
bcpe-1790	95	2	the	the	DET
bcpe-1790	95	3	coordinates	coordinate	NOUN
bcpe-1790	95	4	of	of	ADP
bcpe-1790	95	5	the	the	DET
bcpe-1790	95	6	points	point	NOUN
bcpe-1790	95	7	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	95	8	and	and	CCONJ
bcpe-1790	95	9	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	95	10	satisfy	satisfy	VERB
bcpe-1790	95	11	the	the	DET
bcpe-1790	95	12	equation	equation	NOUN
bcpe-1790	95	13	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	95	14	=	=	NOUN
bcpe-1790	95	15	1	1	NUM
bcpe-1790	95	16	2	2	NUM
bcpe-1790	95	17	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	95	18	−	−	PROPN
bcpe-1790	95	19	𝑝𝑝.	𝑝𝑝.	PROPN
bcpe-1790	95	20	∴	∴	PROPN
bcpe-1790	95	21	the	the	DET
bcpe-1790	95	22	equation	equation	NOUN
bcpe-1790	95	23	of	of	ADP
bcpe-1790	95	24	the	the	DET
bcpe-1790	95	25	straight	straight	ADJ
bcpe-1790	95	26	line	line	NOUN
bcpe-1790	95	27	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	95	28	is	be	AUX
bcpe-1790	95	29	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	95	30	=	=	SYM
bcpe-1790	95	31	1	1	NUM
bcpe-1790	95	32	2	2	NUM
bcpe-1790	95	33	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	95	34	−	−	NOUN
bcpe-1790	95	35	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	95	36	,	,	PUNCT
bcpe-1790	95	37	that	that	ADV
bcpe-1790	95	38	is	is	ADV
bcpe-1790	95	39	,	,	PUNCT
bcpe-1790	95	40	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	95	41	=	=	NOUN
bcpe-1790	95	42	1	1	NUM
bcpe-1790	95	43	2	2	NUM
bcpe-1790	95	44	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	95	45	−	−	PROPN
bcpe-1790	95	46	𝑝𝑝0	𝑝𝑝0	PROPN
bcpe-1790	95	47	.	.	PROPN
bcpe-1790	95	48	combine	combine	VERB
bcpe-1790	95	49	�	�	PROPN
bcpe-1790	95	50	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	95	51	=	=	SYM
bcpe-1790	95	52	1	1	NUM
bcpe-1790	95	53	2	2	NUM
bcpe-1790	95	54	𝑥𝑥0𝑥𝑥	𝑥𝑥0𝑥𝑥	NOUN
bcpe-1790	95	55	−	−	ADP
bcpe-1790	95	56	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	95	57	𝑝𝑝	𝑝𝑝	NOUN
bcpe-1790	95	58	=	=	NOUN
bcpe-1790	95	59	1	1	NUM
bcpe-1790	95	60	4	4	NUM
bcpe-1790	95	61	𝑥𝑥2	𝑥𝑥2	ADV
bcpe-1790	95	62	,	,	PUNCT
bcpe-1790	95	63	get	get	VERB
bcpe-1790	95	64	𝑥𝑥2	𝑥𝑥2	PRON
bcpe-1790	95	65	−	−	ADP
bcpe-1790	95	66	2𝑥𝑥0𝑥𝑥	2𝑥𝑥0𝑥𝑥	NUM
bcpe-1790	95	67	+	+	NOUN
bcpe-1790	95	68	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	95	69	=	=	SYM
bcpe-1790	95	70	0	0	NUM
bcpe-1790	95	71	,	,	PUNCT
bcpe-1790	95	72	according	accord	VERB
bcpe-1790	95	73	to	to	ADP
bcpe-1790	95	74	veda	veda	PROPN
bcpe-1790	95	75	's	's	PART
bcpe-1790	95	76	theorem	theorem	NOUN
bcpe-1790	95	77	,	,	PUNCT
bcpe-1790	95	78	we	we	PRON
bcpe-1790	95	79	can	can	AUX
bcpe-1790	95	80	get	get	VERB
bcpe-1790	95	81	𝑥𝑥1	𝑥𝑥1	NOUN
bcpe-1790	95	82	+	+	CCONJ
bcpe-1790	95	83	𝑥𝑥2	𝑥𝑥2	NOUN
bcpe-1790	95	84	=	=	SYM
bcpe-1790	95	85	2𝑥𝑥0	2𝑥𝑥0	ADJ
bcpe-1790	95	86	,	,	PUNCT
bcpe-1790	95	87	𝑥𝑥1𝑥𝑥2	𝑥𝑥1𝑥𝑥2	ADV
bcpe-1790	95	88	=	=	SYM
bcpe-1790	95	89	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	95	90	.	.	PUNCT
bcpe-1790	96	1	therefore	therefore	ADV
bcpe-1790	96	2	,	,	PUNCT
bcpe-1790	96	3	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	96	4	=	=	SYM
bcpe-1790	96	5	�	�	PROPN
bcpe-1790	96	6	1	1	NUM
bcpe-1790	96	7	+	+	CCONJ
bcpe-1790	96	8	(	(	PUNCT
bcpe-1790	96	9	1	1	NUM
bcpe-1790	96	10	2	2	NUM
bcpe-1790	96	11	𝑥𝑥0)2	𝑥𝑥0)2	PROPN
bcpe-1790	96	12	�	�	NOUN
bcpe-1790	96	13	(𝑥𝑥1	(𝑥𝑥1	PUNCT
bcpe-1790	96	14	+	+	CCONJ
bcpe-1790	96	15	𝑥𝑥2)2	𝑥𝑥2)2	ADP
bcpe-1790	96	16	−	−	NOUN
bcpe-1790	96	17	4𝑥𝑥1𝑥𝑥2	4𝑥𝑥1𝑥𝑥2	NUM
bcpe-1790	96	18	=	=	SYM
bcpe-1790	96	19	�	�	PROPN
bcpe-1790	96	20	1	1	NUM
bcpe-1790	96	21	+	+	CCONJ
bcpe-1790	96	22	(	(	PUNCT
bcpe-1790	96	23	1	1	NUM
bcpe-1790	96	24	2	2	NUM
bcpe-1790	96	25	𝑥𝑥0)2	𝑥𝑥0)2	PROPN
bcpe-1790	96	26	�	�	PROPN
bcpe-1790	96	27	4𝑥𝑥02	4𝑥𝑥02	NUM
bcpe-1790	96	28	−	−	PROPN
bcpe-1790	96	29	16𝑝𝑝0	16𝑝𝑝0	PROPN
bcpe-1790	96	30	=	=	SYM
bcpe-1790	96	31	�	�	PROPN
bcpe-1790	96	32	4	4	NUM
bcpe-1790	96	33	+	+	CCONJ
bcpe-1790	96	34	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	96	35	�	�	NOUN
bcpe-1790	96	36	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	96	37	−	−	PROPN
bcpe-1790	96	38	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	96	39	.	.	PUNCT
bcpe-1790	97	1	bcp	bcp	PROPN
bcpe-1790	97	2	education	education	PROPN
bcpe-1790	97	3	&	&	CCONJ
bcpe-1790	97	4	psychology	psychology	PROPN
bcpe-1790	97	5	erhss	erhss	PROPN
bcpe-1790	97	6	2022	2022	NUM
bcpe-1790	97	7	volume	volume	NOUN
bcpe-1790	97	8	6	6	NUM
bcpe-1790	97	9	(	(	PUNCT
bcpe-1790	97	10	2022	2022	NUM
bcpe-1790	97	11	)	)	PUNCT
bcpe-1790	97	12	204	204	NUM
bcpe-1790	97	13	the	the	DET
bcpe-1790	97	14	distance	distance	NOUN
bcpe-1790	97	15	from	from	ADP
bcpe-1790	97	16	the	the	DET
bcpe-1790	97	17	𝑑𝑑	𝑑𝑑	PROPN
bcpe-1790	97	18	=	=	SYM
bcpe-1790	97	19	�	�	PROPN
bcpe-1790	97	20	𝑥𝑥02−4𝑦𝑦0	𝑥𝑥02−4𝑦𝑦0	PROPN
bcpe-1790	97	21	�	�	PROPN
bcpe-1790	97	22	�	�	PROPN
bcpe-1790	97	23	4+𝑥𝑥02	4+𝑥𝑥02	PROPN
bcpe-1790	97	24	from	from	ADP
bcpe-1790	97	25	the	the	DET
bcpe-1790	97	26	point	point	NOUN
bcpe-1790	97	27	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	97	28	to	to	ADP
bcpe-1790	97	29	the	the	DET
bcpe-1790	97	30	straight	straight	ADJ
bcpe-1790	97	31	line	line	NOUN
bcpe-1790	97	32	𝑃𝑃𝑃𝑃.	𝑃𝑃𝑃𝑃.	X
bcpe-1790	98	1	∴	∴	NOUN
bcpe-1790	98	2	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	98	3	=	=	SYM
bcpe-1790	98	4	1	1	NUM
bcpe-1790	98	5	2	2	NUM
bcpe-1790	98	6	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	98	7	∙	∙	PROPN
bcpe-1790	98	8	𝑑𝑑	𝑑𝑑	VERB
bcpe-1790	99	1	=	=	SYM
bcpe-1790	99	2	1	1	NUM
bcpe-1790	99	3	2	2	NUM
bcpe-1790	99	4	|𝑥𝑥02	|𝑥𝑥02	NOUN
bcpe-1790	99	5	−	−	NOUN
bcpe-1790	99	6	4𝑝𝑝0|	4𝑝𝑝0|	NOUN
bcpe-1790	99	7	�	�	ADP
bcpe-1790	99	8	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	99	9	−	−	NOUN
bcpe-1790	99	10	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	99	11	=	=	SYM
bcpe-1790	99	12	1	1	NUM
bcpe-1790	99	13	2	2	NUM
bcpe-1790	99	14	(	(	PUNCT
bcpe-1790	99	15	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	99	16	−	−	PROPN
bcpe-1790	99	17	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	99	18	)	)	PUNCT
bcpe-1790	99	19	3	3	NUM
bcpe-1790	99	20	2	2	NUM
bcpe-1790	99	21	.	.	PUNCT
bcpe-1790	99	22	and	and	CCONJ
bcpe-1790	99	23	𝑥𝑥02	𝑥𝑥02	PROPN
bcpe-1790	99	24	−	−	PROPN
bcpe-1790	99	25	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	99	26	=	=	SYM
bcpe-1790	99	27	1	1	NUM
bcpe-1790	99	28	−	−	PROPN
bcpe-1790	99	29	(	(	PUNCT
bcpe-1790	99	30	𝑝𝑝0	𝑝𝑝0	X
bcpe-1790	99	31	+	+	NOUN
bcpe-1790	99	32	4)2	4)2	PROPN
bcpe-1790	99	33	−	−	NOUN
bcpe-1790	99	34	4𝑝𝑝0	4𝑝𝑝0	NUM
bcpe-1790	100	1	=	=	PUNCT
bcpe-1790	100	2	−𝑝𝑝02	−𝑝𝑝02	VERB
bcpe-1790	100	3	−	−	PROPN
bcpe-1790	101	1	12𝑝𝑝0	12𝑝𝑝0	NOUN
bcpe-1790	101	2	−	−	PROPN
bcpe-1790	101	3	15	15	NUM
bcpe-1790	102	1	=	=	NOUN
bcpe-1790	102	2	−(𝑝𝑝0	−(𝑝𝑝0	NOUN
bcpe-1790	102	3	+	+	CCONJ
bcpe-1790	102	4	6)2	6)2	NUM
bcpe-1790	102	5	+	+	NOUN
bcpe-1790	102	6	21	21	NUM
bcpe-1790	102	7	.	.	PUNCT
bcpe-1790	103	1	it	it	PRON
bcpe-1790	103	2	is	be	AUX
bcpe-1790	103	3	known	know	VERB
bcpe-1790	103	4	that	that	SCONJ
bcpe-1790	103	5	−5	−5	ADV
bcpe-1790	103	6	≤	≤	X
bcpe-1790	103	7	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	103	8	≤	≤	PROPN
bcpe-1790	103	9	−3	−3	ADV
bcpe-1790	103	10	,	,	PUNCT
bcpe-1790	103	11	so	so	CCONJ
bcpe-1790	103	12	when	when	SCONJ
bcpe-1790	103	13	𝑝𝑝0	𝑝𝑝0	NOUN
bcpe-1790	103	14	=	=	NOUN
bcpe-1790	103	15	−5	−5	NOUN
bcpe-1790	103	16	,	,	PUNCT
bcpe-1790	103	17	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	103	18	obtains	obtain	VERB
bcpe-1790	103	19	the	the	DET
bcpe-1790	103	20	maximum	maximum	ADJ
bcpe-1790	103	21	value	value	NOUN
bcpe-1790	103	22	of	of	ADP
bcpe-1790	103	23	1	1	NUM
bcpe-1790	103	24	2	2	NUM
bcpe-1790	103	25	×	×	NOUN
bcpe-1790	103	26	20	20	NUM
bcpe-1790	103	27	3	3	NUM
bcpe-1790	103	28	2	2	NUM
bcpe-1790	103	29	=	=	SYM
bcpe-1790	103	30	20√5	20√5	NUM
bcpe-1790	103	31	.	.	PUNCT
bcpe-1790	104	1	(	(	PUNCT
bcpe-1790	104	2	4	4	X
bcpe-1790	104	3	)	)	PUNCT
bcpe-1790	104	4	reviewing	review	VERB
bcpe-1790	104	5	and	and	CCONJ
bcpe-1790	104	6	doubting	doubt	VERB
bcpe-1790	104	7	the	the	DET
bcpe-1790	104	8	teacher	teacher	NOUN
bcpe-1790	104	9	organizes	organize	VERB
bcpe-1790	104	10	students	student	NOUN
bcpe-1790	104	11	to	to	PART
bcpe-1790	104	12	review	review	VERB
bcpe-1790	104	13	the	the	DET
bcpe-1790	104	14	problem	problem	NOUN
bcpe-1790	104	15	-	-	PUNCT
bcpe-1790	104	16	solving	solve	VERB
bcpe-1790	104	17	method	method	NOUN
bcpe-1790	104	18	of	of	ADP
bcpe-1790	104	19	this	this	DET
bcpe-1790	104	20	sub	sub	NOUN
bcpe-1790	104	21	-	-	NOUN
bcpe-1790	104	22	question	question	NOUN
bcpe-1790	104	23	and	and	CCONJ
bcpe-1790	104	24	asks	ask	VERB
bcpe-1790	104	25	s2	s2	VERB
bcpe-1790	104	26	to	to	PART
bcpe-1790	104	27	review	review	VERB
bcpe-1790	104	28	how	how	SCONJ
bcpe-1790	104	29	this	this	DET
bcpe-1790	104	30	problem	problem	NOUN
bcpe-1790	104	31	-	-	PUNCT
bcpe-1790	104	32	solving	solve	VERB
bcpe-1790	104	33	idea	idea	NOUN
bcpe-1790	104	34	was	be	AUX
bcpe-1790	104	35	born	bear	VERB
bcpe-1790	104	36	.	.	PUNCT
bcpe-1790	105	1	s2	s2	PROPN
bcpe-1790	105	2	:	:	PUNCT
bcpe-1790	105	3	i	i	PRON
bcpe-1790	105	4	used	use	VERB
bcpe-1790	105	5	reverse	reverse	NOUN
bcpe-1790	105	6	thinking	think	VERB
bcpe-1790	105	7	to	to	PART
bcpe-1790	105	8	think	think	VERB
bcpe-1790	105	9	about	about	ADP
bcpe-1790	105	10	the	the	DET
bcpe-1790	105	11	problem	problem	NOUN
bcpe-1790	105	12	you	you	PRON
bcpe-1790	105	13	mentioned	mention	VERB
bcpe-1790	105	14	,	,	PUNCT
bcpe-1790	105	15	guessing	guess	VERB
bcpe-1790	105	16	and	and	CCONJ
bcpe-1790	105	17	assuming	assume	VERB
bcpe-1790	105	18	that	that	SCONJ
bcpe-1790	105	19	the	the	DET
bcpe-1790	105	20	expression	expression	NOUN
bcpe-1790	105	21	of	of	ADP
bcpe-1790	105	22	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	105	23	can	can	AUX
bcpe-1790	105	24	be	be	AUX
bcpe-1790	105	25	written	write	VERB
bcpe-1790	105	26	out	out	ADP
bcpe-1790	105	27	,	,	PUNCT
bcpe-1790	105	28	and	and	CCONJ
bcpe-1790	105	29	then	then	ADV
bcpe-1790	105	30	the	the	DET
bcpe-1790	105	31	maximum	maximum	ADJ
bcpe-1790	105	32	value	value	NOUN
bcpe-1790	105	33	can	can	AUX
bcpe-1790	105	34	be	be	AUX
bcpe-1790	105	35	obtained	obtain	VERB
bcpe-1790	105	36	according	accord	VERB
bcpe-1790	105	37	to	to	ADP
bcpe-1790	105	38	the	the	DET
bcpe-1790	105	39	properties	property	NOUN
bcpe-1790	105	40	of	of	ADP
bcpe-1790	105	41	the	the	DET
bcpe-1790	105	42	function	function	NOUN
bcpe-1790	105	43	.	.	PUNCT
bcpe-1790	106	1	and	and	CCONJ
bcpe-1790	106	2	it	it	PRON
bcpe-1790	106	3	is	be	AUX
bcpe-1790	106	4	known	know	VERB
bcpe-1790	106	5	that	that	SCONJ
bcpe-1790	106	6	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	106	7	=	=	SYM
bcpe-1790	106	8	1	1	NUM
bcpe-1790	106	9	2	2	NUM
bcpe-1790	106	10	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	106	11	∙	∙	PROPN
bcpe-1790	106	12	𝑑𝑑	𝑑𝑑	PROPN
bcpe-1790	106	13	,	,	PUNCT
bcpe-1790	106	14	then	then	ADV
bcpe-1790	106	15	the	the	DET
bcpe-1790	106	16	problem	problem	NOUN
bcpe-1790	106	17	direction	direction	NOUN
bcpe-1790	106	18	is	be	AUX
bcpe-1790	106	19	transformed	transform	VERB
bcpe-1790	106	20	into	into	ADP
bcpe-1790	106	21	an	an	DET
bcpe-1790	106	22	expression	expression	NOUN
bcpe-1790	106	23	for	for	ADP
bcpe-1790	106	24	finding	find	VERB
bcpe-1790	106	25	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	106	26	and	and	CCONJ
bcpe-1790	106	27	𝑑𝑑.	𝑑𝑑.	PROPN
bcpe-1790	106	28	t	t	PROPN
bcpe-1790	106	29	:	:	PUNCT
bcpe-1790	106	30	a	a	DET
bcpe-1790	106	31	"	"	PUNCT
bcpe-1790	106	32	good	good	ADJ
bcpe-1790	106	33	thought	thought	NOUN
bcpe-1790	106	34	"	"	PUNCT
bcpe-1790	106	35	produces	produce	VERB
bcpe-1790	106	36	good	good	ADJ
bcpe-1790	106	37	results	result	NOUN
bcpe-1790	106	38	.	.	PUNCT
bcpe-1790	107	1	during	during	ADP
bcpe-1790	107	2	the	the	DET
bcpe-1790	107	3	inspection	inspection	NOUN
bcpe-1790	107	4	,	,	PUNCT
bcpe-1790	107	5	i	i	PRON
bcpe-1790	107	6	saw	see	VERB
bcpe-1790	107	7	that	that	SCONJ
bcpe-1790	107	8	s3	s3	PROPN
bcpe-1790	107	9	used	use	VERB
bcpe-1790	107	10	different	different	ADJ
bcpe-1790	107	11	methods	method	NOUN
bcpe-1790	107	12	and	and	CCONJ
bcpe-1790	107	13	obtained	obtain	VERB
bcpe-1790	107	14	results	result	NOUN
bcpe-1790	107	15	.	.	PUNCT
bcpe-1790	108	1	we	we	PRON
bcpe-1790	108	2	asked	ask	VERB
bcpe-1790	108	3	him	he	PRON
bcpe-1790	108	4	to	to	PART
bcpe-1790	108	5	share	share	VERB
bcpe-1790	108	6	his	his	PRON
bcpe-1790	108	7	ideas	idea	NOUN
bcpe-1790	108	8	.	.	PUNCT
bcpe-1790	109	1	s3	s3	NOUN
bcpe-1790	109	2	:	:	PUNCT
bcpe-1790	109	3	in	in	ADP
bcpe-1790	109	4	daily	daily	ADJ
bcpe-1790	109	5	problem	problem	NOUN
bcpe-1790	109	6	solving	solving	NOUN
bcpe-1790	109	7	,	,	PUNCT
bcpe-1790	109	8	i	i	PRON
bcpe-1790	109	9	am	be	AUX
bcpe-1790	109	10	accustomed	accustomed	ADJ
bcpe-1790	109	11	to	to	PART
bcpe-1790	109	12	directly	directly	ADV
bcpe-1790	109	13	setting	set	VERB
bcpe-1790	109	14	the	the	DET
bcpe-1790	109	15	equation	equation	NOUN
bcpe-1790	109	16	of	of	ADP
bcpe-1790	109	17	the	the	DET
bcpe-1790	109	18	straight	straight	ADJ
bcpe-1790	109	19	line	line	NOUN
bcpe-1790	109	20	𝑃𝑃𝑃𝑃.	𝑃𝑃𝑃𝑃.	NOUN
bcpe-1790	109	21	according	accord	VERB
bcpe-1790	109	22	to	to	ADP
bcpe-1790	109	23	the	the	DET
bcpe-1790	109	24	meaning	meaning	NOUN
bcpe-1790	109	25	of	of	ADP
bcpe-1790	109	26	the	the	DET
bcpe-1790	109	27	question	question	NOUN
bcpe-1790	109	28	,	,	PUNCT
bcpe-1790	109	29	i	i	PRON
bcpe-1790	109	30	know	know	VERB
bcpe-1790	109	31	that	that	SCONJ
bcpe-1790	109	32	the	the	DET
bcpe-1790	109	33	straight	straight	ADJ
bcpe-1790	109	34	line	line	NOUN
bcpe-1790	109	35	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	109	36	and	and	CCONJ
bcpe-1790	109	37	the	the	DET
bcpe-1790	109	38	parabola	parabola	PROPN
bcpe-1790	109	39	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	109	40	must	must	AUX
bcpe-1790	109	41	have	have	VERB
bcpe-1790	109	42	two	two	NUM
bcpe-1790	109	43	intersections	intersection	NOUN
bcpe-1790	109	44	,	,	PUNCT
bcpe-1790	109	45	so	so	ADV
bcpe-1790	109	46	set	set	VERB
bcpe-1790	109	47	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	𝑙𝑙𝑃𝑃𝑃𝑃:𝑝𝑝	X
bcpe-1790	110	1	=	=	PUNCT
bcpe-1790	110	2	𝑘𝑘𝑥𝑥	𝑘𝑘𝑥𝑥	PROPN
bcpe-1790	110	3	+	+	CCONJ
bcpe-1790	110	4	𝑏𝑏	𝑏𝑏	NOUN
bcpe-1790	110	5	,	,	PUNCT
bcpe-1790	110	6	according	accord	VERB
bcpe-1790	110	7	to	to	ADP
bcpe-1790	110	8	the	the	DET
bcpe-1790	110	9	slope	slope	NOUN
bcpe-1790	110	10	of	of	ADP
bcpe-1790	110	11	the	the	DET
bcpe-1790	110	12	straight	straight	ADJ
bcpe-1790	110	13	line	line	NOUN
bcpe-1790	110	14	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	110	15	and	and	CCONJ
bcpe-1790	110	16	the	the	DET
bcpe-1790	110	17	straight	straight	ADJ
bcpe-1790	110	18	line	line	NOUN
bcpe-1790	110	19	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	110	20	can	can	AUX
bcpe-1790	110	21	be	be	AUX
bcpe-1790	110	22	solved	solve	VERB
bcpe-1790	110	23	.	.	PUNCT
bcpe-1790	111	1	the	the	DET
bcpe-1790	111	2	coordinates	coordinate	NOUN
bcpe-1790	111	3	of	of	ADP
bcpe-1790	111	4	point	point	NOUN
bcpe-1790	111	5	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	111	6	is	be	AUX
bcpe-1790	111	7	(	(	PUNCT
bcpe-1790	111	8	𝑥𝑥1+𝑥𝑥2	𝑥𝑥1+𝑥𝑥2	ADP
bcpe-1790	111	9	2	2	NUM
bcpe-1790	111	10	,	,	PUNCT
bcpe-1790	111	11	𝑥𝑥1𝑥𝑥2	𝑥𝑥1𝑥𝑥2	ADV
bcpe-1790	111	12	4	4	NUM
bcpe-1790	111	13	)	)	PUNCT
bcpe-1790	111	14	,	,	PUNCT
bcpe-1790	111	15	and	and	CCONJ
bcpe-1790	111	16	then	then	ADV
bcpe-1790	111	17	the	the	DET
bcpe-1790	111	18	coordinates	coordinate	NOUN
bcpe-1790	111	19	of	of	ADP
bcpe-1790	111	20	point	point	NOUN
bcpe-1790	111	21	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	111	22	can	can	AUX
bcpe-1790	111	23	be	be	AUX
bcpe-1790	111	24	simplified	simplify	VERB
bcpe-1790	111	25	to	to	ADP
bcpe-1790	111	26	(	(	PUNCT
bcpe-1790	111	27	2𝑘𝑘,−𝑏𝑏	2𝑘𝑘,−𝑏𝑏	PROPN
bcpe-1790	111	28	)	)	PUNCT
bcpe-1790	111	29	.	.	PUNCT
bcpe-1790	112	1	simultaneous	simultaneous	ADJ
bcpe-1790	112	2	equations	equation	NOUN
bcpe-1790	112	3	,	,	PUNCT
bcpe-1790	112	4	veda	veda	PROPN
bcpe-1790	112	5	's	's	PART
bcpe-1790	112	6	theorem	theorem	ADJ
bcpe-1790	112	7	,	,	PUNCT
bcpe-1790	112	8	and	and	CCONJ
bcpe-1790	112	9	other	other	ADJ
bcpe-1790	112	10	methods	method	NOUN
bcpe-1790	112	11	are	be	AUX
bcpe-1790	112	12	shown	show	VERB
bcpe-1790	112	13	as	as	SCONJ
bcpe-1790	112	14	follows	follow	VERB
bcpe-1790	112	15	.	.	PUNCT
bcpe-1790	113	1	it	it	PRON
bcpe-1790	113	2	is	be	AUX
bcpe-1790	113	3	similar	similar	ADJ
bcpe-1790	113	4	to	to	ADP
bcpe-1790	113	5	the	the	DET
bcpe-1790	113	6	above	above	ADJ
bcpe-1790	113	7	.	.	PUNCT
bcpe-1790	114	1	the	the	DET
bcpe-1790	114	2	expression	expression	NOUN
bcpe-1790	114	3	of	of	ADP
bcpe-1790	114	4	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	𝑆𝑆∆𝑃𝑃𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	114	5	is	be	AUX
bcpe-1790	114	6	finally	finally	ADV
bcpe-1790	114	7	converted	convert	VERB
bcpe-1790	114	8	into	into	ADP
bcpe-1790	114	9	an	an	DET
bcpe-1790	114	10	expression	expression	NOUN
bcpe-1790	114	11	containing	contain	VERB
bcpe-1790	114	12	𝑏𝑏.	𝑏𝑏.	PROPN
bcpe-1790	114	13	t	t	NOUN
bcpe-1790	114	14	:	:	PUNCT
bcpe-1790	114	15	very	very	ADV
bcpe-1790	114	16	good	good	ADJ
bcpe-1790	114	17	!	!	PUNCT
bcpe-1790	115	1	the	the	DET
bcpe-1790	115	2	above	above	ADJ
bcpe-1790	115	3	two	two	NUM
bcpe-1790	115	4	students	student	NOUN
bcpe-1790	115	5	have	have	AUX
bcpe-1790	115	6	made	make	VERB
bcpe-1790	115	7	a	a	DET
bcpe-1790	115	8	brief	brief	ADJ
bcpe-1790	115	9	exposition	exposition	NOUN
bcpe-1790	115	10	of	of	ADP
bcpe-1790	115	11	their	their	PRON
bcpe-1790	115	12	problem	problem	NOUN
bcpe-1790	115	13	-	-	PUNCT
bcpe-1790	115	14	solving	solve	VERB
bcpe-1790	115	15	ideas	idea	NOUN
bcpe-1790	115	16	,	,	PUNCT
bcpe-1790	115	17	i	i	PRON
bcpe-1790	115	18	believe	believe	VERB
bcpe-1790	115	19	some	some	PRON
bcpe-1790	115	20	should	should	AUX
bcpe-1790	115	21	inspire	inspire	VERB
bcpe-1790	115	22	everyone	everyone	PRON
bcpe-1790	115	23	.	.	PUNCT
bcpe-1790	116	1	what	what	PRON
bcpe-1790	116	2	do	do	AUX
bcpe-1790	116	3	you	you	PRON
bcpe-1790	116	4	gain	gain	VERB
bcpe-1790	116	5	from	from	ADP
bcpe-1790	116	6	solving	solve	VERB
bcpe-1790	116	7	this	this	DET
bcpe-1790	116	8	problem	problem	NOUN
bcpe-1790	116	9	?	?	PUNCT
bcpe-1790	117	1	s4	s4	PROPN
bcpe-1790	117	2	:	:	PUNCT
bcpe-1790	118	1	i	i	PRON
bcpe-1790	118	2	feel	feel	VERB
bcpe-1790	118	3	that	that	SCONJ
bcpe-1790	118	4	reverse	reverse	ADJ
bcpe-1790	118	5	thinking	thinking	NOUN
bcpe-1790	118	6	is	be	AUX
bcpe-1790	118	7	a	a	DET
bcpe-1790	118	8	common	common	ADJ
bcpe-1790	118	9	way	way	NOUN
bcpe-1790	118	10	of	of	ADP
bcpe-1790	118	11	thinking	thinking	NOUN
bcpe-1790	118	12	in	in	ADP
bcpe-1790	118	13	problem	problem	NOUN
bcpe-1790	118	14	-	-	PUNCT
bcpe-1790	118	15	solving	solving	NOUN
bcpe-1790	118	16	.	.	PUNCT
bcpe-1790	119	1	this	this	DET
bcpe-1790	119	2	question	question	NOUN
bcpe-1790	119	3	is	be	AUX
bcpe-1790	119	4	not	not	PART
bcpe-1790	119	5	limited	limit	VERB
bcpe-1790	119	6	to	to	ADP
bcpe-1790	119	7	looking	look	VERB
bcpe-1790	119	8	for	for	ADP
bcpe-1790	119	9	a	a	DET
bcpe-1790	119	10	"	"	PUNCT
bcpe-1790	119	11	bridge	bridge	NOUN
bcpe-1790	119	12	"	"	PUNCT
bcpe-1790	119	13	between	between	ADP
bcpe-1790	119	14	conditions	condition	NOUN
bcpe-1790	119	15	and	and	CCONJ
bcpe-1790	119	16	problems	problem	NOUN
bcpe-1790	119	17	but	but	CCONJ
bcpe-1790	119	18	inversely	inversely	ADV
bcpe-1790	119	19	looking	look	VERB
bcpe-1790	119	20	for	for	ADP
bcpe-1790	119	21	a	a	DET
bcpe-1790	119	22	"	"	PUNCT
bcpe-1790	119	23	bridge	bridge	NOUN
bcpe-1790	119	24	"	"	PUNCT
bcpe-1790	119	25	between	between	ADP
bcpe-1790	119	26	problems	problem	NOUN
bcpe-1790	119	27	and	and	CCONJ
bcpe-1790	119	28	conditions	condition	NOUN
bcpe-1790	119	29	,	,	PUNCT
bcpe-1790	119	30	which	which	PRON
bcpe-1790	119	31	is	be	AUX
bcpe-1790	119	32	the	the	DET
bcpe-1790	119	33	breakthrough	breakthrough	ADJ
bcpe-1790	119	34	point	point	NOUN
bcpe-1790	119	35	of	of	ADP
bcpe-1790	119	36	this	this	DET
bcpe-1790	119	37	problem	problem	NOUN
bcpe-1790	119	38	.	.	PUNCT
bcpe-1790	120	1	s5	s5	NOUN
bcpe-1790	120	2	:	:	PUNCT
bcpe-1790	120	3	i	i	PRON
bcpe-1790	120	4	feel	feel	VERB
bcpe-1790	120	5	that	that	SCONJ
bcpe-1790	120	6	it	it	PRON
bcpe-1790	120	7	is	be	AUX
bcpe-1790	120	8	essential	essential	ADJ
bcpe-1790	120	9	to	to	PART
bcpe-1790	120	10	transform	transform	VERB
bcpe-1790	120	11	your	your	PRON
bcpe-1790	120	12	mind	mind	NOUN
bcpe-1790	120	13	.	.	PUNCT
bcpe-1790	121	1	this	this	DET
bcpe-1790	121	2	problem	problem	NOUN
bcpe-1790	121	3	transforms	transform	VERB
bcpe-1790	121	4	the	the	DET
bcpe-1790	121	5	problem	problem	NOUN
bcpe-1790	121	6	of	of	ADP
bcpe-1790	121	7	the	the	DET
bcpe-1790	121	8	maximum	maximum	ADJ
bcpe-1790	121	9	value	value	NOUN
bcpe-1790	121	10	of	of	ADP
bcpe-1790	121	11	the	the	DET
bcpe-1790	121	12	∆𝑃𝑃𝑃𝑃𝑃𝑃	∆𝑃𝑃𝑃𝑃𝑃𝑃	PROPN
bcpe-1790	121	13	area	area	NOUN
bcpe-1790	121	14	into	into	ADP
bcpe-1790	121	15	a	a	DET
bcpe-1790	121	16	function	function	NOUN
bcpe-1790	121	17	to	to	PART
bcpe-1790	121	18	find	find	VERB
bcpe-1790	121	19	the	the	DET
bcpe-1790	121	20	maximum	maximum	ADJ
bcpe-1790	121	21	value	value	NOUN
bcpe-1790	121	22	and	and	CCONJ
bcpe-1790	121	23	converts	convert	VERB
bcpe-1790	121	24	it	it	PRON
bcpe-1790	121	25	into	into	ADP
bcpe-1790	121	26	the	the	DET
bcpe-1790	121	27	maximum	maximum	ADJ
bcpe-1790	121	28	value	value	NOUN
bcpe-1790	121	29	of	of	ADP
bcpe-1790	121	30	the	the	DET
bcpe-1790	121	31	expression	expression	NOUN
bcpe-1790	121	32	to	to	PART
bcpe-1790	121	33	find	find	VERB
bcpe-1790	121	34	the	the	DET
bcpe-1790	121	35	∆𝑃𝑃𝑃𝑃𝑃𝑃	∆𝑃𝑃𝑃𝑃𝑃𝑃	NUM
bcpe-1790	121	36	area	area	NOUN
bcpe-1790	121	37	.	.	PUNCT
bcpe-1790	122	1	continuous	continuous	ADJ
bcpe-1790	122	2	transformation	transformation	NOUN
bcpe-1790	122	3	is	be	AUX
bcpe-1790	122	4	the	the	DET
bcpe-1790	122	5	key	key	NOUN
bcpe-1790	122	6	to	to	ADP
bcpe-1790	122	7	the	the	DET
bcpe-1790	122	8	success	success	NOUN
bcpe-1790	122	9	of	of	ADP
bcpe-1790	122	10	the	the	DET
bcpe-1790	122	11	problem	problem	NOUN
bcpe-1790	122	12	.	.	PUNCT
bcpe-1790	123	1	(	(	PUNCT
bcpe-1790	123	2	5	5	X
bcpe-1790	123	3	)	)	PUNCT
bcpe-1790	123	4	reflecting	reflect	VERB
bcpe-1790	123	5	and	and	CCONJ
bcpe-1790	123	6	doubting	doubt	VERB
bcpe-1790	123	7	t	t	PROPN
bcpe-1790	123	8	:	:	PUNCT
bcpe-1790	123	9	please	please	INTJ
bcpe-1790	123	10	review	review	VERB
bcpe-1790	123	11	the	the	DET
bcpe-1790	123	12	problem	problem	NOUN
bcpe-1790	123	13	-	-	PUNCT
bcpe-1790	123	14	solving	solve	VERB
bcpe-1790	123	15	process	process	NOUN
bcpe-1790	123	16	again	again	ADV
bcpe-1790	123	17	after	after	ADP
bcpe-1790	123	18	completing	complete	VERB
bcpe-1790	123	19	the	the	DET
bcpe-1790	123	20	problem	problem	NOUN
bcpe-1790	123	21	-	-	PUNCT
bcpe-1790	123	22	solving	solve	VERB
bcpe-1790	123	23	process	process	NOUN
bcpe-1790	123	24	.	.	PUNCT
bcpe-1790	124	1	maybe	maybe	ADV
bcpe-1790	124	2	you	you	PRON
bcpe-1790	124	3	will	will	AUX
bcpe-1790	124	4	generate	generate	VERB
bcpe-1790	124	5	some	some	DET
bcpe-1790	124	6	new	new	ADJ
bcpe-1790	124	7	ideas	idea	NOUN
bcpe-1790	124	8	and	and	CCONJ
bcpe-1790	124	9	gain	gain	VERB
bcpe-1790	124	10	some	some	DET
bcpe-1790	124	11	unexpected	unexpected	ADJ
bcpe-1790	124	12	discoveries	discovery	NOUN
bcpe-1790	124	13	.	.	PUNCT
bcpe-1790	125	1	s6	s6	PROPN
bcpe-1790	125	2	:	:	PUNCT
bcpe-1790	125	3	(	(	PUNCT
bcpe-1790	125	4	raises	raise	VERB
bcpe-1790	125	5	his	his	PRON
bcpe-1790	125	6	hand	hand	NOUN
bcpe-1790	125	7	)	)	PUNCT
bcpe-1790	125	8	in	in	ADP
bcpe-1790	125	9	this	this	DET
bcpe-1790	125	10	question	question	NOUN
bcpe-1790	125	11	,	,	PUNCT
bcpe-1790	125	12	the	the	DET
bcpe-1790	125	13	focus	focus	NOUN
bcpe-1790	125	14	of	of	ADP
bcpe-1790	125	15	the	the	DET
bcpe-1790	125	16	parabola	parabola	NOUN
bcpe-1790	125	17	and	and	CCONJ
bcpe-1790	125	18	the	the	DET
bcpe-1790	125	19	center	center	NOUN
bcpe-1790	125	20	of	of	ADP
bcpe-1790	125	21	the	the	DET
bcpe-1790	125	22	circle	circle	NOUN
bcpe-1790	125	23	are	be	AUX
bcpe-1790	125	24	both	both	PRON
bcpe-1790	125	25	on	on	ADP
bcpe-1790	125	26	the	the	DET
bcpe-1790	125	27	𝑝𝑝axis	𝑝𝑝axis	NOUN
bcpe-1790	125	28	.	.	PUNCT
bcpe-1790	126	1	if	if	SCONJ
bcpe-1790	126	2	they	they	PRON
bcpe-1790	126	3	are	be	AUX
bcpe-1790	126	4	replaced	replace	VERB
bcpe-1790	126	5	by	by	ADP
bcpe-1790	126	6	both	both	PRON
bcpe-1790	126	7	on	on	ADP
bcpe-1790	126	8	the	the	DET
bcpe-1790	126	9	𝑥𝑥axis	𝑥𝑥axis	NOUN
bcpe-1790	126	10	,	,	PUNCT
bcpe-1790	126	11	the	the	DET
bcpe-1790	126	12	method	method	NOUN
bcpe-1790	126	13	must	must	AUX
bcpe-1790	126	14	be	be	AUX
bcpe-1790	126	15	the	the	DET
bcpe-1790	126	16	same	same	ADJ
bcpe-1790	126	17	.	.	PUNCT
bcpe-1790	127	1	t	t	NOUN
bcpe-1790	127	2	:	:	PUNCT
bcpe-1790	127	3	if	if	SCONJ
bcpe-1790	127	4	you	you	PRON
bcpe-1790	127	5	change	change	VERB
bcpe-1790	127	6	it	it	PRON
bcpe-1790	127	7	according	accord	VERB
bcpe-1790	127	8	to	to	ADP
bcpe-1790	127	9	the	the	DET
bcpe-1790	127	10	method	method	NOUN
bcpe-1790	127	11	you	you	PRON
bcpe-1790	127	12	said	say	VERB
bcpe-1790	127	13	,	,	PUNCT
bcpe-1790	127	14	can	can	AUX
bcpe-1790	127	15	a	a	DET
bcpe-1790	127	16	new	new	ADJ
bcpe-1790	127	17	topic	topic	NOUN
bcpe-1790	127	18	be	be	AUX
bcpe-1790	127	19	adapted	adapt	VERB
bcpe-1790	127	20	so	so	SCONJ
bcpe-1790	127	21	that	that	SCONJ
bcpe-1790	127	22	the	the	DET
bcpe-1790	127	23	final	final	ADJ
bcpe-1790	127	24	result	result	NOUN
bcpe-1790	127	25	remains	remain	VERB
bcpe-1790	127	26	the	the	DET
bcpe-1790	127	27	same	same	ADJ
bcpe-1790	127	28	?	?	PUNCT
bcpe-1790	128	1	s6	s6	PROPN
bcpe-1790	128	2	:	:	PUNCT
bcpe-1790	128	3	bcp	bcp	PROPN
bcpe-1790	128	4	education	education	PROPN
bcpe-1790	128	5	&	&	CCONJ
bcpe-1790	128	6	psychology	psychology	PROPN
bcpe-1790	128	7	erhss	erhss	PROPN
bcpe-1790	128	8	2022	2022	NUM
bcpe-1790	128	9	volume	volume	NOUN
bcpe-1790	128	10	6	6	NUM
bcpe-1790	128	11	(	(	PUNCT
bcpe-1790	128	12	2022	2022	NUM
bcpe-1790	128	13	)	)	PUNCT
bcpe-1790	128	14	205	205	NUM
bcpe-1790	128	15	t	t	NOUN
bcpe-1790	128	16	:	:	PUNCT
bcpe-1790	128	17	very	very	ADV
bcpe-1790	128	18	good	good	ADJ
bcpe-1790	128	19	!	!	PUNCT
bcpe-1790	129	1	did	do	AUX
bcpe-1790	129	2	any	any	PRON
bcpe-1790	129	3	of	of	ADP
bcpe-1790	129	4	the	the	DET
bcpe-1790	129	5	other	other	ADJ
bcpe-1790	129	6	students	student	NOUN
bcpe-1790	129	7	have	have	VERB
bcpe-1790	129	8	any	any	DET
bcpe-1790	129	9	surprises	surprise	NOUN
bcpe-1790	129	10	?	?	PUNCT
bcpe-1790	130	1	try	try	VERB
bcpe-1790	130	2	to	to	PART
bcpe-1790	130	3	adapt	adapt	VERB
bcpe-1790	130	4	a	a	DET
bcpe-1790	130	5	question	question	NOUN
bcpe-1790	130	6	.	.	PUNCT
bcpe-1790	131	1	s7	s7	NOUN
bcpe-1790	131	2	:	:	PUNCT
bcpe-1790	131	3	replacing	replace	VERB
bcpe-1790	131	4	the	the	DET
bcpe-1790	131	5	circle	circle	NOUN
bcpe-1790	131	6	𝑀𝑀with	𝑀𝑀with	PROPN
bcpe-1790	131	7	a	a	DET
bcpe-1790	131	8	straight	straight	ADJ
bcpe-1790	131	9	line	line	NOUN
bcpe-1790	131	10	can	can	AUX
bcpe-1790	131	11	also	also	ADV
bcpe-1790	131	12	be	be	AUX
bcpe-1790	131	13	solved	solve	VERB
bcpe-1790	131	14	by	by	ADP
bcpe-1790	131	15	using	use	VERB
bcpe-1790	131	16	the	the	DET
bcpe-1790	131	17	method	method	NOUN
bcpe-1790	131	18	of	of	ADP
bcpe-1790	131	19	this	this	DET
bcpe-1790	131	20	problem	problem	NOUN
bcpe-1790	131	21	.	.	PUNCT
bcpe-1790	132	1	t	t	PROPN
bcpe-1790	132	2	:	:	PUNCT
bcpe-1790	132	3	have	have	AUX
bcpe-1790	132	4	you	you	PRON
bcpe-1790	132	5	ever	ever	ADV
bcpe-1790	132	6	thought	think	VERB
bcpe-1790	132	7	about	about	ADP
bcpe-1790	132	8	extending	extend	VERB
bcpe-1790	132	9	and	and	CCONJ
bcpe-1790	132	10	promoting	promote	VERB
bcpe-1790	132	11	the	the	DET
bcpe-1790	132	12	method	method	NOUN
bcpe-1790	132	13	of	of	ADP
bcpe-1790	132	14	solving	solve	VERB
bcpe-1790	132	15	this	this	DET
bcpe-1790	132	16	problem	problem	NOUN
bcpe-1790	132	17	?	?	PUNCT
bcpe-1790	133	1	how	how	SCONJ
bcpe-1790	133	2	did	do	AUX
bcpe-1790	133	3	this	this	DET
bcpe-1790	133	4	idea	idea	NOUN
bcpe-1790	133	5	come	come	VERB
bcpe-1790	133	6	about	about	ADP
bcpe-1790	133	7	?	?	PUNCT
bcpe-1790	134	1	s8	s8	NOUN
bcpe-1790	134	2	:	:	PUNCT
bcpe-1790	134	3	i	i	PRON
bcpe-1790	134	4	am	be	AUX
bcpe-1790	134	5	thinking	think	VERB
bcpe-1790	134	6	,	,	PUNCT
bcpe-1790	134	7	is	be	AUX
bcpe-1790	134	8	the	the	DET
bcpe-1790	134	9	solution	solution	NOUN
bcpe-1790	134	10	of	of	ADP
bcpe-1790	134	11	the	the	DET
bcpe-1790	134	12	area	area	NOUN
bcpe-1790	134	13	maximum	maximum	PROPN
bcpe-1790	134	14	value	value	NOUN
bcpe-1790	134	15	problem	problem	NOUN
bcpe-1790	134	16	in	in	ADP
bcpe-1790	134	17	the	the	DET
bcpe-1790	134	18	conic	conic	ADJ
bcpe-1790	134	19	section	section	NOUN
bcpe-1790	134	20	transformed	transform	VERB
bcpe-1790	134	21	into	into	ADP
bcpe-1790	134	22	the	the	DET
bcpe-1790	134	23	function	function	NOUN
bcpe-1790	134	24	maximum	maximum	ADJ
bcpe-1790	134	25	value	value	NOUN
bcpe-1790	134	26	problem	problem	NOUN
bcpe-1790	134	27	?	?	PUNCT
bcpe-1790	135	1	in	in	ADP
bcpe-1790	135	2	the	the	DET
bcpe-1790	135	3	past	past	ADJ
bcpe-1790	135	4	problems	problem	NOUN
bcpe-1790	135	5	of	of	ADP
bcpe-1790	135	6	conic	conic	ADJ
bcpe-1790	135	7	sections	section	NOUN
bcpe-1790	135	8	,	,	PUNCT
bcpe-1790	135	9	i	i	PRON
bcpe-1790	135	10	found	find	VERB
bcpe-1790	135	11	that	that	SCONJ
bcpe-1790	135	12	the	the	DET
bcpe-1790	135	13	area	area	NOUN
bcpe-1790	135	14	formula	formula	NOUN
bcpe-1790	135	15	was	be	AUX
bcpe-1790	135	16	often	often	ADV
bcpe-1790	135	17	written	write	VERB
bcpe-1790	135	18	𝑆𝑆	𝑆𝑆	PROPN
bcpe-1790	135	19	=	=	SYM
bcpe-1790	135	20	1	1	NUM
bcpe-1790	135	21	2	2	NUM
bcpe-1790	135	22	|𝑃𝑃𝑃𝑃|	|𝑃𝑃𝑃𝑃|	PROPN
bcpe-1790	135	23	∙	∙	PROPN
bcpe-1790	135	24	𝑑𝑑	𝑑𝑑	PROPN
bcpe-1790	135	25	,	,	PUNCT
bcpe-1790	135	26	and	and	CCONJ
bcpe-1790	135	27	then	then	ADV
bcpe-1790	135	28	the	the	DET
bcpe-1790	135	29	most	most	ADJ
bcpe-1790	135	30	value	value	NOUN
bcpe-1790	135	31	was	be	AUX
bcpe-1790	135	32	calculated	calculate	VERB
bcpe-1790	135	33	based	base	VERB
bcpe-1790	135	34	on	on	ADP
bcpe-1790	135	35	the	the	DET
bcpe-1790	135	36	knowledge	knowledge	NOUN
bcpe-1790	135	37	of	of	ADP
bcpe-1790	135	38	the	the	DET
bcpe-1790	135	39	function	function	NOUN
bcpe-1790	135	40	.	.	PUNCT
bcpe-1790	136	1	t	t	PROPN
bcpe-1790	136	2	:	:	PUNCT
bcpe-1790	136	3	that	that	PRON
bcpe-1790	136	4	is	be	AUX
bcpe-1790	136	5	right	right	ADJ
bcpe-1790	136	6	!	!	PUNCT
bcpe-1790	137	1	in	in	ADP
bcpe-1790	137	2	other	other	ADJ
bcpe-1790	137	3	words	word	NOUN
bcpe-1790	137	4	,	,	PUNCT
bcpe-1790	137	5	it	it	PRON
bcpe-1790	137	6	is	be	AUX
bcpe-1790	137	7	aimed	aim	VERB
bcpe-1790	137	8	at	at	ADP
bcpe-1790	137	9	solving	solve	VERB
bcpe-1790	137	10	the	the	DET
bcpe-1790	137	11	problem	problem	NOUN
bcpe-1790	137	12	of	of	ADP
bcpe-1790	137	13	the	the	DET
bcpe-1790	137	14	maximum	maximum	ADJ
bcpe-1790	137	15	value	value	NOUN
bcpe-1790	137	16	of	of	ADP
bcpe-1790	137	17	the	the	DET
bcpe-1790	137	18	area	area	NOUN
bcpe-1790	137	19	in	in	ADP
bcpe-1790	137	20	the	the	DET
bcpe-1790	137	21	conic	conic	ADJ
bcpe-1790	137	22	curve	curve	NOUN
bcpe-1790	137	23	,	,	PUNCT
bcpe-1790	137	24	which	which	PRON
bcpe-1790	137	25	is	be	AUX
bcpe-1790	137	26	the	the	DET
bcpe-1790	137	27	problem	problem	NOUN
bcpe-1790	137	28	of	of	ADP
bcpe-1790	137	29	"	"	PUNCT
bcpe-1790	137	30	the	the	DET
bcpe-1790	137	31	maximum	maximum	ADJ
bcpe-1790	137	32	value	value	NOUN
bcpe-1790	137	33	of	of	ADP
bcpe-1790	137	34	the	the	DET
bcpe-1790	137	35	function	function	NOUN
bcpe-1790	137	36	.	.	PUNCT
bcpe-1790	137	37	"	"	PUNCT
bcpe-1790	138	1	this	this	PRON
bcpe-1790	138	2	will	will	AUX
bcpe-1790	138	3	provide	provide	VERB
bcpe-1790	138	4	a	a	DET
bcpe-1790	138	5	solution	solution	NOUN
bcpe-1790	138	6	path	path	NOUN
bcpe-1790	138	7	for	for	ADP
bcpe-1790	138	8	subsequent	subsequent	ADJ
bcpe-1790	138	9	model	model	NOUN
bcpe-1790	138	10	-	-	PUNCT
bcpe-1790	138	11	like	like	ADJ
bcpe-1790	138	12	problems	problem	NOUN
bcpe-1790	138	13	.	.	PUNCT
bcpe-1790	139	1	4	4	X
bcpe-1790	139	2	.	.	X
bcpe-1790	139	3	summary	summary	NOUN
bcpe-1790	139	4	and	and	CCONJ
bcpe-1790	139	5	outlook	outlook	VERB
bcpe-1790	139	6	this	this	DET
bcpe-1790	139	7	paper	paper	NOUN
bcpe-1790	139	8	explores	explore	VERB
bcpe-1790	139	9	the	the	DET
bcpe-1790	139	10	problems	problem	NOUN
bcpe-1790	139	11	existing	exist	VERB
bcpe-1790	139	12	in	in	ADP
bcpe-1790	139	13	mathematics	mathematics	NOUN
bcpe-1790	139	14	teaching	teaching	NOUN
bcpe-1790	139	15	thinking	thinking	NOUN
bcpe-1790	139	16	and	and	CCONJ
bcpe-1790	139	17	the	the	DET
bcpe-1790	139	18	current	current	ADJ
bcpe-1790	139	19	problem	problem	NOUN
bcpe-1790	139	20	-	-	PUNCT
bcpe-1790	139	21	solving	solve	VERB
bcpe-1790	139	22	teaching	teaching	NOUN
bcpe-1790	139	23	using	use	VERB
bcpe-1790	139	24	polya	polya	PROPN
bcpe-1790	139	25	's	's	PART
bcpe-1790	139	26	problem	problem	NOUN
bcpe-1790	139	27	-	-	PUNCT
bcpe-1790	139	28	solving	solve	VERB
bcpe-1790	139	29	thought	thought	NOUN
bcpe-1790	139	30	and	and	CCONJ
bcpe-1790	139	31	puts	put	VERB
bcpe-1790	139	32	forward	forward	ADV
bcpe-1790	139	33	the	the	DET
bcpe-1790	139	34	innovative	innovative	ADJ
bcpe-1790	139	35	thinking	thinking	NOUN
bcpe-1790	139	36	of	of	ADP
bcpe-1790	139	37	"	"	PUNCT
bcpe-1790	139	38	whether	whether	SCONJ
bcpe-1790	139	39	it	it	PRON
bcpe-1790	139	40	is	be	AUX
bcpe-1790	139	41	possible	possible	ADJ
bcpe-1790	139	42	to	to	PART
bcpe-1790	139	43	're	're	AUX
bcpe-1790	139	44	-	-	PUNCT
bcpe-1790	139	45	create	create	VERB
bcpe-1790	139	46	'	'	PUNCT
bcpe-1790	139	47	the	the	DET
bcpe-1790	139	48	'	'	NUM
bcpe-1790	139	49	four	four	NUM
bcpe-1790	139	50	steps	step	NOUN
bcpe-1790	139	51	of	of	ADP
bcpe-1790	139	52	problem	problem	NOUN
bcpe-1790	139	53	-	-	PUNCT
bcpe-1790	139	54	solving	solving	NOUN
bcpe-1790	139	55	.	.	PUNCT
bcpe-1790	139	56	"	"	PUNCT
bcpe-1790	140	1	on	on	ADP
bcpe-1790	140	2	the	the	DET
bcpe-1790	140	3	original	original	ADJ
bcpe-1790	140	4	basis	basis	NOUN
bcpe-1790	140	5	,	,	PUNCT
bcpe-1790	140	6	the	the	DET
bcpe-1790	140	7	step	step	NOUN
bcpe-1790	140	8	of	of	ADP
bcpe-1790	140	9	"	"	PUNCT
bcpe-1790	140	10	looking	look	VERB
bcpe-1790	140	11	back	back	ADV
bcpe-1790	140	12	"	"	PUNCT
bcpe-1790	140	13	is	be	AUX
bcpe-1790	140	14	innovatively	innovatively	ADV
bcpe-1790	140	15	decomposed	decompose	VERB
bcpe-1790	140	16	into	into	ADP
bcpe-1790	140	17	two	two	NUM
bcpe-1790	140	18	steps	step	NOUN
bcpe-1790	140	19	of	of	ADP
bcpe-1790	140	20	"	"	PUNCT
bcpe-1790	140	21	reviewing	review	VERB
bcpe-1790	140	22	and	and	CCONJ
bcpe-1790	140	23	doubting	doubt	VERB
bcpe-1790	140	24	"	"	PUNCT
bcpe-1790	140	25	and	and	CCONJ
bcpe-1790	140	26	"	"	PUNCT
bcpe-1790	140	27	reflecting	reflect	VERB
bcpe-1790	140	28	and	and	CCONJ
bcpe-1790	140	29	doubting	doubt	VERB
bcpe-1790	140	30	,	,	PUNCT
bcpe-1790	140	31	"	"	PUNCT
bcpe-1790	140	32	and	and	CCONJ
bcpe-1790	140	33	the	the	DET
bcpe-1790	140	34	names	name	NOUN
bcpe-1790	140	35	of	of	ADP
bcpe-1790	140	36	the	the	DET
bcpe-1790	140	37	remaining	remain	VERB
bcpe-1790	140	38	three	three	NUM
bcpe-1790	140	39	steps	step	NOUN
bcpe-1790	140	40	are	be	AUX
bcpe-1790	140	41	changed	change	VERB
bcpe-1790	140	42	.	.	PUNCT
bcpe-1790	141	1	finally	finally	ADV
bcpe-1790	141	2	,	,	PUNCT
bcpe-1790	141	3	taking	take	VERB
bcpe-1790	141	4	the	the	DET
bcpe-1790	141	5	topic	topic	NOUN
bcpe-1790	141	6	of	of	ADP
bcpe-1790	141	7	the	the	DET
bcpe-1790	141	8	conic	conic	ADJ
bcpe-1790	141	9	section	section	NOUN
bcpe-1790	141	10	of	of	ADP
bcpe-1790	141	11	the	the	DET
bcpe-1790	141	12	college	college	NOUN
bcpe-1790	141	13	entrance	entrance	NOUN
bcpe-1790	141	14	examination	examination	NOUN
bcpe-1790	141	15	as	as	ADP
bcpe-1790	141	16	an	an	DET
bcpe-1790	141	17	example	example	NOUN
bcpe-1790	141	18	,	,	PUNCT
bcpe-1790	141	19	this	this	DET
bcpe-1790	141	20	paper	paper	NOUN
bcpe-1790	141	21	discusses	discuss	VERB
bcpe-1790	141	22	the	the	DET
bcpe-1790	141	23	innovative	innovative	ADJ
bcpe-1790	141	24	application	application	NOUN
bcpe-1790	141	25	of	of	ADP
bcpe-1790	141	26	polya	polya	NOUN
bcpe-1790	141	27	's	's	PART
bcpe-1790	141	28	problem	problem	NOUN
bcpe-1790	141	29	-	-	PUNCT
bcpe-1790	141	30	solving	solve	VERB
bcpe-1790	141	31	thought	thought	NOUN
bcpe-1790	141	32	in	in	ADP
bcpe-1790	141	33	mathematical	mathematical	ADJ
bcpe-1790	141	34	problem	problem	NOUN
bcpe-1790	141	35	-	-	PUNCT
bcpe-1790	141	36	solving	solving	NOUN
bcpe-1790	141	37	,	,	PUNCT
bcpe-1790	141	38	which	which	PRON
bcpe-1790	141	39	strengthens	strengthen	VERB
bcpe-1790	141	40	the	the	DET
bcpe-1790	141	41	process	process	NOUN
bcpe-1790	141	42	of	of	ADP
bcpe-1790	141	43	students	student	NOUN
bcpe-1790	141	44	inferring	infer	VERB
bcpe-1790	141	45	others	other	NOUN
bcpe-1790	141	46	from	from	ADP
bcpe-1790	141	47	one	one	NUM
bcpe-1790	141	48	case	case	NOUN
bcpe-1790	141	49	.	.	PUNCT
bcpe-1790	142	1	it	it	PRON
bcpe-1790	142	2	provides	provide	VERB
bcpe-1790	142	3	a	a	DET
bcpe-1790	142	4	method	method	NOUN
bcpe-1790	142	5	reference	reference	NOUN
bcpe-1790	142	6	for	for	ADP
bcpe-1790	142	7	optimizing	optimize	VERB
bcpe-1790	142	8	high	high	ADJ
bcpe-1790	142	9	school	school	NOUN
bcpe-1790	142	10	mathematics	mathematic	NOUN
bcpe-1790	142	11	teachers	teacher	NOUN
bcpe-1790	142	12	'	'	PART
bcpe-1790	142	13	teaching	teaching	NOUN
bcpe-1790	142	14	thinking	thinking	NOUN
bcpe-1790	142	15	.	.	PUNCT
bcpe-1790	143	1	polya	polya	PROPN
bcpe-1790	143	2	's	's	PART
bcpe-1790	143	3	problem	problem	NOUN
bcpe-1790	143	4	-	-	PUNCT
bcpe-1790	143	5	solving	solve	VERB
bcpe-1790	143	6	thought	thought	NOUN
bcpe-1790	143	7	provides	provide	VERB
bcpe-1790	143	8	a	a	DET
bcpe-1790	143	9	method	method	NOUN
bcpe-1790	143	10	for	for	SCONJ
bcpe-1790	143	11	students	student	NOUN
bcpe-1790	143	12	to	to	PART
bcpe-1790	143	13	do	do	VERB
bcpe-1790	143	14	problems	problem	NOUN
bcpe-1790	143	15	and	and	CCONJ
bcpe-1790	143	16	for	for	SCONJ
bcpe-1790	143	17	teachers	teacher	NOUN
bcpe-1790	143	18	to	to	PART
bcpe-1790	143	19	give	give	VERB
bcpe-1790	143	20	lectures	lecture	NOUN
bcpe-1790	143	21	.	.	PUNCT
bcpe-1790	144	1	whether	whether	SCONJ
bcpe-1790	144	2	it	it	PRON
bcpe-1790	144	3	is	be	AUX
bcpe-1790	144	4	past	past	ADJ
bcpe-1790	144	5	,	,	PUNCT
bcpe-1790	144	6	present	present	ADJ
bcpe-1790	144	7	,	,	PUNCT
bcpe-1790	144	8	or	or	CCONJ
bcpe-1790	144	9	future	future	NOUN
bcpe-1790	144	10	,	,	PUNCT
bcpe-1790	144	11	research	research	NOUN
bcpe-1790	144	12	on	on	ADP
bcpe-1790	144	13	this	this	DET
bcpe-1790	144	14	problem	problem	NOUN
bcpe-1790	144	15	-	-	PUNCT
bcpe-1790	144	16	solving	solve	VERB
bcpe-1790	144	17	thought	thought	NOUN
bcpe-1790	144	18	has	have	VERB
bcpe-1790	144	19	a	a	DET
bcpe-1790	144	20	particular	particular	ADJ
bcpe-1790	144	21	value	value	NOUN
bcpe-1790	144	22	.	.	PUNCT
bcpe-1790	145	1	with	with	ADP
bcpe-1790	145	2	the	the	DET
bcpe-1790	145	3	changing	change	VERB
bcpe-1790	145	4	times	time	NOUN
bcpe-1790	145	5	and	and	CCONJ
bcpe-1790	145	6	the	the	DET
bcpe-1790	145	7	emergence	emergence	NOUN
bcpe-1790	145	8	of	of	ADP
bcpe-1790	145	9	various	various	ADJ
bcpe-1790	145	10	problems	problem	NOUN
bcpe-1790	145	11	in	in	ADP
bcpe-1790	145	12	teaching	teaching	NOUN
bcpe-1790	145	13	,	,	PUNCT
bcpe-1790	145	14	polya	polya	PROPN
bcpe-1790	145	15	's	's	PART
bcpe-1790	145	16	problem	problem	NOUN
bcpe-1790	145	17	-	-	PUNCT
bcpe-1790	145	18	solving	solve	VERB
bcpe-1790	145	19	thought	thought	NOUN
bcpe-1790	145	20	will	will	AUX
bcpe-1790	145	21	continue	continue	VERB
bcpe-1790	145	22	to	to	PART
bcpe-1790	145	23	evolve	evolve	VERB
bcpe-1790	145	24	,	,	PUNCT
bcpe-1790	145	25	radiate	radiate	VERB
bcpe-1790	145	26	and	and	CCONJ
bcpe-1790	145	27	generate	generate	VERB
bcpe-1790	145	28	new	new	ADJ
bcpe-1790	145	29	values	value	NOUN
bcpe-1790	145	30	in	in	ADP
bcpe-1790	145	31	the	the	DET
bcpe-1790	145	32	future	future	NOUN
bcpe-1790	145	33	.	.	PUNCT
bcpe-1790	146	1	references	reference	NOUN
bcpe-1790	146	2	[	[	X
bcpe-1790	146	3	1	1	X
bcpe-1790	146	4	]	]	PUNCT
bcpe-1790	146	5	[	[	PUNCT
bcpe-1790	146	6	america	america	PROPN
bcpe-1790	146	7	]	]	PUNCT
bcpe-1790	146	8	d.	d.	PROPN
bcpe-1790	146	9	a.	a.	PROPN
bcpe-1790	146	10	grouse	grouse	PROPN
bcpe-1790	146	11	.	.	PUNCT
bcpe-1790	147	1	research	research	NOUN
bcpe-1790	147	2	handbook	handbook	NOUN
bcpe-1790	147	3	of	of	ADP
bcpe-1790	147	4	mathematics	mathematic	NOUN
bcpe-1790	147	5	teaching	teaching	NOUN
bcpe-1790	147	6	and	and	CCONJ
bcpe-1790	147	7	learning	learn	VERB
bcpe-1790	148	1	[	[	X
bcpe-1790	149	1	m	m	X
bcpe-1790	149	2	]	]	X
bcpe-1790	149	3	.	.	PUNCT
bcpe-1790	150	1	shanghai	shanghai	PROPN
bcpe-1790	150	2	education	education	PROPN
bcpe-1790	150	3	press	press	PROPN
bcpe-1790	150	4	,	,	PUNCT
bcpe-1790	150	5	1999	1999	NUM
bcpe-1790	150	6	.	.	PUNCT
bcpe-1790	151	1	[	[	X
bcpe-1790	151	2	2	2	NUM
bcpe-1790	151	3	]	]	PUNCT
bcpe-1790	151	4	yang	yang	PROPN
bcpe-1790	151	5	qian	qian	PROPN
bcpe-1790	151	6	.	.	PUNCT
bcpe-1790	152	1	the	the	DET
bcpe-1790	152	2	modern	modern	ADJ
bcpe-1790	152	3	enlightenment	enlightenment	NOUN
bcpe-1790	152	4	of	of	ADP
bcpe-1790	152	5	polya	polya	PROPN
bcpe-1790	152	6	's	's	PART
bcpe-1790	152	7	mathematical	mathematical	ADJ
bcpe-1790	152	8	education	education	NOUN
bcpe-1790	152	9	theory[j	theory[j	NOUN
bcpe-1790	152	10	]	]	PUNCT
bcpe-1790	152	11	.	.	PUNCT
bcpe-1790	153	1	journal	journal	PROPN
bcpe-1790	153	2	of	of	ADP
bcpe-1790	153	3	mathematical	mathematical	ADJ
bcpe-1790	153	4	education	education	NOUN
bcpe-1790	153	5	,	,	PUNCT
bcpe-1790	153	6	2002(02):18	2002(02):18	NUM
bcpe-1790	153	7	-	-	SYM
bcpe-1790	153	8	20	20	NUM
bcpe-1790	153	9	.	.	PUNCT
bcpe-1790	154	1	[	[	X
bcpe-1790	154	2	3	3	X
bcpe-1790	154	3	]	]	X
bcpe-1790	154	4	ren	ren	PROPN
bcpe-1790	154	5	zhanghui	zhanghui	PROPN
bcpe-1790	154	6	.	.	PROPN
bcpe-1790	154	7	theory	theory	NOUN
bcpe-1790	154	8	of	of	ADP
bcpe-1790	154	9	mathematical	mathematical	ADJ
bcpe-1790	154	10	thinking	thinking	NOUN
bcpe-1790	155	1	[	[	X
bcpe-1790	155	2	m	m	X
bcpe-1790	155	3	]	]	X
bcpe-1790	155	4	.	.	PUNCT
bcpe-1790	156	1	nanning	nanne	VERB
bcpe-1790	156	2	:	:	PUNCT
bcpe-1790	156	3	guangxi	guangxi	PROPN
bcpe-1790	156	4	education	education	PROPN
bcpe-1790	156	5	press	press	NOUN
bcpe-1790	156	6	,	,	PUNCT
bcpe-1790	156	7	1990	1990	NUM
bcpe-1790	156	8	.	.	PUNCT
bcpe-1790	157	1	[	[	X
bcpe-1790	157	2	4	4	NUM
bcpe-1790	157	3	]	]	X
bcpe-1790	157	4	li	li	PROPN
bcpe-1790	157	5	changguan	changguan	PROPN
bcpe-1790	157	6	.	.	PUNCT
bcpe-1790	158	1	the	the	DET
bcpe-1790	158	2	way	way	NOUN
bcpe-1790	158	3	of	of	ADP
bcpe-1790	158	4	thinking	thinking	NOUN
bcpe-1790	158	5	:	:	PUNCT
bcpe-1790	158	6	the	the	DET
bcpe-1790	158	7	focus	focus	NOUN
bcpe-1790	158	8	of	of	ADP
bcpe-1790	158	9	mathematics	mathematic	NOUN
bcpe-1790	158	10	teaching	teach	VERB
bcpe-1790	158	11	[	[	X
bcpe-1790	158	12	j	j	X
bcpe-1790	158	13	]	]	X
bcpe-1790	158	14	.	.	PUNCT
bcpe-1790	159	1	mathematics	mathematic	NOUN
bcpe-1790	159	2	bulletin	bulletin	NOUN
bcpe-1790	159	3	,	,	PUNCT
bcpe-1790	159	4	2022	2022	NUM
bcpe-1790	159	5	,	,	PUNCT
bcpe-1790	159	6	61(01	61(01	NUM
bcpe-1790	159	7	):	):	PUNCT
bcpe-1790	159	8	20	20	NUM
bcpe-1790	159	9	-	-	SYM
bcpe-1790	159	10	23	23	NUM
bcpe-1790	159	11	.	.	PUNCT
bcpe-1790	160	1	[	[	X
bcpe-1790	160	2	5	5	X
bcpe-1790	160	3	]	]	PUNCT
bcpe-1790	160	4	zhang	zhang	PROPN
bcpe-1790	160	5	jianyue	jianyue	PROPN
bcpe-1790	160	6	.	.	PUNCT
bcpe-1790	161	1	establish	establish	VERB
bcpe-1790	161	2	curriculum	curriculum	NOUN
bcpe-1790	161	3	awareness	awareness	NOUN
bcpe-1790	161	4	and	and	CCONJ
bcpe-1790	161	5	implement	implement	VERB
bcpe-1790	161	6	core	core	NOUN
bcpe-1790	161	7	literacy	literacy	NOUN
bcpe-1790	162	1	[	[	X
bcpe-1790	162	2	j	j	X
bcpe-1790	162	3	]	]	X
bcpe-1790	162	4	.	.	PUNCT
bcpe-1790	163	1	mathematics	mathematic	NOUN
bcpe-1790	163	2	bulletin	bulletin	NOUN
bcpe-1790	163	3	,	,	PUNCT
bcpe-1790	163	4	2016(5	2016(5	NUM
bcpe-1790	163	5	):	):	PUNCT
bcpe-1790	163	6	1	1	NUM
bcpe-1790	163	7	-	-	SYM
bcpe-1790	163	8	4,14	4,14	PRON
bcpe-1790	164	1	[	[	X
bcpe-1790	164	2	6	6	NUM
bcpe-1790	164	3	]	]	PUNCT
bcpe-1790	164	4	g.	g.	PROPN
bcpe-1790	164	5	polya	polya	PROPN
bcpe-1790	164	6	.	.	PUNCT
bcpe-1790	165	1	how	how	SCONJ
bcpe-1790	165	2	to	to	PART
bcpe-1790	165	3	solve	solve	VERB
bcpe-1790	165	4	it	it	PRON
bcpe-1790	165	5	[	[	X
bcpe-1790	165	6	m	m	X
bcpe-1790	165	7	]	]	X
bcpe-1790	165	8	.	.	PUNCT
bcpe-1790	166	1	princeton	princeton	PROPN
bcpe-1790	166	2	university	university	PROPN
bcpe-1790	166	3	press	press	NOUN
bcpe-1790	166	4	,	,	PUNCT
bcpe-1790	166	5	1973	1973	NUM
bcpe-1790	166	6	.	.	PUNCT
bcpe-1790	167	1	it	it	PRON
bcpe-1790	167	2	is	be	AUX
bcpe-1790	167	3	known	know	VERB
bcpe-1790	167	4	that	that	SCONJ
bcpe-1790	167	5	the	the	DET
bcpe-1790	167	6	focus	focus	NOUN
bcpe-1790	167	7	𝐹𝐹	𝐹𝐹	NOUN
bcpe-1790	167	8	of	of	ADP
bcpe-1790	167	9	the	the	DET
bcpe-1790	167	10	parabola	parabola	PROPN
bcpe-1790	167	11	𝐶𝐶:𝑝𝑝2	𝐶𝐶:𝑝𝑝2	ADV
bcpe-1790	167	12	=	=	SYM
bcpe-1790	167	13	2𝑝𝑝𝑥𝑥(𝑝𝑝	2𝑝𝑝𝑥𝑥(𝑝𝑝	NUM
bcpe-1790	167	14	>	>	X
bcpe-1790	167	15	0	0	NUM
bcpe-1790	167	16	)	)	PUNCT
bcpe-1790	167	17	,	,	PUNCT
bcpe-1790	167	18	and	and	CCONJ
bcpe-1790	167	19	the	the	DET
bcpe-1790	167	20	minimum	minimum	ADJ
bcpe-1790	167	21	distance	distance	NOUN
bcpe-1790	167	22	between	between	ADP
bcpe-1790	167	23	𝐹𝐹	𝐹𝐹	PROPN
bcpe-1790	167	24	and	and	CCONJ
bcpe-1790	167	25	the	the	DET
bcpe-1790	167	26	point	point	NOUN
bcpe-1790	167	27	on	on	ADP
bcpe-1790	167	28	the	the	DET
bcpe-1790	167	29	circle	circle	NOUN
bcpe-1790	167	30	𝑀𝑀	𝑀𝑀	PROPN
bcpe-1790	167	31	:	:	PUNCT
bcpe-1790	167	32	(	(	PUNCT
bcpe-1790	167	33	𝑥𝑥	𝑥𝑥	ADP
bcpe-1790	167	34	+	+	X
bcpe-1790	167	35	4)2	4)2	PROPN
bcpe-1790	167	36	+	+	PUNCT
bcpe-1790	167	37	𝑝𝑝2	𝑝𝑝2	ADV
bcpe-1790	167	38	=	=	SYM
bcpe-1790	167	39	1	1	NUM
bcpe-1790	167	40	is	be	AUX
bcpe-1790	167	41	4	4	NUM
bcpe-1790	167	42	.	.	PUNCT
bcpe-1790	168	1	if	if	SCONJ
bcpe-1790	168	2	the	the	DET
bcpe-1790	168	3	point	point	NOUN
bcpe-1790	168	4	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	168	5	is	be	AUX
bcpe-1790	168	6	on	on	ADP
bcpe-1790	168	7	𝑀𝑀	𝑀𝑀	PROPN
bcpe-1790	168	8	,	,	PUNCT
bcpe-1790	168	9	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	168	10	and	and	CCONJ
bcpe-1790	168	11	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	168	12	are	be	AUX
bcpe-1790	168	13	the	the	DET
bcpe-1790	168	14	two	two	NUM
bcpe-1790	168	15	tangents	tangent	NOUN
bcpe-1790	168	16	of	of	ADP
bcpe-1790	168	17	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	168	18	,	,	PUNCT
bcpe-1790	168	19	and	and	CCONJ
bcpe-1790	168	20	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	168	21	and	and	CCONJ
bcpe-1790	168	22	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	168	23	are	be	AUX
bcpe-1790	168	24	the	the	DET
bcpe-1790	168	25	tangent	tangent	ADJ
bcpe-1790	168	26	points	point	NOUN
bcpe-1790	168	27	,	,	PUNCT
bcpe-1790	168	28	find	find	VERB
bcpe-1790	168	29	the	the	DET
bcpe-1790	168	30	maximum	maximum	ADJ
bcpe-1790	168	31	value	value	NOUN
bcpe-1790	168	32	of	of	ADP
bcpe-1790	168	33	the	the	DET
bcpe-1790	168	34	area	area	NOUN
bcpe-1790	168	35	of	of	ADP
bcpe-1790	168	36	∆𝑃𝑃𝑃𝑃𝑃𝑃.	∆𝑃𝑃𝑃𝑃𝑃𝑃.	PROPN
bcpe-1790	168	37	it	it	PRON
bcpe-1790	168	38	is	be	AUX
bcpe-1790	168	39	known	know	VERB
bcpe-1790	168	40	that	that	SCONJ
bcpe-1790	168	41	the	the	DET
bcpe-1790	168	42	focus	focus	NOUN
bcpe-1790	168	43	𝐹𝐹	𝐹𝐹	NOUN
bcpe-1790	168	44	of	of	ADP
bcpe-1790	168	45	the	the	DET
bcpe-1790	168	46	parabola	parabola	PROPN
bcpe-1790	168	47	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	168	48	:	:	PUNCT
bcpe-1790	168	49	𝑥𝑥2	𝑥𝑥2	X
bcpe-1790	168	50	=	=	SYM
bcpe-1790	168	51	2𝑝𝑝𝑝𝑝(𝑝𝑝	2𝑝𝑝𝑝𝑝(𝑝𝑝	NUM
bcpe-1790	168	52	>	>	X
bcpe-1790	168	53	0	0	NUM
bcpe-1790	168	54	)	)	PUNCT
bcpe-1790	168	55	,	,	PUNCT
bcpe-1790	168	56	and	and	CCONJ
bcpe-1790	168	57	the	the	DET
bcpe-1790	168	58	minimum	minimum	ADJ
bcpe-1790	168	59	distance	distance	NOUN
bcpe-1790	168	60	between	between	ADP
bcpe-1790	168	61	𝐹𝐹	𝐹𝐹	PROPN
bcpe-1790	168	62	and	and	CCONJ
bcpe-1790	168	63	the	the	DET
bcpe-1790	168	64	point	point	NOUN
bcpe-1790	168	65	on	on	ADP
bcpe-1790	168	66	the	the	DET
bcpe-1790	168	67	line	line	NOUN
bcpe-1790	168	68	𝑙𝑙	𝑙𝑙	NOUN
bcpe-1790	168	69	:	:	PUNCT
bcpe-1790	169	1	3𝑥𝑥	3𝑥𝑥	ADJ
bcpe-1790	169	2	−	−	PROPN
bcpe-1790	169	3	4𝑝𝑝	4𝑝𝑝	NOUN
bcpe-1790	169	4	−	−	NOUN
bcpe-1790	170	1	16	16	NUM
bcpe-1790	170	2	=	=	SYM
bcpe-1790	170	3	0	0	NUM
bcpe-1790	170	4	is	be	AUX
bcpe-1790	170	5	4	4	NUM
bcpe-1790	170	6	.	.	PUNCT
bcpe-1790	171	1	if	if	SCONJ
bcpe-1790	171	2	the	the	DET
bcpe-1790	171	3	point	point	NOUN
bcpe-1790	171	4	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	171	5	is	be	AUX
bcpe-1790	171	6	on	on	ADP
bcpe-1790	171	7	𝑀𝑀	𝑀𝑀	PROPN
bcpe-1790	171	8	,	,	PUNCT
bcpe-1790	171	9	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	171	10	and	and	CCONJ
bcpe-1790	171	11	𝑃𝑃𝑃𝑃	𝑃𝑃𝑃𝑃	NOUN
bcpe-1790	171	12	are	be	AUX
bcpe-1790	171	13	the	the	DET
bcpe-1790	171	14	two	two	NUM
bcpe-1790	171	15	tangents	tangent	NOUN
bcpe-1790	171	16	of	of	ADP
bcpe-1790	171	17	𝐶𝐶	𝐶𝐶	PROPN
bcpe-1790	171	18	,	,	PUNCT
bcpe-1790	171	19	and	and	CCONJ
bcpe-1790	171	20	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	171	21	and	and	CCONJ
bcpe-1790	171	22	𝑃𝑃	𝑃𝑃	ADV
bcpe-1790	171	23	are	be	AUX
bcpe-1790	171	24	the	the	DET
bcpe-1790	171	25	tangent	tangent	ADJ
bcpe-1790	171	26	points	point	NOUN
bcpe-1790	171	27	,	,	PUNCT
bcpe-1790	171	28	find	find	VERB
bcpe-1790	171	29	the	the	DET
bcpe-1790	171	30	maximum	maximum	ADJ
bcpe-1790	171	31	value	value	NOUN
bcpe-1790	171	32	of	of	ADP
bcpe-1790	171	33	the	the	DET
bcpe-1790	171	34	area	area	NOUN
bcpe-1790	171	35	of	of	ADP
bcpe-1790	171	36	∆𝑃𝑃𝑃𝑃𝑃𝑃.	∆𝑃𝑃𝑃𝑃𝑃𝑃.	PROPN
bcpe-1790	171	37	bcp	bcp	PROPN
bcpe-1790	171	38	education	education	PROPN
bcpe-1790	171	39	&	&	CCONJ
bcpe-1790	171	40	psychology	psychology	PROPN
bcpe-1790	171	41	erhss	erhss	PROPN
bcpe-1790	171	42	2022	2022	NUM
bcpe-1790	171	43	volume	volume	NOUN
bcpe-1790	171	44	6	6	NUM
bcpe-1790	171	45	(	(	PUNCT
bcpe-1790	171	46	2022	2022	NUM
bcpe-1790	171	47	)	)	PUNCT
bcpe-1790	171	48	206	206	NUM
bcpe-1790	172	1	[	[	X
bcpe-1790	172	2	10	10	NUM
bcpe-1790	172	3	]	]	X
bcpe-1790	172	4	zhou	zhou	PROPN
bcpe-1790	172	5	chenchen	chenchen	ADV
bcpe-1790	172	6	.	.	PUNCT
bcpe-1790	173	1	talking	talk	VERB
bcpe-1790	173	2	about	about	ADP
bcpe-1790	173	3	the	the	DET
bcpe-1790	173	4	application	application	NOUN
bcpe-1790	173	5	of	of	ADP
bcpe-1790	173	6	polya	polya	PROPN
bcpe-1790	173	7	's	's	PART
bcpe-1790	173	8	four	four	NUM
bcpe-1790	173	9	-	-	PUNCT
bcpe-1790	173	10	step	step	NOUN
bcpe-1790	173	11	problem	problem	NOUN
bcpe-1790	173	12	-	-	PUNCT
bcpe-1790	173	13	solving	solve	VERB
bcpe-1790	173	14	method	method	NOUN
bcpe-1790	173	15	in	in	ADP
bcpe-1790	173	16	mathematical	mathematical	ADJ
bcpe-1790	173	17	problem	problem	NOUN
bcpe-1790	173	18	-	-	PUNCT
bcpe-1790	173	19	solving	solving	NOUN
bcpe-1790	173	20	—	—	PUNCT
bcpe-1790	173	21	taking	take	VERB
bcpe-1790	173	22	a	a	DET
bcpe-1790	173	23	conic	conic	ADJ
bcpe-1790	173	24	section	section	NOUN
bcpe-1790	173	25	question	question	NOUN
bcpe-1790	173	26	in	in	ADP
bcpe-1790	173	27	the	the	DET
bcpe-1790	173	28	college	college	NOUN
bcpe-1790	173	29	entrance	entrance	NOUN
bcpe-1790	173	30	examination	examination	NOUN
bcpe-1790	173	31	as	as	ADP
bcpe-1790	173	32	an	an	DET
bcpe-1790	173	33	example	example	NOUN
bcpe-1790	173	34	[	[	X
bcpe-1790	173	35	j	j	X
bcpe-1790	173	36	]	]	X
bcpe-1790	173	37	.	.	PUNCT
bcpe-1790	174	1	mathematics	mathematic	NOUN
bcpe-1790	174	2	learning	learning	NOUN
bcpe-1790	174	3	and	and	CCONJ
bcpe-1790	174	4	research	research	NOUN
bcpe-1790	174	5	,	,	PUNCT
bcpe-1790	174	6	2020(05):133	2020(05):133	NUM
bcpe-1790	174	7	-	-	SYM
bcpe-1790	174	8	134	134	NUM
bcpe-1790	174	9	.	.	NOUN
bcpe-1790	175	1	1	1	X
bcpe-1790	175	2	.	.	PUNCT
bcpe-1790	176	1	the	the	DET
bcpe-1790	176	2	question	question	NOUN
bcpe-1790	176	3	is	be	AUX
bcpe-1790	176	4	raised	raise	VERB
bcpe-1790	176	5	1.1	1.1	NUM
bcpe-1790	176	6	mathematical	mathematical	ADJ
bcpe-1790	176	7	teaching	teaching	NOUN
bcpe-1790	176	8	thinking	thinking	NOUN
bcpe-1790	176	9	and	and	CCONJ
bcpe-1790	176	10	its	its	PRON
bcpe-1790	176	11	present	present	ADJ
bcpe-1790	176	12	situation	situation	NOUN
bcpe-1790	176	13	1.2	1.2	NUM
bcpe-1790	176	14	polya	polya	NOUN
bcpe-1790	176	15	's	's	PART
bcpe-1790	176	16	problem	problem	NOUN
bcpe-1790	176	17	-	-	PUNCT
bcpe-1790	176	18	solving	solve	VERB
bcpe-1790	176	19	ideas	idea	NOUN
bcpe-1790	176	20	and	and	CCONJ
bcpe-1790	176	21	new	new	ADJ
bcpe-1790	176	22	mathematical	mathematical	ADJ
bcpe-1790	176	23	problems	problem	NOUN
bcpe-1790	176	24	2	2	NUM
bcpe-1790	176	25	.	.	PUNCT
bcpe-1790	177	1	the	the	DET
bcpe-1790	177	2	innovative	innovative	ADJ
bcpe-1790	177	3	thinking	thinking	NOUN
bcpe-1790	177	4	of	of	ADP
bcpe-1790	177	5	polya	polya	PROPN
bcpe-1790	177	6	's	's	PART
bcpe-1790	177	7	problem	problem	NOUN
bcpe-1790	177	8	-	-	PUNCT
bcpe-1790	177	9	solving	solve	VERB
bcpe-1790	177	10	ideas	idea	NOUN
bcpe-1790	177	11	2.1	2.1	NUM
bcpe-1790	177	12	observing	observe	VERB
bcpe-1790	177	13	and	and	CCONJ
bcpe-1790	177	14	doubting	doubt	VERB
bcpe-1790	177	15	finding	find	VERB
bcpe-1790	177	16	the	the	DET
bcpe-1790	177	17	crux	crux	NOUN
bcpe-1790	177	18	of	of	ADP
bcpe-1790	177	19	the	the	DET
bcpe-1790	177	20	problem	problem	NOUN
bcpe-1790	177	21	2.2	2.2	NUM
bcpe-1790	177	22	multivariate	multivariate	NOUN
bcpe-1790	177	23	analyzing	analyzing	NOUN
bcpe-1790	177	24	doubts	doubt	NOUN
bcpe-1790	177	25	exploring	explore	VERB
bcpe-1790	177	26	the	the	DET
bcpe-1790	177	27	path	path	NOUN
bcpe-1790	177	28	of	of	ADP
bcpe-1790	177	29	solving	solve	VERB
bcpe-1790	177	30	the	the	DET
bcpe-1790	177	31	problem	problem	NOUN
bcpe-1790	177	32	2.3	2.3	NUM
bcpe-1790	177	33	accurately	accurately	ADV
bcpe-1790	177	34	solving	solve	VERB
bcpe-1790	177	35	doubts	doubt	NOUN
bcpe-1790	177	36	accurately	accurately	ADV
bcpe-1790	177	37	implementing	implement	VERB
bcpe-1790	177	38	the	the	DET
bcpe-1790	177	39	plan	plan	NOUN
bcpe-1790	177	40	2.4	2.4	NUM
bcpe-1790	177	41	reviewing	reviewing	NOUN
bcpe-1790	177	42	and	and	CCONJ
bcpe-1790	177	43	doubting	doubt	VERB
bcpe-1790	177	44	perceive	perceive	NOUN
bcpe-1790	177	45	the	the	DET
bcpe-1790	177	46	problem	problem	NOUN
bcpe-1790	177	47	-	-	PUNCT
bcpe-1790	177	48	solving	solve	VERB
bcpe-1790	177	49	process	process	NOUN
bcpe-1790	177	50	2.5	2.5	NUM
bcpe-1790	177	51	reflecting	reflect	VERB
bcpe-1790	177	52	and	and	CCONJ
bcpe-1790	177	53	doubting	doubt	VERB
bcpe-1790	177	54	the	the	DET
bcpe-1790	177	55	problem	problem	NOUN
bcpe-1790	177	56	of	of	ADP
bcpe-1790	177	57	sublimation	sublimation	NOUN
bcpe-1790	177	58	and	and	CCONJ
bcpe-1790	177	59	expansion	expansion	NOUN
bcpe-1790	177	60	3	3	NUM
bcpe-1790	177	61	.	.	PUNCT
bcpe-1790	177	62	practical	practical	ADJ
bcpe-1790	177	63	applications	application	NOUN
bcpe-1790	177	64	4	4	NUM
bcpe-1790	177	65	.	.	PUNCT
bcpe-1790	177	66	summary	summary	NOUN
bcpe-1790	177	67	and	and	CCONJ
bcpe-1790	177	68	outlook	outlook	NOUN
bcpe-1790	177	69	references	reference	NOUN