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eurus2_ground_truth_with_system

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AI2 Adapt Dev 40

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "A caravan of camels, 1 km in length, is moving uniformly across the desert. A rider traveled from the end of the caravan to the front and back to the end of the caravan. During this time, the caravan covered 1 km. What distance did the rider travel if his speed was constant?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "From the statements given by the students, determine the actual number of students who studied math yesterday.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If $g(x)=\\dfrac{x-2}{x-5}$, determine for what value of $x$ is $g^{-1}(x)$ undefined.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the smallest whole number that is larger than the sum\n\\[3\\dfrac{1}{3} + 4\\dfrac{1}{4} + 5\\dfrac{1}{5} + 6\\dfrac{1}{6}.\\]\nA) 18\nB) 19\nC) 20\nD) 21\nE) 22\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "(a) A and B move clockwise with equal angular speed along circles centered at P and Q, respectively. C moves continuously so that AB = BC = CA. Establish C's locus and speed.\n\n(b) ABC is an equilateral triangle and P satisfies AP = 2, BP = 3. Establish the maximum possible value of CP.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If 10 bahs are equal to 16 rahs, and 9 rahs are equal in value to 15 yahs, how many bahs are equal in value to 1500 yahs?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the function\n\\[ x(t)=5(t+1)^{2}+\\frac{a}{(t+1)^{5}}, \\]\nwhere \\( a \\) is a constant. Find the minimum value of \\( a \\) such that \\( x(t) \\geqslant 24 \\) for all \\( t \\geqslant 0 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the equation of the line $l$ that is parallel to the line $3x + 4y + 12 = 0$ and forms a triangle with the coordinate axes such that the area of the triangle is 24.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the largest value of $n$ such that $3x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Write out the set $S$ of angles that have the same terminal side as $-\\frac{\\pi}{3}$, and list the angles in $S$ that are between $-4\\pi$ and $4\\pi$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Express \\( 2.\\overline{3} \\) as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{2 \\sqrt{2}} \\frac{x^{4} \\, dx}{\\left(16-x^{2}\\right) \\sqrt{16-x^{2}}}\n$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The quadratic $3x^2 + 6x - 9$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In the trapezoid shown, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $5:2$. If $AB + CD = 280$ cm and the height of the trapezoid from $A$ to $DC$ is 20 cm less than the length of $AB$, how long is segment $\\overline{AB}$? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8));\npair A = (0,0), B = (5,0), C = (3,2), D = (1,2);\ndraw(A--B--C--D--cycle--C);\nlabel(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,NE); label(\"$D$\",D,NW);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The solution set of the inequality $\\left| \\frac{x-2}{x} \\right| > \\frac{x-2}{x}$ is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In $\\triangle ABC$, if $\\tan A= \\frac {1}{2}$ and $\\cos B= \\frac {3 \\sqrt {10}}{10}$, then $\\tan C=$ ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that a natural number \\( N \\) ends with the digit 0 and has 8 divisors, what is the smallest value of \\( N \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the units digit of the result of \\( \\underbrace{2 \\times 2 \\times \\ldots \\times 2}_{20 \\uparrow 2}-1 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "A spy with a radio transmitter parachutes into a completely random point in a perfectly square forest with dimensions $10 \\times 10$ km. Upon landing, he immediately starts broadcasting important information to the enemy.\n\nAt each of the four corners of the forest, there are radio direction finders, each capable of detecting the transmitter's signal at a distance of up to 10 km. For the counterintelligence to accurately determine the coordinates of the active transmitter, the signal must be detected by at least two of these direction finders.\n\nThe spy is lucky: one of the radio direction finders is not working today. What is the probability that the counterintelligence will not be able to determine the coordinates of the spy's landing site?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that the function $f(x)=a^{x}+b$ ($a > 0,a\\neq 1$) has both its domain and range as $[-1,0]$, find $a+b=$     .\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The function $f(x) = x^2 - 4x + 3$ is a decreasing function on $(-\\infty, 2)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In the geometric sequence $\\{a_n\\}$, if $a_3$ and $a_7$ are the two roots of the equation $3x^2-11x+9=0$, then the value of $a_5$ is \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Let \\( p, \\) \\( q, \\) and \\( r \\) be real numbers, and let \\( P, \\) \\( Q, \\) \\( R \\) be points such that the midpoint of \\( \\overline{QR} \\) is \\( (p,0,0), \\) the midpoint of \\( \\overline{PR} \\) is \\( (0,q,0), \\) and the midpoint of \\( \\overline{PQ} \\) is \\( (0,0,r). \\) Find\n\\[\n\\frac{PQ^2 + PR^2 + QR^2}{p^2 + q^2 + r^2}.\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2b\\cos A=c\\cos A+a\\cos C$. \n$(1)$ Find the value of angle $A$. \n$(2)$ If $a=4$, find the maximum area of triangle $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that $i$ is the imaginary unit, if the complex number $\\frac{1+ai}{2-i}$ is purely imaginary, find the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If $$\\tan\\left(\\alpha+ \\frac {\\pi}{4}\\right)=2$$, determine the value of $$\\frac {\\sin\\alpha-\\cos\\alpha}{\\sin\\alpha +\\cos\\alpha }$$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Let $a_1 ,a_2 ,\\ldots, a_n$ be a permutation of the integers $1,2,\\ldots, n.$ Call $a_i$ a *big* integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In the sequence $\\{a_n\\}$, $S_n$ denotes the sum of the first $n$ terms. Given that $a_2 = 4, a_3 = 15$, and the sequence $\\{a_n+n\\}$ forms a geometric progression, find the expression for $S_n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given a triangle \\( \\triangle ABC \\) with interior angles \\( A, B, \\) and \\( C \\) opposite sides \\( a, b, \\) and \\( c \\) respectively, and the equation \\( \\sqrt{3} b \\cos \\frac{A+B}{2} = c \\sin B \\):\n\n1. Find the measure of \\( \\angle C \\).\n2. If \\( a + b = \\sqrt{3} c \\), find \\( \\sin A \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given complex numbers $z_{1}=m+2i$ and $z_{2}=3-4i$, if $\\frac{z_{1}}{z_{2}}$ is a real number, then the real number $m=$ ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find all pairs of primes $(p,q)$ such that $6pq$ divides $$ p^3+q^2+38 $$ \n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "What is the least common multiple of 5, 8, 12, and 20?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Compute\n\\[\n\\sum_{n=1}^\\infty \\frac{1}{n(n+3)}.\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Five positive integers $E$, $F$, $G$, $H$, and $I$ have a sum of 110. If $E \\div 2 = F \\div 3 = G \\times 4 = H \\times 2 = I - 5$, what is the value of the product $E \\times F \\times G \\times H \\times I$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In $\\triangle XYZ$, the sum of $\\angle X$ and $\\angle Y$ is $90^\\circ.$ What is the measure of $\\angle Z?$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The arithmetic square root of $\\sqrt{81}$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the magnitude of the sixth power of the complex number \\(5 + 2\\sqrt{3}i\\):\n\\[ \\left| \\left( 5 + 2\\sqrt{3}i \\right)^6 \\right| \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "How many integers satisfy \\( -6 \\leq 3x + 2 \\leq 9 \\) excluding \\( x = 0 \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If the directrix of the parabola $y^{2}=2px(p > 0)$ passes through a focus of the ellipse $\\dfrac{x^{2}}{9}+\\dfrac{y^{2}}{5}=1$, then the equation of the directrix of this parabola is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the parabola $y=x^{2}$ and the line $x-y-2=0$, the shortest distance from a point on the parabola to the line is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If $\\log_{16} (x-3)=\\frac{1}{2}$, find $\\log_{256} (x+1)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "For any real number $x$, the notation $\\lfloor x \\rfloor$ represents the integer part of $x$, meaning $\\lfloor x \\rfloor$ is the largest integer not exceeding $x$, such as $\\lfloor 2 \\rfloor = 2$, $\\lfloor 2.1 \\rfloor = 2$, $\\lfloor -2.2 \\rfloor = -3$. This function $\\lfloor x \\rfloor$ is called the \"floor function,\" which has a wide range of applications in mathematics itself and practical production. Given a function $f(x)$ $(x \\in R)$ that satisfies $f(x) = f(2-x)$, and when $x \\geq 1$, $f(x) = \\log_2{x}$. Find the value of $\\lfloor f(-16) \\rfloor + \\lfloor f(-15) \\rfloor + \\ldots + \\lfloor f(15) \\rfloor + \\lfloor f(16) \\rfloor$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the sets, $A=\\{x \\mid |x|<1\\}$, $B=\\{x \\mid x^2-2x\\leq0\\}$, find the intersection of the sets $A$ and $B$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given an geometric sequence $\\{a\\_n\\}$ with the sum of the first $n$ terms denoted as $S\\_n$. It is known that $S\\_2 = \\frac{3}{2}a\\_2 - 1$ and $S\\_3 = \\frac{3}{2}a\\_3 - 1$.\n\n1. Find the general formula for the $n$-th term of the sequence.\n2. Insert $n$ numbers between $a\\_n$ and $a\\_{n+1}$ to form an arithmetic sequence with a common difference of $d\\_n$. Denote $T\\_n$ as the sum of the first $n$ terms of the sequence $\\{ \\frac{1}{d\\_n} \\}$. Find the maximum positive integer value of $n$ that satisfies $\\frac{8}{5}T\\_n + \\frac{n}{5 \\times 3^{n-1}} \\leqslant \\frac{40}{27}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Ms. Hamilton's eighth-grade class wants to participate in the annual five-person-team hockey tournament. The losing team of each game is eliminated from the tournament. If thirty-two teams compete, how many games will be played to determine the winner?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "What is the nonnegative difference between the roots for the equation $x^2 + 42x + 420 = -48$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "$(1)$ If the solution of the equation $m(1-x)=x+3$ is the same as the solution of the equation $2-x=x+4$, find the value of $m$. \n$(2)$ Under the condition of $(1)$, find the solution of the system of equations in terms of $x$ and $y$: $\\left\\{\\begin{array}{l}{3x+2m=-y}\\\\{2x+2y=m-1}\\end{array}\\right.$. \n$(3)$ A diligent student, Xiao Yi, found that regardless of the value of $m$, there is a relationship between the solutions $x$ and $y$ of the system of equations in $(2)$. Find this relationship.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "How many perfect squares are three-digit and divisible by $4$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given non-negative real numbers \\( a, b, c, x, y, z \\) that satisfy \\( a+b+c = x+y+z = 1 \\), find the maximum value of \\( \\left(a-x^{2}\\right)\\left(b-y^{2}\\right)\\left(c-z^{2}\\right) \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given $x+m≤{e}^{\\frac{2x}{m}+n}$ holds for any $x\\in \\left(-m,+\\infty \\right)$, then the minimum value of $m\\cdot n$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "We wrote distinct real numbers at the vertices of an $n$-sided polygon such that any number is equal to the product of the numbers written at the two neighboring vertices.\n\nDetermine the value of $n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Distinct lines $l_1, l_2, ..., l_m$ intersect the interior of a regular tetrahedron T. Let $S'$ be the union of the faces of T and let $L = \\bigcup_{i=1}^{m} l_i$. Each line $l_i$ intersects the edges of $T$ precisely at their midpoints. What is the difference between the maximum and minimum possible values of $m$?\nA) 0\nB) 1\nC) 2\nD) 3\nE) 4\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the inequality $x^{2}+5x-24 \\lt 0$, find the solution set.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Calculate the product of $0.8 \\cdot 0.25 \\cdot 0.2$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The length of the line parallel to the base that divides the triangle into sections of area 1 and 2, counting from the top, can be found if the height of the triangle is known, using properties of similar triangles and the given ratio of the areas of the three sections within the triangle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the sets $P={x|x\\leqslant a}$ and $Q={y|y=\\sin \\theta, \\theta \\in \\mathbb{R}}$. If $P \\supseteq Q$, find the range of values for the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that all terms of the geometric sequence $\\{a_n\\}$ are positive, and the common ratio is $q$, the sum of the first $n$ terms is $S_n$. If for any $n\\in\\mathbb{N}^*$, it holds that $S_{2n} < 3S_{n}$, determine the range of the common ratio $q$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The side $AC$ of triangle $ABC$ is extended at segment $CD = AB = 1$ . It is known that $\\angle ABC = 90^o$ , $\\angle CBD = 30^o$ . Calculate $AC$ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "From the set $\\{-9, -5, -1, 1, 3, 5, 8\\}$, calculate the minimum possible product that can be formed using three different numbers.\nA) -200\nB) -360\nC) -45\nD) 360\nE) 15\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Straw returning to the field is a widely valued measure for increasing soil fertility and production in the world today, which eliminates the air pollution caused by straw burning and also has the effect of increasing fertility and production. A farmer spent $137,600 to buy a new type of combine harvester to achieve straw returning to the field while harvesting. The annual income from harvesting is $60,000 (fuel costs deducted); the harvester requires regular maintenance, with the first year's maintenance being free of charge provided by the manufacturer, and from the second year onwards, the farmer pays for the maintenance, with the cost $y$ (in yuan) related to the number of years $n$ used as: $y=kn+b$ ($n\\geqslant 2$, and $n\\in N^{\\ast}$), knowing that the second year's maintenance costs $1,800 yuan, and the fifth year's costs $6,000 yuan. \n(Ⅰ) Try to find the relationship between the maintenance cost $f(n)$ (in yuan) and the number of years $n$ ($n\\in N^{\\ast}$) used; \n(Ⅱ) How many years should this harvester be used to maximize the average profit? (Profit = Income - Maintenance cost - Cost of machinery)\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Determine the biggest possible value of $ m$ for which the equation $ 2005x \\plus{} 2007y \\equal{} m$ has unique solution in natural numbers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour.\nWhat is the largest possible value of $n$ ?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The function \\( f \\) satisfies\n\\[\nf(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1\n\\]\nfor all real numbers \\( x \\) and \\( y \\). Determine the value of \\( f(10) \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "A five-digit number with digits 1, 2, 3, 4, 5 without repetition is a \"wave number\" if its digits follow the alternative pattern \"less-greater-less-greater\", i.e., satisfying a_1 < a_2 > a_3 < a_4 > a_5 < a_6... Find the total number of such \"wave numbers\".\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Simplify $120x - 55x$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In the expansion of the binomial ${\\left( {{x}^{2}}-{\\frac{a}{x}} \\right)}^{5}$, the coefficient of the $x$ term is $-10$. Find the value of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "How many positive integer values of $x$ are solutions to the inequality $8 < -2x + 20$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Let $f(x) = \\sin x + mx$ where $m \\in \\mathbb{R}$. Four conclusions are given as follows:<br/>① Regardless of the value of $m$, the curve $y = f(x)$ always has two parallel tangent lines;<br/>② Regardless of the value of $m$, the curve $y = f(x)$ always has two perpendicular tangent lines;<br/>③ Regardless of the value of $m$, there always exists an infinite sequence $\\{a_{n}\\}$ such that the tangent lines of the curve $y = f(x)$ at $x = a_{n}$ $(n = 1, 2, 3, \\cdots)$ are parallel to each other;<br/>④ Regardless of the value of $m$, there always exists an infinite sequence $\\{a_{n}\\}$ such that the tangent lines of the curve $y = f(x)$ at $x = a_{n}$ $(n = 1, 2, 3, \\cdots)$ are the same straight line.<br/>The correct numbers of the conclusions are ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "A bus company's parking lot has 15 buses. Starting from 6 a.m., the first bus departs, and thereafter, a bus departs every 6 minutes. Three minutes after the first bus departs, a bus enters the parking lot, and thereafter, a bus enters every 8 minutes. The buses that enter will depart after the original 15 buses in sequence. When will the parking lot be empty for the first time?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that line $l$ is perpendicular to the line $x-y-1=0$, then the slope angle $\\alpha$ of line $l$ is \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Two circles, one of radius 7 inches, the other of radius 3 inches, are tangent at point P. Two ants start crawling at the same time from point P, one along the bigger circle at $5\\pi$ inches per minute, the other along the smaller circle at $3\\pi$ inches per minute. How many minutes before they next meet at point P?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "What is the greatest prime factor of 247?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Determine the order of $tan(1), tan(2), tan(3)$ from greatest to smallest.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "An equilateral triangle has sides 10 units long. An equilateral triangle with sides 5 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $A:B:C=1:2:3$, then $a:b:c=$ _______ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Olivia's Omelette Oasis offers a range of omelettes that include various fillings: cheese, ham, mushrooms, peppers, onions, tomatoes, spinach, and olives. A customer can choose omelette egg base ranging from one to four eggs, and any combination of fillings. How many different kinds of omelettes can be ordered?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The function $f(x)=x^{3}-3x^{2}+4$ attains its minimum value at $x=$ \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given that $\\alpha$ is an acute angle and $\\tan \\alpha = 2$, find $cos(\\frac{π}{2}-2\\alpha)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Two concentric circles have radii $2$ and $4$. Find the probability that the chord joining two points chosen independently and uniformly at random on the outer circle does not intersect the inner circle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find the product of all real $x$ such that the ratio $\\frac{3x+5}{6x+5}$ is equal to the ratio $\\frac{5x+4}{9x+4}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In triangle \\(ABC\\), angle \\(B\\) is \\(120^\\circ\\), and \\(AB = 2BC\\). The perpendicular bisector of side \\(AB\\) intersects \\(AC\\) at point \\(D\\). Find the ratio \\(AD:DC\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "From point $P$, there are two ways to get to city $B$: either walk the $PB$ route, or walk to city $A$ and then take the train to $B$. Which points make it preferable to choose the first option, and which points make it preferable to choose the second option if you want to reach $B$ as quickly as possible? The walking speed is $5 \\mathrm{~km/h}$, the train speed is $50 \\mathrm{~km/h}$, and the distance between $A$ and $B$ is $20 \\mathrm{~km}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Solve the inequality\n\\[\\frac{(x - 3)(x - 5)(x - 7)}{(x - 2)(x - 6)(x - 8)} > 0.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The distance from the foci of the hyperbola $\\dfrac {x^{2}}{a^{2}} - \\dfrac {y^{2}}{b^{2}} = 1$ to the corresponding directrix is equal to the length of the real axis. Find the eccentricity of the hyperbola.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Let the line $p$ be the perpendicular bisector of points $A = (30, 10)$ and $B = (6, 3)$. Given that $AB$ meets $p$ at point $C = (x, y)$, what is $2x - 4y$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Evaluate the expression $\\frac{2024}{2023} - \\frac{2023}{2024}$ and express it as a reduced fraction $\\frac{p}{q}$. What is $p$?\nA) 4046\nB) 4047\nC) 4048\nD) 4049\nE) 4050\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Find all pairs of integers \\( m \\leq n \\) such that the following quantity is an integer:\n\n$$\n\\frac{1}{m} + \\frac{1}{m+1} + \\ldots + \\frac{1}{n}\n$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Simplify $$\\frac{13!}{10! + 3\\cdot 9!}$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Rationalize the denominator of $\\frac{7}{3+\\sqrt{8}}$. The answer can be expressed as $\\frac{A\\sqrt{B} + C}{D}$, where $A$, $B$, $C$, and $D$ are integers, with $D$ positive and $B$ not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, find $A+B+C+D$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "The distance from home to work is \\( s = 3 \\kappa \\) meters. At the moment when Ivan left work, his beloved dog rushed out of the house and ran to meet him. They met at a quarter of the way from work. The dog instantly turned around and ran back home. Reaching home, he instantly turned around again and ran back to his owner, and so on. Assuming Ivan and his dog move at constant speeds, determine the distance the dog will have run by the time Ivan gets home.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "We cover a $4 \\times 4$ chessboard with dominoes without gaps and overlaps (exactly once). Each domino covers two adjacent squares. How many ways are there to cover the board? (Two coverings are not considered different if one can be transformed into the other by rotating or reflecting the board.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Let the complex number $z$ satisfy $z = 3 + 4i$ (where $i$ is the imaginary unit). Then, the modulus of $z$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the function $f(x)=2x^{2}-\\ln x$, determine the range of the real number $k$ such that the function is not monotonic in the subinterval $(k-1,k+1)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Assign four teachers, A, B, C, and D, to three different schools, with each school receiving at least one teacher. Teachers A and B cannot be assigned to the same school. The number of different ways to distribute the teachers is\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "When the square root of $y$ is raised to the fourth power, the result is 256. What is the value of $y$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If the solutions of the equations $2x+1=3$ and $2-\\frac{a-x}{3}=1$ with respect to $x$ are the same, then the value of $a$ is ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given $$\\sin\\left(\\theta+ \\frac {\\pi}{3}\\right)= \\frac {2}{3}$$, then $$\\cos\\left(\\theta- \\frac {\\pi}{6}\\right)$$ equals \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "If the difference between the maximum and minimum values of the exponential function $y = a^x$ on the interval $[-1, 1]$ is 1, calculate the base $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given $$\\frac {2\\sin^{2}\\alpha+\\sin 2\\alpha}{1+\\tan\\alpha }=k(0＜\\alpha＜ \\frac {\\pi}{4})$$, find the relationship between $\\sin\\left(\\alpha- \\frac {\\pi}{4}\\right)$ and $k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]