This paper addresses two persistent challenges in sequential recommendation: (i) evidence insufficiency-cold-start sparsity together with noisy, length-varying item texts; and (ii) opaque modeling of dynamic, multi-faceted intents across long/short horizons. We propose R3-REC (Reasoning-Retrieval-Recommendation), a prompt-centric, retrieval-augmented framework that unifies Multi-level User Intent Reasoning, Item Semantic Extraction, Long-Short Interest Polarity Mining, Similar User Collaborative Enhancement, and Reasoning-based Interest Matching and Scoring. Across ML-1M, Games, and Bundle, R3-REC consistently surpasses strong neural and LLM baselines, yielding improvements up to +10.2% (HR@1) and +6.4% (HR@5) with manageable end-to-end latency. Ablations corroborate complementary gains of all modules.