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Large Language Models (LLMs) demonstrate strong generalization and reasoning abilities, making them well-suited for complex decision-making tasks such as medical consultation (MC). However, existing LLM-based methods often fail to capture the dual nature of MC, which entails two distinct sub-tasks: symptom inquiry, a sequential decision-making process, and disease diagnosis, a classification problem. This mismatch often results in ineffective symptom inquiry and unreliable disease diagnosis. To address this, we propose DDO, a novel LLM-based framework that performs Dual-Decision Optimization by decoupling the two sub-tasks and optimizing them with distinct objectives through a collaborative multi-agent workflow. Experiments on three real-world MC datasets show that DDO consistently outperforms existing LLM-based approaches and achieves competitive performance with state-of-the-art generation-based methods, demonstrating its effectiveness in the MC task. The code is available at https://github.com/zh-jia/DDO.