In this paper, we investigate the usage of large language models (LLMs) to improve the performance of competitive speech recognition systems. Different from previous LLM-based ASR error correction methods, we propose a novel multi-stage approach that utilizes uncertainty estimation of ASR outputs and reasoning capability of LLMs. Specifically, the proposed approach has two stages: the first stage is about ASR uncertainty estimation and exploits N-best list hypotheses to identify less reliable transcriptions; The second stage works on these identified transcriptions and performs LLM-based corrections. This correction task is formulated as a multi-step rule-based LLM reasoning process, which uses explicitly written rules in prompts to decompose the task into concrete reasoning steps. Our experimental results demonstrate the effectiveness of the proposed method by showing 10% ~ 20% relative improvement in WER over competitive ASR systems -- across multiple test domains and in zero-shot settings.