Despite the syntactic fluency of Large Language Models (LLMs), ensuring their logical correctness in high-stakes domains remains a fundamental challenge. We present VERGE, a neurosymbolic framework that combines LLMs with SMT solvers to produce verification-guided answers through iterative refinement. VERGE decomposes LLM outputs into atomic claims, autoformalizes them into first-order logic, and verifies their logical consistency using automated theorem proving. VERGE introduce three key innovations: (1) multi-model consensus via formal semantic equivalence checking to ensure logic-level alignment between candidates, eliminating the syntactic bias of surface-form metrics, (2) semantic routing that directs different claim types to appropriate verification strategies: symbolic solvers for logical claims and LLM ensembles for commonsense reasoning, and (3) precise logical error localization via Minimal Correction Subsets (MCS), which pinpoint the exact subset of claims to revise, transforming binary failure signals into actionable feedback. VERGE framework classifies claims by their logical status and aggregates multiple verification signals into a unified score with variance-based penalty. The system iteratively refines answers using structured feedback until acceptance criteria are met or convergence is achieved. This hybrid approach delivers formal guarantees where possible and consensus verification elsewhere, advancing trustworthy AI. With the GPT-OSS-120B model, VERGE demonstrates an average performance uplift of 18.7% at convergence across a set of reasoning benchmarks compared to single-pass approaches.

Reasoning language models are controlled through explicit modes such as Think and No-think, yet we find that these behaviors are largely governed by a few token-level triggers rather than high-level instructions. Through attention analysis and controlled prompting experiments, we show that a leading “Okay” token induces reasoning behavior, while the newline pattern following ‘</think>‘ suppresses it. Based on this observation, we propose Mid-Think, a simple training-free prompting format that combines these triggers to achieve intermediate-budget reasoning, consistently outperforming fixed-token and prompt-based baselines in terms of the accuracy–length trade-off. Furthermore, applying Mid-Think to RL training after SFT reduces training time by approximately 15% while improving final performance of Qwen3-8B on AIME from 69.8% to 72.4% and on GPQA from 58.5% to 61.1%, demonstrating its effectiveness for both inference-time control and RL-based reasoning training.