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Anonymizing text that contains sensitive information is crucial for a wide range of applications. Existing techniques face the emerging challenges of the re-identification ability of large language models (LLMs), which have shown advanced capability in memorizing detailed information and reasoning over dispersed pieces of patterns to draw conclusions. When defending against LLM-based re-identification, anonymization could jeopardize the utility of the resulting anonymized data in downstream tasks. In general, the interaction between anonymization and data utility requires a deeper understanding within the context of LLMs. In this paper, we propose a framework composed of three key LLM-based components: a privacy evaluator, a utility evaluator and an optimization component, which work collaboratively to perform anonymization. Extensive experiments demonstrate that the proposed model outperforms existing baselines, showing robustness in reducing the risk of re-identification while preserving greater data utility in downstream tasks. We provide detailed studies on these core modules. To consider large-scale and real-time applications, we investigate the distillation of the anonymization capabilities into lightweight models. All of our code and datasets will be made publicly available at [Github URL].