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Humans use a remarkable set of strategies to manipulate objects in clutter. We pick up, push, slide, and sweep with our hands and arms to rearrange clutter surrounding our primary task. But our robots treat the world like the Tower of Hanoi — moving with pick-and-place actions and fearful to interact with it with anything but rigid grasps. This produces inefficient plans and is often inapplicable with heavy, large, or otherwise ungraspable objects. We introduce a framework for planning in clutter that uses a library of actions inspired by human strategies. The action library is derived analytically from the mechanics of pushing and is provably conservative. The framework reduces the problem to one of combinatorial search, and demonstrates planning times on the order of seconds. With the extra functionality, our planner succeeds where traditional grasp planners fail, and works under high uncertainty by utilizing the funneling effect of pushing. We demonstrate our results with experiments in simulation and on HERB, a robotic platform developed at the Personal Robotics Lab at Carnegie Mellon University.