this is my full latex , create a bibilography to add in bibtex from the given sources as well as customize this  % !TEX root =  ../Dissertation.tex  \chapter{Introduction}  Large language models LLMs have shown a remarkable ability to perform complex reasoning tasks when prompted to generate intermediate reasoning steps, a technique known as chain-of-thought prompting. This approach has been shown to be effective for a range of tasks, including arithmetic, commonsense, and symbolic reasoning. Chain-of-thought prompting involves providing the LLM with a few examples of a reasoning task, along with the corresponding chain of thought and the final answer. This allows the LLM to learn the process of decomposing complex problems into smaller, more manageable steps. However, the reasoning paths generated by LLMs can be sensitive to the specific examples used in the prompt, as well as the model's training data. This can lead to a lack of diversity in the generated reasoning traces, which can limit the ability of the LLM to generalize to new problems. This project focuses on a novel approach to enhance the diversity and robustness of mathematical reasoning in LLMs. Instead of relying on a single LLM or a single set of chain-of-thought examples, we propose to leverage the collective intelligence of multiple specialized math LLMs trained on different corpora.    Specifically, our approach involves the following steps: \begin{itemize}     \item Sampling Multiple Chain-of-Thought Answers: We will employ different math LLMs, each trained on diverse datasets, to generate multiple chain-of-thought answers for a given mathematical problem. Each LLM's unique training data and architectural biases will contribute to a distinct reasoning trace.     \item Creating a Union of Reasoning Traces: The diverse chain-of-thought answers from different LLMs will be combined to create a rich and comprehensive set of reasoning traces. This union will capture a wider range of problem-solving approaches and logical steps.     \item Finetuning a Base LLM: A base LLM will be finetuned on the combined set of reasoning traces. This finetuning process will allow the base model to learn from the diverse reasoning patterns and improve its ability to generate coherent and accurate solutions.     \item By combining the strengths of multiple specialized math LLMs, we aim to overcome the limitations of individual models and achieve more robust and generalizable mathematical reasoning capabilities. This approach has the potential to significantly improve the performance of LLMs on a wide range of mathematical tasks and contribute to the development of more intelligent and reliable AI systems. \end{itemize}   In the context of reinforcement learning (RL), where the LLM learns to generate reasoning traces as actions within a mathematical problem-solving environment, the reward function plays a critical role in guiding the model towards generating correct and efficient solutions. A verifier, as discussed in the MuMath-Code paper, is crucial for training a robust reward model.  A strong reward model is essential for effective RL training. A verifier, trained to assess the correctness of individual reasoning steps or the overall solution, can provide a reliable signal to the reward model. The more accurate the verifier, the stronger the reward model, and consequently, the better the performance of the LLM in generating correct and coherent reasoning traces.  \begin{itemize}     \item Chain of Thought with Python Code and Self-Verification: Incorporating Python code excecution interleaved with rationale of natural language, as proposed in the MuMath-Code paper, offers a powerful mechanism for self-verification and error pinpointing. By integrating code execution within the reasoning process, the LLM can leverage the precision and logical consistency of code to validate its reasoning.     \item For instance, the LLM can generate a chain of thought that interleaves natural language reasoning with Python code snippets. Each code snippet can be executed using an external Python interpreter, and the results can be used to verify the correctness of the preceding reasoning steps.     \item Utilizing Python Variables for Error Pinpointing: The variables used within the Python code snippets provide a precise means of pinpointing errors within the reasoning trace. If a code snippet throws an error or produces an unexpected output, the values of the variables at that point in the chain of thought can be examined to identify the specific step where the reasoning went wrong.     \item Synergy of Multiple Reasoning Traces and Verification: The use of multiple reasoning traces, generated by different specialized math LLMs, can further enhance the effectiveness of the verifier. By comparing and contrasting the verification results across multiple traces, the model can gain a more robust understanding of the potential pitfalls and develop more sophisticated error correction strategies.       \end{itemize} We will show a method to generate dataset for training of self correcting LLM especially for mathametical domain where the problems can be verified on the fly and granular level feedback can be given at which step the solution goes wrong and then can be backtracked.