Large reasoning models achieve impressive performance on challenging tasks but often incur substantial computational and latency costs due to excessive reasoning. This talk presents a principled framework for safe and efficient online reasoning that dynamically routes queries between thinking and non-thinking models while rigorously controlling performance degradation. The proposed Betting PAC (B-PAC) framework combines uncertainty-aware routing, inverse propensity scoring, and betting-based supermartingale methods to provide anytime-valid guarantees under partial feedback and non-stationary data streams. Theoretical results establish distribution-free control of performance loss together with efficient adaptive threshold selection. Empirical studies on benchmarks including MATH, MMLU-Pro, BIG-Bench Hard, and Magpie demonstrate substantial reductions in reasoning cost while maintaining user-specified reliability guarantees.