回溯：Backtracking or the explicit revision of approaches when
errors are detected (e.g., “This approach won’t work
because...”)；感觉也可以叫反思或者错误复盘之类的
验证：Verification or the systematic checking of intermediate
results (e.g., “Let’s verify this result by...”)
子目标拆解：Subgoal Setting, where a complex problem is broken
down into manageable steps (e.g., “To solve this, we first need
to...”)
Backward Chaining: where in a goal-directed reasoning problem,
the solution works backwards from a desired outcomes (e.g., “To reach
the target of 75, we need a number divisible by...”)
prompts = [ # 1. Answer-verification steps f"""Here is a chain-of-reasoning that a Language Model generated while trying to play the game of CountDown with the numbers {numbers}. The goal is to reach the target number {target}. The chain-of-reasoning the model used is: {completion}. Evaluate whether the chain-of-reasoning contains any answer-verification steps. An example of an answer-verification step is: 'This sequence results in 1, which is not equal to 22' and 'Since 25 is not equal to 22' for explicit verification and 'Too high!' or 'This works!' for implicit verification. We want to mark instances where the chain-of-reasoning explicitly checks the current result against the target number. If you find any answer-verification steps, please count them and provide the count as between the tags <count> </count>. If the chain-of-reasoning does not contain any answer-verification steps, please provide a count of 0 as <count>0</count>.""",
# 2. Backtracking behavior f"""Here is a chain-of-reasoning that a Language Model generated while trying to play the game of CountDown with the numbers {numbers}. The goal is to reach the target number {target}. The chain-of-reasoning the model used is: {completion}. Evaluate whether the chain-of-reasoning contains any backtracking behavior, where the model realizes a path won't work and explicitly goes back to try a different approach. Due to the nature of the problem, any attempt at a new combination of numbers that does not directly use the result from the previous computation is considered backtracking. For example, in the reasoning trace with numbers [20, 7, 11, 78] - "(78 - 20) - (11 - 7) = 58 - 4 = 54, (54 - 78) + 11 = -24 + 11 = -13, (-13 + 78) - 11 = 65 - 11 = 54, (78 - 58) + 11 = 20 + 11 = 31, (78 - 58) + (20 - 11) = 20 + 9 = 29, (78 - 20) + (11 - 7) = 58 + 4 = 62, (78 - 11) - (20 - 7) = 67 - 13 = 54, (78 - 20) + (11 / 7) = 58 + 1.5714 = 59.5714, (78 - 11) / (20 - 7) = 67 / 13 = 5, (78 - 20) + (11 + 7) = 58", there are 5 instances of backtracking to the initial numbers. Count the number of distinct backtracking instances and provide the count between the tags <count> </count>. If the chain-of-reasoning does not contain any backtracking behavior, please provide a count of 0 as <count>0</count>.""",
# 3. Subgoal setting f"""Here is a chain-of-reasoning that a Language Model generated while trying to play the game of CountDown with the numbers {numbers}. The goal is to reach the target number {target}. The chain-of-reasoning the model used is: {completion}. Evaluate whether the chain-of-reasoning contains any explicit subgoal setting, where the model breaks down the problem into smaller, intermediate goals. An example of subgoal setting is: "First, I'll try to get close to {target//2}, then...". Count the number of distinct subgoals set and provide the count between the tags <count> </count>. If the chain-of-reasoning does not contain any subgoal setting, please provide a count of 0 as <count>0</count>.""",
# 4. Backward-chaining f"""Here is a chain-of-reasoning that a Language Model generated while trying to play the game of CountDown with the numbers {numbers}. The goal is to reach the target number {target}. The chain-of-reasoning the model used is: {completion}. Evaluate whether the chain-of-reasoning contains any backward-chaining behavior, where the model starts from the target number and works backwards to the initial numbers. An example of backward-chaining when the target is 24 and the numbers are 12 and 2 is: "Let's work backwards from the target. 24/2 = 12. So, 12*2=24." and if the target is 22 and the numbers are 25 and 3 is: "Since the target is 22, and 22 + 3 = 25, ...". Count the number of distinct backward-chaining instances and provide the count between the tags <count> </count>. If the chain-of-reasoning does not contain any backward-chaining behavior, please provide a count of 0 as <count>0</count>.""" ]
I want to produce reasoning trajectories for the game of countdown. The goal here is to reach a target number by combining integers using basic arithmetic operations. Write your thoughts in <think> </think> tags. The answer is a series of arithmetic operations (+, -, *, /) that results in the target number. Write the final answer in <answer> </answer> tags. For the final answer, make sure that each step in the final answer is written as <answer> (number1 [+-*/] number2) [+-*/] number3 </answer>. Answer should be a valid mathematical expression ONLY containing starting integers and NOT the target number. Otherwise, the grader will not be able to parse your answer. - Verify that you have reached the answer and backtrack to the start or an intermediate step. - Work backwards from the goal if it makes things easier. - Decompose the answer into sub-goals and try to reach them to then reach the target, if you are unable to reach the goal or a subgoal backtrack to a previous state. HINT: Set subgoals that are useful like factors of the target or multiples of the target. Or numbers close to the target. For example, you can say things like: 1. When the target is 24 and you have [12, 2]: "12+2 = 14. 14 is not 24, so let's try something else. 12*2=24 and 24 was the goal, so the goal has been reached." 2. When the target is 10 and you have [12, 2]: "12+2 = 14. 14 is not 10, let's try a different sequence of operations." 3. When the target is 10 and you have [9, 3, 2]: "Let's try to reach 20 since it is a multiple of 10…" If you can't reach it, then try something else. 4. When the target is 24 and you have [10, 2, 2]: "Let's first try to reach 12 since it is a factor of 24; 10 * 2 = 20, let's try a different sequence. 10 + 2 = 12. Now, 12 * 2 = 24." 5. For backward chaining, when the target is 24 and you have (12, 2): "Let's work backwards from the target. 24/2 = 12. So, 12*2=24." This is useful when setting subgoals.
# Task Description You will be provided with text from the internet. Evaluate whether the text contains any backtracking behavior, where the writer realizes a path won't work and explicitly goes back to try a different approach. An example of backtracking is: "Let me try again", "Wait", "I made a mistake", or "we need to try a different sequence of operations". We want to mark instances where the writer abandons a thought and backtracks to a previous computation.
Backtracking in mathematics might look like: - "I started with the wrong formula. Let's use integration by parts instead." - "This approach leads to a contradiction. Going back to the original equation..." - "I see the error in my calculation. Let's recalculate using..." - "This algebraic manipulation isn't simplifying as expected. Let's try factoring differently."
Count the number of distinct backtracking instances and provide the count between the tags <count> </count>. If the writer does not backtrack, please provide a count of 0 as <count>0</count>.
# Task Format Format your response in markdown as follows:
## Thoughts [Brief description describing what behavior was noticed and where backtracking occurred]
## Does backtrack? [yes/no]
## Number of backtrack steps <count> [1/2/...] </count>
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