This paper presents SPARK_AI, a prompt-orchestrated system architecture for governing how large language models (LLMs) conduct structured and adaptive reasoning in human–AI interaction. The framework mitigates ad hoc LLM use by replacing direct answer generation with a process-oriented, step-by-step reasoning workflow. We focus on SPARK_AI_MATH, a domain module that supports learners in solving non-routine problem-solving tasks by operationalizing well-established problem-solving phases and guided questioning dialog strategies (Socratic-style prompts), with an optional tool-mediated visualization layer (e.g., GeoGebra). The module implements a five-phase conversational protocol consisting of problem interpretation, analysis of givens, planning, execution, and reflection, together with a controlled hint policy. This design is realized through a stateful system architecture in which each problem instance is maintained as an independent interaction track with a persistent reasoning state. User acceptance was evaluated by first-year mechanical engineering students (N = 108) using an expanded Technology Acceptance Model instrument, and the results were analyzed via PLS-SEM. The findings indicate overall favorable perceptions, with perceived usefulness and learning support emerging as key predictors of intention for continued use. Beyond this specific domain, the SPARK_AI framework enables efficient domain adaptation through localized prompt strategies while preserving a shared cognitive control layer for reasoning-centered human–LLM interaction.