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{
"log_file": [
"step_104-b_0.log",
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"query": [
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\n",
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\nSEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/444.json\n\n#### Similar Question\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n#### User Answer\nThe ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = 275$ feet total.\n\n--- RESULTS END ---\n\n",
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\n",
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\nSEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1347.json\n\n#### Similar Question\nA ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it travelled?\n\n#### User Answer\nThe ball traveled $16+16\\cdot\\frac34+16\\cdot\\left(\\frac34\\right)^2 = 16+ 12+9 = 37$ meters on its three descents. The ball also traveled $16\\cdot\\frac34+16\\cdot\\left(\\frac34\\right)^2+16\\cdot\\left(\\frac34\\right)^3 = 12+9+\\frac{27}4 = 27.75$ meters on its three ascents. Thus, the ball traveled $37+27.75 = 64.75 \\approx 65$ meters total.\n\n--- RESULTS END ---\n\n",
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\n",
"<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nA super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?\n\n### Response:\nSEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6068.json\n\n#### Similar Question\nLet $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\\overline{BG}$ and 5 units from $\\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n$.\n\n#### User Answer\nWhen a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for the cube vertices. The coordinates will all involve 0's and 12's only, so that means that the X, Y, and Z distance traveled by the light must all be divisible by 12. Since the light's Y changes by 5 and the X changes by 7 (the Z changes by 12, don't worry about that), and 5 and 7 are relatively prime to 12, the light must make 12 reflections onto the XY plane or the face parallel to the XY plane.\nIn each reflection, the distance traveled by the light is $\\sqrt{ (12^2) + (5^2) + (7^2) }$ = $\\sqrt{218}$. This happens 12 times, so the total distance is $12\\sqrt{218}$. $m=12$ and $n=218$, so therefore, the answer is $m+n=230$.\n\n--- RESULTS END ---\n\n"
],
"response": [
" SEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]</s>",
" The result might be helpful, I will try using it to answer the question only if it is useful.\n\nLet $d$ be the total distance traveled by the ball. Then, we have:\n\n$d = 100 + 50 + 25 + 12.5 = 187.5$ feet\n\nWe can rewrite this as:\n\n$d = 100 + (1/2) \\cdot 100 + (1/4) \\cdot 100 + (1/8) \\cdot 100$\n\nSimplifying, we have:\n\n$d = 100 + 50 + 25 + 12.5 = 187.5$ feet\n\nThus, the ball traveled $\\boxed{187.5}$ feet when it hit the ground the fourth time.</s>",
" SEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]</s>",
" The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe calculate the total distance traveled by the ball by summing the distance traveled in each bounce. The ball traveled $100+50+25+12.5 = \\boxed{187.5}$ feet.</s>",
" SEARCH[\"super ball\", \"dropped\", \"100 feet\", \"rebounds\", \"half\", \"distance\", \"it falls\", \"time\", \"ball\", \"traveled\", \"ground\", \"fourth time\"]</s>",
" The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents.\n\nOn the first descent, the ball traveled $100$ feet.\n\nOn the second descent, the ball traveled $50$ feet.\n\nOn the third descent, the ball traveled $25$ feet.\n\nOn the fourth descent, the ball traveled $12.5$ feet.\n\nAdding them up, the ball traveled $100+50+25+12.5 = \\boxed{187.5}$ feet.</s>"
],
"src_path": [
"train/algebra/444.json",
"train/algebra/444.json",
"train/algebra/444.json",
"train/algebra/444.json",
"train/algebra/444.json",
"train/algebra/444.json"
],
"input": [
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?",
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?",
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?",
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?",
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?",
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?"
],
"output": [
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total.",
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total.",
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total.",
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total.",
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total.",
"The ball traveled $100+50+25+12.5 = 187.5$ feet on its four descents. The ball also traveled $50+25+12.5 = 87.5$ feet on its three ascents. Thus, the ball traveled $187.5+87.5 = \\boxed{275}$ feet total."
],
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1.0,
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