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Deep search agents, which aim to answer complex questions requiring reasoning across multiple documents, can significantly speed up the information-seeking process. Collecting human annotations for this application is prohibitively expensive due to long and complex exploration trajectories. We propose an agentic pipeline that automatically generates high quality, difficulty-controlled deep search question-answer pairs for a given corpus and a target difficulty level. Our pipeline, SAGE, consists of a data generator which proposes QA pairs and a search agent which attempts to solve the generated question and provide execution feedback for the data generator. The two components interact over multiple rounds to iteratively refine the question-answer pairs until they satisfy the target difficulty level. Our intrinsic evaluation shows SAGE generates questions that require diverse reasoning strategies, while significantly increases the correctness and difficulty of the generated data. Our extrinsic evaluation demonstrates up to 23% relative performance gain on popular deep search benchmarks by training deep search agents with our synthetic data. Additional experiments show that agents trained on our data can adapt from fixed-corpus retrieval to Google Search at inference time, without further training.