The "Reason-Then-Respond" paradigm, enhanced by Reinforcement Learning, has shown great promise in advancing Multimodal Large Language Models. However, its application to the video domain has led to specialized models that excel at either question answering (QA) or captioning tasks, but struggle to master both. Naively combining reward signals from these tasks results in mutual performance degradation, which we attribute to a conflict between their opposing task natures. To address this challenge, we propose a novel training framework built upon two intermediate proxy tasks: DarkEventInfer, which presents videos with masked event segments, requiring models to infer the obscured content based on contextual video cues; and MixVidQA, which presents interleaved video sequences composed of two distinct clips, challenging models to isolate and reason about one while disregarding the other. These proxy tasks compel the model to simultaneously develop both holistic, divergent understanding and precise, convergent reasoning capabilities. Embodying this framework, we present VidBridge-R1, the first versatile video reasoning model that effectively bridges the paradigm conflict. Extensive experiments show that VidBridge-R1 achieves significant performance gains on both QA and captioning within one model, demonstrating the efficacy of our approach in fostering more generalizable and powerful video understanding models.