Recent advances in large language models (LLMs), make it potentially feasible to automatically refactor source code with LLMs. However, it remains unclear how well LLMs perform compared to human experts in conducting refactorings automatically and accurately. To fill this gap, in this paper, we conduct an empirical study to investigate the potential of LLMs in automated software refactoring, focusing on the identification of refactoring opportunities and the recommendation of refactoring solutions. We first construct a high-quality refactoring dataset comprising 180 real-world refactorings from 20 projects, and conduct the empirical study on the dataset. With the to-be-refactored Java documents as input, ChatGPT and Gemini identified only 28 and 7 respectively out of the 180 refactoring opportunities. However, explaining the expected refactoring subcategories and narrowing the search space in the prompts substantially increased the success rate of ChatGPT from 15.6% to 86.7%. Concerning the recommendation of refactoring solutions, ChatGPT recommended 176 refactoring solutions for the 180 refactorings, and 63.6% of the recommended solutions were comparable to (even better than) those constructed by human experts. However, 13 out of the 176 solutions suggested by ChatGPT and 9 out of the 137 solutions suggested by Gemini were unsafe in that they either changed the functionality of the source code or introduced syntax errors, which indicate the risk of LLM-based refactoring. To this end, we propose a detect-and-reapply tactic, called RefactoringMirror, to avoid such unsafe refactorings. By reapplying the identified refactorings to the original code using thoroughly tested refactoring engines, we can effectively mitigate the risks associated with LLM-based automated refactoring while still leveraging LLM's intelligence to obtain valuable refactoring recommendations.