OpenAI o1 has shown that applying reinforcement learning to integrate reasoning steps directly during inference can significantly improve a model's reasoning capabilities. This result is exciting as the field transitions from the conventional autoregressive method of generating answers to a more deliberate approach that models the slow-thinking process through step-by-step reasoning training. Reinforcement learning plays a key role in both the model's training and decoding processes. In this article, we present a comprehensive formulation of reasoning problems and investigate the use of both model-based and model-free approaches to better support this slow-thinking framework.