Chest X-ray (CXR) imaging is one of the most widely used diagnostic modalities in clinical practice, encompassing a broad spectrum of diagnostic tasks. Recent advancements have seen the extensive application of reasoning-based multimodal large language models (MLLMs) in medical imaging to enhance diagnostic efficiency and interpretability. However, existing multimodal models predominantly rely on "one-time" diagnostic approaches, lacking verifiable supervision of the reasoning process. This leads to challenges in multi-task CXR diagnosis, including lengthy reasoning, sparse rewards, and frequent hallucinations. To address these issues, we propose CX-Mind, the first generative model to achieve interleaved "think-answer" reasoning for CXR tasks, driven by curriculum-based reinforcement learning and verifiable process rewards (CuRL-VPR). Specifically, we constructed an instruction-tuning dataset, CX-Set, comprising 708,473 images and 2,619,148 samples, and generated 42,828 high-quality interleaved reasoning data points supervised by clinical reports. Optimization was conducted in two stages under the Group Relative Policy Optimization framework: initially stabilizing basic reasoning with closed-domain tasks, followed by transfer to open-domain diagnostics, incorporating rule-based conditional process rewards to bypass the need for pretrained reward models. Extensive experimental results demonstrate that CX-Mind significantly outperforms existing medical and general-domain MLLMs in visual understanding, text generation, and spatiotemporal alignment, achieving an average performance improvement of 25.1% over comparable CXR-specific models. On real-world clinical dataset (Rui-CXR), CX-Mind achieves a mean recall@1 across 14 diseases that substantially surpasses the second-best results, with multi-center expert evaluations further confirming its clinical utility across multiple dimensions.