Recently, Long Chain-of-Thoughts (CoTs) have gained widespread attention for improving the reasoning capabilities of Large Language Models (LLMs). This necessitates that existing LLMs, which lack the ability to generate Long CoTs, to acquire such capability through post-training methods. Without additional training, LLMs typically enhance their mathematical reasoning abilities through inference scaling methods such as MCTS. However, they are hindered by the large action space and inefficient search strategies, making it challenging to generate Long CoTs effectively. To tackle this issue, we propose constraining the action space and guiding the emergence of Long CoTs through a refined search strategy. In our proposed Constrained Monte Carlo Tree Search (C-MCTS) framework, we limit the actions selected from a constrained action space, which is divided into five disjoint subsets: \emph{understanding}, \emph{planning}, \emph{reflection}, \emph{coding}, and \emph{summary}. Each subset is further constrained to a small number of predefined prompts, rather than allowing LLMs to generate actions arbitrarily. Additionally, we refine the search strategy by incorporating prior knowledge about the action sets, such as a human-like partial order of the action subsets and the pretrained process reward models. These strategies work together to significantly reduce the vast search space of Long CoTs. Extensive evaluations on mathematical reasoning benchmarks show that, under zero-shot settings, our method enables the 7B model to achieve reasoning capabilities that surpass those of the 72B model.

Addressing the challenge of high annotation costs in solving Math Word Problems (MWPs) through full supervision with intermediate equations, recent works have proposed weakly supervised task settings that rely solely on the final answer as a supervised signal. Existing leading approaches typically employ various search techniques to infer intermediate equations, but cannot ensure their semantic consistency with natural language descriptions. The rise of Large Language Models (LLMs) like ChatGPT has opened up new possibilities for addressing MWPs directly. However, the computational demands of LLMs make them less than ideal for use in settings where resources are tight. In light of these challenges, we introduce an innovative two-stage framework that adeptly transfers mathematical Expertise from large to tiny language models. In \emph{Distillation Stage}, we propose a series of extraction processes that satisfy the properties of MWPs to distill mathematical knowledge from LLMs to construct problem-equation pairs required for supervised training. In \emph{Refinement Stage}, Due to Knowledge distilling method cannot guarantee the full utilization of all data, we further utilize the unsuccessfully searched data effectively by Knowledge Refine method. Finally, We train a small model using distilled data generated through two-stage methods. As our method fully leverages the semantic understanding capabilities during the searching 'problem-equation' pair, it demonstrates significantly improved performance on the Math23K and Weak12K datasets compared to existing small model methods, while maintaining a much lower computational cost than ChatGPT.