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Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is $6$, then find the volume of the resulting polyhedron.

Donatello starts with a marble cube He then slices a pyramid off each corner so that in the resulting polyhedron all the edges have the same side length If the class=