PRIME (Planning and Retrieval-Integrated Memory for Enhanced Reasoning) is a multi-agent inference framework inspired by the dual-processing theory of human cognitive processes that dynamically integrates System 1 (fast, intuitive thinking) and System 2 (slow, deliberate thinking). PRIME first uses a Quick Thinking Agent (System 1) to generate rapid answers. When uncertainty is detected, it triggers a structured System 2 inference pipeline comprised of specialized agents for planning, hypothesis generation, retrieval, information integration, and decision-making. Experimental results using the LLaMA-3 model demonstrate that PRIME enables open-source LLM to perform competitively with state-of-the-art closed-source models such as GPT-4 and GPT-4o.