Process supervision has emerged as a promising approach for enhancing LLM reasoning, yet existing methods fail to distinguish meaningful progress from mere verbosity, leading to limited reasoning capabilities and unresolved token inefficiency.To address this, we propose Stage-aware Hierarchical Advantage via Potential Estimation (SHAPE), a framework that formalizes reasoning as a trajectory through a state space of empirical solvability.SHAPE introduces a hierarchical credit assignment mechanism: at the segment level, it employs a stageaware advantage function to prioritize efficient breakthroughs in low-potential states; at the token level, it utilizes entropy-driven redistribution to sharpen execution signals.Extensive experiments in math reasoning across three base models and five benchmarks demonstrate that SHAPE achieves an average accuracy gain of 3% with 30% reduced token consumption.

This paper studies the problem of using an autonomous underwater vehicle (AUV) to seek the source of some signal in the underwater environment. To avoid the redundant travels for getting the local gradient in existing methods, we propose a novel source-seeking strategy in which the AUV keeps updating an estimated field model using measurements along its gradient-climbing path. Based on the convergence results of the sequential least-squares field estimation algorithm and the path planning method by nonlinear programming, the AUV will finally reach a maximum of the field which indicates a potential source. The way-point tracking control of the AUV is realized by a nonlinear model predictive control (NMPC) scheme embedded with the line-of-sight (LOS) guidance law. The effectiveness and efficiency of the proposed source-seeking strategy is validated in the simulation experiment with a real AUV model.