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REVIEW 4 major objections 4 minor 28 references

Prior Prompt Engineering for Reinforcement Fine-Tuning

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Prior prompt choices during reinforcement fine-tuning steer model behavior, and a null-example prompt matches or beats reasoning prompts on average.

desk verdict A genuinely new empirical axis for RFT—systematic prior prompt comparison with a surprising null-example result—but the design bundles prompt wording with format reward, so the causal claim outruns the evidence. read the letter →

arxiv 2505.14157 v2 pith:YUNIGC5C submitted 2025-05-20 cs.CL cs.AI

Pittawat Taveekitworachai , Potsawee Manakul , Sarana Nutanong , Kunat Pipatanakul This is my paper · ORCID
classification cs.CLcs.AI
keywords approachesbehaviorsengineeringmodelsperformancepriorpromptaime2024
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Language models that think before answering are usually trained with reinforcement fine-tuning (RFT), where the model gets a reward for correct answers and is free to discover its own reasoning style. This paper asks whether the exact wording of the instruction shown before each question during training matters. The authors take five styles of prompting that work at inference time (step-by-step reasoning, planning, code-based reasoning, knowledge recall, and asking for examples) and prepend each one to math questions while training a Qwen2.5-7B model with reinforcement learning. They then test the five trained models on math, coding, and question-answering benchmarks.

The main result is that all five trained models beat their non-trained counterparts that are simply prompted at inference time. More surprisingly, the model trained with an examples instruction, which asks the model to list illustrative examples before solving, achieved the highest average gain, slightly ahead of the usual reasoning prompt. The models also developed visibly different styles: the planning-trained model produces a numbered plan, the code-trained coder model writes Python, and so on. However, the training setup includes a formatting reward that requires the exact output tag structure, so the observed behavioral differences are not purely due to the prompt wording. All numbers come from a single run with no error bars, and the headline gaps are small, within a few questions on benchmarks with hundreds or fewer items. Smaller models also failed to replicate the benefit, sometimes collapsing to tag-only outputs.

The paper positions prior prompt engineering as a cheap and overlooked lever in RFT.

Extended reading notes

Core claim

The central assertion is that different prior prompts used during reinforcement fine-tuning lead language models to internalize distinct behavioral styles, and that the null-example (examples) prior prompt achieves the largest average performance gain and the highest improvement on AIME2024 and GPQA-Diamond, surpassing the commonly used reasoning prompt. If correct, pPE is an effective and understudied axis for RFT.

Load-bearing premise

The load-bearing premise is that the observed behavioral and performance differences are caused by the semantic content of the prior prompt, not by the co-varying format reward that gives 0.5 reward for producing exactly the expected XML tags. The experiments vary pPE and format reward jointly (Section 3.2), so if the format reward alone drives the tag-structured behaviors, the conclusion that prompt wording shapes post-RFT behavior would not follow.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces prior prompt engineering (pPE) for reinforcement fine-tuning, adapting five inference-time prompting strategies (chain-of-thought, plan-and-solve, program-of-thought, generated knowledge, and null-shot) into prior prompts with corresponding XML tags. It trains Qwen2.5-7B with GRPO on a math dataset under each pPE, compares to iPE-prompted baselines and a No PP RFT baseline, and evaluates on AIME, AMC, MATH-500, HumanEval+, and GPQA-Diamond. The paper reports that all pPE models improve over iPE on average, that the null-example pPE achieves the largest average gain and best GPQA performance, and that different pPE approaches induce distinct behavioral styles as measured by an LLM-based classifier. Generalization experiments on Qwen2.5-3B, Llama 3.1-8B, and Qwen2.5-Coder-7B are also reported.

Significance. If the causal attribution is established, the paper identifies a practical and understudied axis for RFT, with a counterintuitive result (null-example prior outperforms reasoning prior) and implications for steering model behavior and test-time compute. The paper is transparent: it ships the training script, hyperparameters, full prior-prompt templates, and qualitative examples, and it reports a negative result (iPE trends do not transfer to pPE) that is useful. However, the central claim currently rests on a confounded design and unreplicated small-benchmark differences, so the significance is conditional on additional control experiments.

major comments (4)
  1. [3.2, D.2] The experimental design varies the prior-prompt instruction and the format reward jointly. Section 3.2 defines a format reward of 0.5 for exactly one pair of the expected XML tag (<think>, <plan>, <code>, <knowledge>, or <examples>), and Section D.2 states this reward is 'crucial' for structured outputs. Each pPE condition therefore differs from the others both in instruction wording and in the specific tag that is rewarded, and the No PP baseline removes both components. The behavioral differences in Figures 6 and 7 could consequently be caused by RL pressure to emit a particular tag token rather than by internalization of the instruction. The reward-hacking collapse in Section 4.3 (Qwen2.5-3B with <plan> pPE producing 9-token responses that collect only the format reward; Table 6 and Figures 25–26) shows the format reward alone can shape output structure. Please add control conditions that fix the format reward and tag structure while varying only the instruction wording, or fix the instruction while varying the tag and reward.
  2. [D.4.1] All evaluations are single runs with a fixed random seed (Section D.4.1), and the benchmarks are small: AIME has 30 problems, AMC has 83, and GPQA-Diamond has 198. The headline differences are within plausible noise: the Examples pPE average (47.60) exceeds Think (46.99) by 0.61 points; AIME is tied at 20.00 for Think, Plan, and Examples; and the GPQA difference (30.81 vs. 28.28) corresponds to about 5 questions out of 198. Without repeated seeds or bootstrap confidence intervals, the superiority of the null-example pPE over the reasoning pPE, and the ranking of pPE variants generally, are not established. Please report multiple seeds or statistical uncertainty estimates for the main Qwen2.5-7B comparison.
  3. [D.4.2, Figures 18–19] The classification prompts for the four fundamental behaviors are swapped in Section D.4.2. Figure 18, captioned 'Subgoal Settings Classification Prompt,' asks the model to detect backward chaining and gives a backward-chaining example, while Figure 19, captioned 'Backward Chaining Classification Prompt,' asks for explicit subgoal setting and gives a subgoal example. As a result, the subgoal-setting and backward-chaining counts in Figure 6 and Table 7 measure the opposite constructs. This undermines the qualitative evidence for distinct behavioral styles and must be corrected and the classification re-run.
  4. [D.4.2] The adapted LLM-based behavior classifier (gpt-4.1-mini) is used to quantify behavior differences that are a central claim of the paper, but no accuracy, agreement, or calibration check is reported for the new prompts. Given the prompt-swap error above and the very small counts in the generalization tables (e.g., Table 8), the reliability of the behavior counts is uncertain. Please validate the classifier on a hand-labeled sample or report inter-annotator agreement.
minor comments (4)
  1. [Abstract, Introduction] The abstract and introduction state that 'all pPE-trained models surpass their iPE-prompted counterparts,' but Table 1 shows this holds only for average performance; per-benchmark exceptions exist (e.g., HE+ Think pPE 70.10 vs. iPE 75.00; GPQA Knowledge pPE 21.72 vs. iPE 24.24). Please qualify the claim.
  2. [Abstract] The abstract states that null-example pPE achieves 'the highest improvement on AIME2024,' but Table 1 shows Think, Plan, and Examples all reach 20.00 on AIME, and the No PP baseline reaches 26.67. The claim should be restricted to average performance and GPQA.
  3. [Figure 4] The reward-level comparison in Figure 4 between No PP and pPE is confounded by different reward scales: No PP uses an accuracy reward maxed at 1.0, while pPE uses accuracy 0.5 plus format 0.5. Normalize or rescale before interpreting the lower reward level of No PP.
  4. [3.2] Section 3.2 contains several language errors, including 'math-only training datasets provides' and 'which inconclusive in implementation standards of the reward function'; these should be copy-edited.
Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the empirical comparison; no new physical entities are introduced. The main ledger items are the format reward and training-sample selection, which are chosen by hand and co-vary with the pPE treatment.

free parameters (3)
  • format reward weight = 0.5
    The reward is a 50/50 sum of accuracy and format components. The format component forces the exact tag structure, so behavioral differences between pPE variants may be driven by this weight rather than prompt semantics.
  • training sample count = 10000
    max_samples is set to 10,000 from the 29,925-problem STILLv3 dataset; the subset selection and ordering are not specified, which can affect trained behavior.
  • KL coefficient = 1e-6
    init_kl_coef chosen in training script; standard but arbitrary, affects exploration and final behavior.
assumptions (3)
  • domain assumption Math-only RFT with verifiable rewards is a representative setting for studying pPE.
    The paper trains only on mathematical problems and generalizes conclusions about pPE to other domains, which is stated as an assumption in Limitations.
  • domain assumption gpt-4.1-mini behavior classifications are reliable proxies for model behavior.
    The adapted classification framework from Gandhi et al. (2025) is used without validation on the pPE-trained models; the paper notes it may not generalize (Section 4.2).
  • domain assumption The format reward does not confound the pPE effect.
    The paper's claim that pPE shapes behavior assumes the prior prompt content, not the co-varying format reward, is responsible for the observed differences. This is not testable from the reported experiments because No PP omits both prompt and format reward.

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Pith. "Pith review of Prior Prompt Engineering for Reinforcement Fine-Tuning." pith.science (2026). https://pith.science/paper/YUNIGC5C

@misc{pith2026250514157,
  author       = {Pith},
  title        = {Pith review of: Prior Prompt Engineering for Reinforcement Fine-Tuning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUNIGC5C}},
  note         = {Machine review of arXiv:2505.14157}
}
read the original abstract

This paper investigates prior prompt engineering (pPE) in the context of reinforcement fine-tuning (RFT), where language models (LMs) are incentivized to exhibit behaviors that maximize performance through reward signals. While existing RFT research has primarily focused on algorithms, reward shaping, and data curation, the design of the prior prompt--the instructions prepended to queries during training to elicit behaviors such as step-by-step reasoning--remains underexplored. We investigate whether different pPE approaches can guide LMs to internalize distinct behaviors after RFT. Inspired by inference-time prompt engineering (iPE), we translate five representative iPE strategies--reasoning, planning, code-based reasoning, knowledge recall, and null-example utilization--into corresponding pPE approaches. We experiment with Qwen2.5-7B using each of the pPE approaches, then evaluate performance on in-domain and out-of-domain benchmarks (e.g., AIME2024, HumanEval+, and GPQA-Diamond). Our results show that all pPE-trained models surpass their iPE-prompted counterparts, with the null-example pPE approach achieving the largest average performance gain and the highest improvement on AIME2024 and GPQA-Diamond, surpassing the commonly used reasoning approach. Furthermore, by adapting a behavior-classification framework, we demonstrate that different pPE strategies instill distinct behavioral styles in the resulting models. These findings position pPE as a powerful yet understudied axis for RFT.

Figures

Figures reproduced from arXiv: 2505.14157 by the authors.

Figure 1
Figure 1. Five generated responses from five distinct models post-RFT with different pPE approaches– [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The prompt used during RFT by DeepSeek-AI et al. (2025). The prior prompt is highlighted in yellow. Non-highlighted content is task content. before producing a final answer; plan-and-solve prompting (PS) (Wang et al., 2023) first gener￾ates a high-level plan before problem solving; and program-of-thought prompting (PoT) (Chen et al., 2023) induces code-based reasoning. These exam￾ples illustrate that different iPE a… view at source ↗
Figure 3
Figure 3. Left: iPE approaches are applied to a prompt during inference, before inputting it into an LM, to elicit desired behaviors in the response. Right: pPE approaches are translated from iPE approaches and applied to the prior prompt to elicit desired behaviors during training. Group Relative Policy Optimization (GRPO) with a pretrained base LM. Our training stack is Open￾RLHF v0.6.4 (Hu et al., 2024) for policy optimiza… view at source ↗
Figures from the paper (28 more)
Figure 5
Figure 5. Figure 5: Evolution of the average response length for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 4
Figure 4. Figure 4: The reward progression of Qwen2.5-7B dur [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: Ratio of the four fundamental cognitive behaviors–backtracking, backward chaining, subgoal setting, and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Ratio of five elicited behavior categories– [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: The prompt used during RFT by Logic-RL (Xie et al., 2025). Open-Reasoner-Zero Prior Prompt A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The as￾sistant first thinks about the reasoning process in the mind and then pro…
Figure 11
Figure 11. Figure 11: The <plan> prior prompt, based on plan-and￾solve prompting (Wang et al., 2023), asks the model to explicitly lay out a plan before solving the problem [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The <code> prior prompt encourages the model to reason through code, inspired by program-of￾thought (PoT) prompting (Chen et al., 2023). Knowledge Prompt You are a helpful assistant. The assistant first re￾calls relevant knowledge used to solve the prob￾lem and then p…
Figure 13
Figure 13. Figure 13: The <knowledge> prior prompt elicits fac￾tual recall relevant to the problem before beginning reasoning, inspired by generated knowledge prompting (Liu et al., 2022). Examples Prompt You are a helpful assistant. The assistant first lists relevant examples used to solv…
Figure 14
Figure 14. Figure 14: The <examples> prior prompt draws on null-shot prompting (Taveekitworachai et al., 2024) to encourage the model to provide illustrative examples before answering. D.2 Reward Design We design our reward function with two equally weighted components: (1) an accuracy rew…
Figure 15
Figure 15. Figure 15: Training script using the OpenRLHF for RFT. This script specifies the model, dataset, GRPO algorithm, reward configuration, and other relevant hy￾perparameters. D.4 Evaluation In this section, we provide additional details on quantitative and qualitative evaluation, m…
Figure 17
Figure 17. Figure 17: Classification prompt used to detect instances [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 19
Figure 19. Figure 19: Classification prompt used to identify back [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 18
Figure 18. Figure 18: Prompt used to detect subgoal setting be [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 20
Figure 20. Figure 20: Prompt used to classify whether the model is [PITH_FULL_IMAGE:figures/full_fig_p020_20.png]
Figure 21
Figure 21. Figure 21: Prompt used to identify whether the model [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 25
Figure 25. Figure 25: Reward progression for Qwen2.5-3B during [PITH_FULL_IMAGE:figures/full_fig_p021_25.png]
Figure 26
Figure 26. Figure 26: Evolution of the average response length for [PITH_FULL_IMAGE:figures/full_fig_p021_26.png]
Figure 27
Figure 27. Figure 27: Reward progression for Llama 3.1-8B during [PITH_FULL_IMAGE:figures/full_fig_p021_27.png]
Figure 28
Figure 28. Figure 28: Evolution of the average response length for [PITH_FULL_IMAGE:figures/full_fig_p022_28.png]
Figure 29
Figure 29. Figure 29: Reward progression for Qwen2.5-Coder-7B during RFT. 0 20 40 60 80 100 120 140 160 Step 0 500 1000 1500 2000 2500 Response Length Code Think [PITH_FULL_IMAGE:figures/full_fig_p022_29.png]
Figure 32
Figure 32. Figure 32: Ratio of the four fundamental cognitive behaviors–backtracking, backward chaining, subgoal setting, and verification–across different prompting (iPE) and RFT (pPE) approaches with Llama 3.1-8B. Base 0 500 1000 1500 2000 2500 Count Base iPE pPE Think iPE pPE Code Backt…
Figure 33
Figure 33. Figure 33: Ratio of the four fundamental cognitive behaviors–backtracking, backward chaining, subgoal setting, and verification–across different prompting (iPE) and RFT (pPE) approaches with Qwen2.5-Coder￾7B [PITH_FULL_IMAGE:figures/full_fig_p022_33.png]
Figure 34
Figure 34. Figure 34: Behavior alignment heatmaps for Qwen2.5 3B: iPE on the left, RFT on the right. Think Plan Code Knowledge Examples Behavior Type Think Plan Approach 116 19 4 49 1 136 23 3 57 0 iPE Think Plan Code Knowledge Examples Behavior Type Think Plan Approach -31 -7 0 -67 -7 -33…
Figure 36
Figure 36. Figure 36: Behavior heatmaps for Qwen2.5-Coder 7B: iPE (left) vs. RFT (right) [PITH_FULL_IMAGE:figures/full_fig_p023_36.png]
Figure 37
Figure 37. Figure 37: The shared prompt used to probe all models [PITH_FULL_IMAGE:figures/full_fig_p024_37.png]
Figure 39
Figure 39. Figure 39: Response from Qwen2.5 7B trained with the [PITH_FULL_IMAGE:figures/full_fig_p025_39.png]
Figure 40
Figure 40. Figure 40: Response from Qwen2.5-Coder 7B trained with the <code> prior prompt. The model uses Python code to assist in its reasoning process [PITH_FULL_IMAGE:figures/full_fig_p025_40.png]
Figure 41
Figure 41. Figure 41: Response from Qwen2.5 7B trained with the [PITH_FULL_IMAGE:figures/full_fig_p026_41.png]

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Reference graph

Works this paper leans on

28 extracted references · 20 canonical work pages

  1. [1]

    Start by dividing 196 by the smallest prime number, which is 2: 196÷2 = 98 So, 196 = 2 x 98

    work page

  2. [2]

    Next, divide 98 by 2: 98÷2 = 49 So, 196 = 2 x 2 x 49

    work page

  3. [3]

    However, 49 is not divisible by 3 either

    Now, 49 is not divisible by 2, so we move to the next smallest prime number, which is 3. However, 49 is not divisible by 3 either. The next prime number is 5, and 49 is not divisible by 5. The next prime number is 7: 49÷7 = 7 So, 196 = 2 x 2 x 7 x 7

    work page

  4. [4]

    We can write this in exponential form: 196 = 22 ×7 2 To find the number of divisors, we use the formula for the number of divisors based on the prime factorization. If a number n has a prime factorization of pe1 1 ×p e2 2 × · · · ×pek k , then the number of divisorsD(n)is given by: D(n) = (e1 + 1)(e2 + 1)· · ·(ek + 1) For 196, the exponents in the prime f...

    work page

  5. [5]

    Jiacheng Liu, Alisa Liu, Ximing Lu, Sean Welleck, Pe- ter West, Ronan Le Bras, Yejin Choi, and Hannaneh Hajishirzi

    Numinamath: The largest public dataset in ai4maths with 860k pairs of competition math prob- lems and solutions.Hugging Face repository, 13:9. Jiacheng Liu, Alisa Liu, Ximing Lu, Sean Welleck, Pe- ter West, Ronan Le Bras, Yejin Choi, and Hannaneh Hajishirzi. 2022. Generated knowledge prompting for commonsense reasoning. InProceedings of the 60th Annual Me...

    arXiv 2022

  6. [6]

    Understanding R1-Zero-Like Training: A Crit- ical Perspective.Preprint, arXiv:2503.20783. Anton Lozhkov, Raymond Li, Loubna Ben Allal, Fed- erico Cassano, Joel Lamy-Poirier, Nouamane Tazi, Ao Tang, Dmytro Pykhtar, Jiawei Liu, Yuxiang Wei, Tianyang Liu, Max Tian, Denis Kocetkov, Arthur Zucker, Younes Belkada, Zijian Wang, Qian Liu, Dmitry Abulkhanov, Indra...

    arXiv

  7. [7]

    Sewon Min, Xinxi Lyu, Ari Holtzman, Mikel Artetxe, Mike Lewis, Hannaneh Hajishirzi, and Luke Zettle- moyer

    StarCoder 2 and The Stack v2: The Next Generation.Preprint, arXiv:2402.19173. Sewon Min, Xinxi Lyu, Ari Holtzman, Mikel Artetxe, Mike Lewis, Hannaneh Hajishirzi, and Luke Zettle- moyer. 2022. Rethinking the role of demonstrations: What makes in-context learning work? InProceed- ings of the 2022 Conference on Empirical Methods in Natural Language Processin...

    arXiv 2022

  8. [8]

    Qwen2.5 Technical Report.Preprint, arXiv:2412.15115. Kiran Ramnath, Kang Zhou, Sheng Guan, Soumya Sm- ruti Mishra, Xuan Qi, Zhengyuan Shen, Shuai Wang, Sangmin Woo, Sullam Jeoung, Yawei Wang, Haozhu Wang, Han Ding, Yuzhe Lu, Zhichao Xu, Yun Zhou, Balasubramaniam Srinivasan, Qiaojing Yan, Yueyan Chen, Haibo Ding, and 2 others. 2025. A System- atic Survey o...

    arXiv 2025

Show all 28 references
  1. [10]

    Let’s think step by step

    SWE-RL: Advancing LLM Reasoning via Re- inforcement Learning on Open Software Evolution. Preprint, arXiv:2502.18449. Tian Xie, Zitian Gao, Qingnan Ren, Haoming Luo, Yuqian Hong, Bryan Dai, Joey Zhou, Kai Qiu, Zhi- rong Wu, and Chong Luo. 2025. Logic-RL: Unleash- ing LLM Reason...

    2025 arXiv

  2. [11]

    Let’s work backwards from the target. 24/2 = 12. So, 12*2=24

    is used to evaluate general coding ability and serves as an out-of-domain probe. • Knowledge-based question answering: GPQA- Diamond (Rein et al., 2024) evaluates factual knowledge and complex reasoning. We include it to assess whether math-centric training with different prio...

    2025

  3. [16]

    Identify the prime factorization of 196

  4. [17]

    </plan> <answer> To find the number of positive whole-number divisors of 196, we first need to determine its prime factorization

    Use the formula for finding the number of divisors from the prime factorization. </plan> <answer> To find the number of positive whole-number divisors of 196, we first need to determine its prime factorization

  5. [18]

    - 98 is also even, so it is divisible by 2: 98÷2 = 49

    Prime factorization of 196: - 196 is even, so it is divisible by 2:196÷2 = 98. - 98 is also even, so it is divisible by 2: 98÷2 = 49. - 49 is not even, but it is divisible by 7:49÷7 = 7. - 7 is a prime number. So, the prime factorization of 196 is2 2 ×7 2

  6. [19]

    ×pek k , then the number of divisors ofnis given by(e 1 + 1)(e2 + 1)

    Using the formula for finding the number of divisors from the prime factorization: If a number n has a prime factorization of pe1 1 ×p e2 2 ×. . .×pek k , then the number of divisors ofnis given by(e 1 + 1)(e2 + 1). . .(ek + 1). For 196, the exponents in the prime factoriza- t...

  7. [20]

    196 is even, so it is divisible by 2: 196÷2 = 98 So, we have196 = 2×98

  8. [21]

    98 is also even, so it is divisible by 2: 98÷2 = 49 So, we have196 = 2×2×49

  9. [22]

    49 is not even, but it is divisible by 7: 49÷7 = 7 So, we have196 = 2×2×7×7

  10. [23]

    The prime factorization of 196 is2 2 ×7 2

    7 is a prime number, so we stop here. The prime factorization of 196 is2 2 ×7 2. To find the number of positive divisors, we use the formula that involves adding 1 to each of the exponents in the prime factorization and then multiplying the results: (2 + 1)(2 + 1) = 3×3 = 9 Th...

  11. [24]

    Start by dividing 196 by the smallest prime number, which is 2: 196÷2 = 98 So, 2 is a prime factor

  12. [25]

    Divide 98 by 2 again: 98÷2 = 49 So, 2 is a prime factor again

  13. [26]

    However, 49 is not divisible by 3 either

    Now, 49 is not divisible by 2, so we move to the next smallest prime number, which is 3. However, 49 is not divisible by 3 either. The next prime number is 5, but 49 is not divisible by 5. The next prime number is 7: 49÷7 = 7 So, 7 is a prime factor

  14. [27]

    Finally, 7 is a prime number itself, so we divide it by 7: 7÷7 = 1 So, 7 is a prime factor again. Thus, the prime factorization of 196 is: 196 = 22 ×7 2 To find the number of positive whole-number divisors, we use the formula for finding the number of divisors from the prime f...

  15. [28]

    </examples> <answer> 9 </answer> Figure 42: Response from Qwen2.5 7B trained with the <examples> prior prompt

    Therefore, the number of positive divisors is: (2 + 1)(2 + 1) = 3×3 = 9 So, 196 has9 positive whole-number divisors. </examples> <answer> 9 </answer> Figure 42: Response from Qwen2.5 7B trained with the <examples> prior prompt. The model introduces illustrative examples to sup...

  16. [1901]

    Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W

    Curran Associates, Inc. Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W. Cohen. 2023. Program of Thoughts Prompting: Disentangling Computation from Rea- soning for Numerical Reasoning Tasks.Transactions on Machine Learning Research. Zhipeng Chen, Yingqian Min, Beichen Zhang...

    2023 arXiv

  17. [2000]

    In Advances in Neural Information Processing Systems, volume 13

    A Neural Probabilistic Language Model. In Advances in Neural Information Processing Systems, volume 13. MIT Press. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini...

    2020

  18. [2023]

    Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y

    Role play with large language models.Nature, 623(7987):493–498. Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y . K. Li, Y . Wu, and Daya Guo. 2024. DeepSeekMath: Pushing the Limits of Mathemat- ical Reasoning in Open Lan...

    2024 arXiv

  19. [2024]

    InThe Twelfth International Conference on Learning Representations

    Safe RLHF: Safe Reinforcement Learning from Human Feedback. InThe Twelfth International Conference on Learning Representations. DeepSeek-AI, Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, Xiaokang Zhang, X...

  20. [2025]

    Jiazhan Feng, Shijue Huang, Xingwei Qu, Ge Zhang, Yujia Qin, Baoquan Zhong, Chengquan Jiang, Jinxin Chi, and Wanjun Zhong

    DeepSeek-R1: Incentivizing Reasoning Capa- bility in LLMs via Reinforcement Learning.Preprint, arXiv:2501.12948. Jiazhan Feng, Shijue Huang, Xingwei Qu, Ge Zhang, Yujia Qin, Baoquan Zhong, Chengquan Jiang, Jinxin Chi, and Wanjun Zhong. 2025. ReTool: Rein- forcement Learning fo...

    2025 arXiv

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