TopsOJ Summer 2025 - Team Round - Problem 7

Denote a pair of opposite faces of a cube with side length 2 as \mathcal{P} and \mathcal{Q}. O is a point outside the cube such that O, the center of \mathcal{P} and the center of \mathcal{Q} are collinear, in that order. Also, O has a distance of 2 from the center of \mathcal{P}. The cube is rotated all the way around point O such that the center of \mathcal{P} always has a distance of 2 to O, and the center of the cube remains in the same plane. The volume swept out by the cube (i.e., the set of all points that are contained in the interior of the cube as it rotates) is a\pi, for some positive integer a. Compute a.