Large Language Models demonstrate strong capabilities in single-turn instruction following but suffer from Lost-in-Conversation (LiC), a degradation in performance as information is revealed progressively in multi-turn settings. Motivated by the current progress on Reinforcement Learning with Verifiable Rewards (RLVR), we propose Curriculum Reinforcement Learning with Verifiable Accuracy and Abstention Rewards (RLAAR), a framework that encourages models not only to generate correct answers, but also to judge the solvability of questions in the multi-turn conversation setting. Our approach employs a competence-gated curriculum that incrementally increases dialogue difficulty (in terms of instruction shards), stabilizing training while promoting reliability. Using multi-turn, on-policy rollouts and a mixed-reward system, RLAAR teaches models to balance problem-solving with informed abstention, reducing premature answering behaviors that cause LiC. Evaluated on LiC benchmarks, RLAAR significantly mitigates LiC performance decay (62.6% to 75.1%) and improves calibrated abstention rates (33.5% to 73.4%). Together, these results provide a practical recipe for building multi-turn reliable and trustworthy LLMs.

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