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Current robots struggle with long-horizon manipulation tasks requiring sequences of prehensile and non-prehensile skills, contact-rich interactions, and long-term reasoning. We present $\texttt{SPIN}$ ($\textbf{S}$kill $\textbf{P}$lanning to $\textbf{IN}$ference), a framework that distills a computationally intensive planning algorithm into a policy via imitation learning. We propose $\texttt{Skill-RRT}$, an extension of RRT that incorporates skill applicability checks and intermediate object pose sampling for solving such long-horizon problems. To chain independently trained skills, we introduce $\textit{connectors}$, goal-conditioned policies trained to minimize object disturbance during transitions. High-quality demonstrations are generated with $\texttt{Skill-RRT}$ and distilled through noise-based replay in order to reduce online computation time. The resulting policy, trained entirely in simulation, transfers zero-shot to the real world and achieves over 80\% success across three challenging long-horizon manipulation tasks and outperforms state-of-the-art hierarchical RL and planning methods.