Agentic search systems must coordinate evidence acquisition and response generation, yet existing approaches either couple both roles under a single agent objective or decompose them without disentangling their respective contributions to the final outcome. We introduce DAC (Divide and Cooperate), a role-decomposed training framework that, given task-specific external verification signals, trains a searcher and a generator with role-specific cross-verification rewards. DAC allows the generator to abstain when the retrieved evidence appears insufficient, and uses this decision together with externally evaluated search sufficiency to assign appropriate credit to each role. To prevent degenerate over-abstention, we further introduce hard-positive evidence augmentation, which discourages abstaining on sufficient but hard evidence. Across seven general and multi-hop QA benchmarks and two model backbones, DAC consistently outperforms strong single-agent and multi-agent baselines. Controlled evaluations show that these gains arise from coordinated improvements in both search and generation. Our results highlight the importance of explicitly and properly assigning credit across interacting roles when training agentic search systems.

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