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hint_DEBUG

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train · 67 rows

hint_chosen stringlengths 102 207 ⌀	hint_completion sequencelengths 16 16	hint_completion_answer sequencelengths 16 16	hint_completion_correct sequencelengths 16 16	hint_completion_succ_rate float64 0 0.81	problem stringclasses 7 values	answer stringclasses 7 values	completion sequencelengths 16 16	completion_answer sequencelengths 16 16	completion_correct sequencelengths 16 16	completion_succ_rate float64 0.13 0.56	domain stringclasses 5 values	context stringclasses 7 values	hint1 stringclasses 7 values	hint2 stringclasses 7 values	hint3 stringclasses 7 values	hint4 stringclasses 7 values	hint5 stringclasses 7 values	hint6 stringclasses 7 values	hint7 stringclasses 7 values	hint8 stringclasses 7 values	hint9 stringclasses 7 values	hint10 stringclasses 7 values
null	["First, let's determine the slope of the given line $l_1: ax - y - a + 2 = 0$. The line equation ca(...TRUNCATED)	["\\sqrt{5}","2","\\sqrt{2}","\\sqrt{5}","\\boxed{\\frac{\\sqrt{5}}{2}} </think> tags and the answer(...TRUNCATED)	[ true, false, false, true, false, false, false, false, false, true, false, false, false, false, true, false ]	0.25	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	["The equation of the line \\(l_1\\) is given as \\(ax - y - a + 2 = 0\\), which we can rearrange to(...TRUNCATED)	["\\sqrt{5}","2","2","2\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","\\sqrt{2}","\\sqrt{2}","1","\\sqrt{5}","S(...TRUNCATED)	[ true, false, false, false, true, false, false, false, false, true, false, true, false, true, true, true ]	0.4375	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	["1. Rewrite the equation $ax - y - a + 2 = 0$ in the slope-intercept form $y = mx + b$.\n - This (...TRUNCATED)	["\\sqrt{2}","\\sqrt{2}","2","\\sqrt{2}","\\sqrt{2}","2","1","2","2","根据数学推导和极限分(...TRUNCATED)	[ false, false, false, false, false, false, false, false, false, false, false, false, false, false, true, false ]	0.0625	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	["Since the lines $l_1: ax - y - a + 2 = 0$ and $l_2$, which passes through the origin, are perpendi(...TRUNCATED)	["\\sqrt{5}","\\sqrt{5}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","(Note:(...TRUNCATED)	[ true, true, false, false, false, true, false, true, false, false, false, false, false, false, false, false ]	0.25	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	["The given line is \\( l_1: ax - y - a + 2 = 0 \\). For the line \\( l_2 \\) passing through the or(...TRUNCATED)	["2","2","\\sqrt{5}","2","2","2","\\sqrt{2}","1","\\boxed{\\frac{2\\sqrt{5}}{5","\\sqrt{2}","2","\\s(...TRUNCATED)	[ false, false, true, false, false, false, false, false, false, false, false, true, false, false, false, false ]	0.125	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	["First, I identify that $l_1$ can be rewritten as $y = ax - a + 2$ which gives us a slope of $a$ fo(...TRUNCATED)	["1","2","\\sqrt{5}","2","\\sqrt{2}","1","最终答案为 \\boxed{\\sqrt{2}}。这一解答综合考(...TRUNCATED)	[ false, false, true, false, false, false, false, true, false, false, false, false, false, false, false, false ]	0.125	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	["To begin with, to find the equation of the line $l_2$ passing through the origin that is perpendic(...TRUNCATED)	["\\sqrt{2}","\\sqrt{2}","1","1","2","\\sqrt{6}","2","2","1","1","\\sqrt{2}","2","\\sqrt{5}","\\sqrt(...TRUNCATED)	[ false, false, false, false, false, false, false, false, false, false, false, false, true, false, false, false ]	0.0625	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	["To solve this problem, we will first find the general equation of $l_1$, which is provided as $ax (...TRUNCATED)	["2\\sqrt{2}","2","\\sqrt{2}","\\frac{\\sqrt{2}}{2}","\\sqrt{5}","\\sqrt{5}","\\sqrt{2}","\\sqrt{5}"(...TRUNCATED)	[ false, false, false, false, true, true, false, true, true, false, false, false, false, false, false, false ]	0.25	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	["To solve this, we need to know that $l_2$ is the line passing through the origin and perpendicular(...TRUNCATED)	["\\sqrt{5}","\\sqrt{5}","2","\\sqrt{5}","\\sqrt{5}","2","y = 1","1","\\sqrt{5}","190","1","2","The (...TRUNCATED)	[ true, true, false, true, true, false, false, false, true, false, false, false, false, true, false, false ]	0.375	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)
"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	["Given that $l_1: ax - y - a + 2 = 0$ and $l_2$ is perpendicular to $l_1$, it can be deduced that $(...TRUNCATED)	["\\sqrt{2}","1","\\sqrt{5}","\\frac{\\sqrt{2}}{2}","2","3","\\sqrt{5}","\\frac{2\\sqrt{5}}{5}","\\f(...TRUNCATED)	[ false, false, true, false, false, false, true, false, false, false, false, false, false, false, true, false ]	0.1875	"For the line $l_1: ax - y - a + 2 = 0$ (where $a \\in \\mathbb{R}$), the line $l_2$ passing through(...TRUNCATED)	\sqrt{5}	["To find the maximum value of \\( \|OM\| \\), we need to determine the distance from the origin \\( O(...TRUNCATED)	["\\sqrt{2}","2","2","\\sqrt{2}","\\sqrt{5}","\\sqrt{2}","1","2","2","\\sqrt{2}","\\sqrt{5}","2","\\(...TRUNCATED)	[ false, false, false, false, true, false, false, false, false, false, true, false, false, false, false, false ]	0.125	Geometry	"A geometry problem involving a parameterized line, finding the foot of the perpendicular from a fix(...TRUNCATED)	"Rewrite the equation of a line in slope-intercept form to clearly identify its slope and intercept;(...TRUNCATED)	"Remember that the slope of a line perpendicular to a given line is the negative reciprocal of the o(...TRUNCATED)	"Solve the system of equations arising from the two lines by using substitution to find the coordina(...TRUNCATED)	"Use the distance formula to express the distance between two points, ensuring that any occurring sq(...TRUNCATED)	"Look for opportunities to factor and simplify expressions, especially when dealing with squares and(...TRUNCATED)	"When an absolute value appears in optimization, split the analysis into different cases to properly(...TRUNCATED)	"Apply differentiation by treating the distance expression as a function of the parameter, then set (...TRUNCATED)	"Consider using alternative methods like completing the square or trigonometric substitutions if the(...TRUNCATED)	"Be cautious when clearing fractions or multiplying through an equation; ensuring all cases are hand(...TRUNCATED)	"Verify your result by analyzing the behavior of the distance function at the boundaries and checkin(...TRUNCATED)

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