REVES introduces a two-stage iterative framework that enhances LLM reasoning through sequential revision and verification. It achieves +6.5 points over RL baselines and +4.0 points over standard multi-turn training on LiveCodeBench, using a 4B base model with fewer rollouts than large evolutionary systems. The method improves error correction and generalizes to out-of-distribution puzzles like n_queens and mini_sudoku.