Sequential recommendation has become indispensable in modern digital services. Prevalent recommendation techniques formulate the recommendation task with a language instruction fed into large language models (LLMs) to generate recommendations. However, the implicit interaction scenario of recommendation task cannot provide explicit reasoning supervision to activate LLM's multi-step reasoning capability. Besides, the manner of reasoning for enhancing recommendation is still underexplored. Therefore, we investigate activating multi-step reasoning with users' interactions and propose a multi-step reasoning-enhanced LLM (MSR-Rec), which tightly integrates reasoning with recommendation from designing reasoning chain to reasoning-based recommendation. A task-decomposed reasoning chain is elaborately designed to imitate users' thinking process, seamlessly involving reasoning into recommendation. Following the reasoning chain, MSR-Rec synthesizes reasoning supervision and fine-tunes LLM to adapt for task-specific reasoning. In inference, bidirectional reasoning is implemented from user and item sides, performing a closed-loop reasoning for recommendation. Comprehensive experiments demonstrate that MSR-Rec achieves the state-of-the-art performance in both recommendation quality and reasoning interpretability, advancing the integration of reasoning and recommendation in LLM-based systems.