Large Language Models (LLMs) are increasingly applied to domains that require reasoning about other agents' behavior, such as negotiation, policy design, and market simulation, yet existing research has mostly evaluated their adherence to equilibrium play or their exhibited depth of reasoning. Whether they display genuine strategic thinking, understood as the coherent formation of beliefs about other agents, evaluation of possible actions, and choice based on those beliefs, remains unexplored. We develop a framework to identify this ability by disentangling beliefs, evaluation, and choice in static, complete-information games, and apply it across a series of non-cooperative environments. By jointly analyzing models' revealed choices and reasoning traces, and introducing a new context-free game to rule out imitation from memorization, we show that current frontier models exhibit belief-coherent best-response behavior at targeted reasoning depths. When unconstrained, they self-limit their depth of reasoning and form differentiated conjectures about human and synthetic opponents, revealing an emergent form of meta-reasoning. Under increasing complexity, explicit recursion gives way to internally generated heuristic rules of choice that are stable, model-specific, and distinct from known human biases. These findings indicate that belief coherence, meta-reasoning, and novel heuristic formation can emerge jointly from language modeling objectives, providing a structured basis for the study of strategic cognition in artificial agents.