Volume of a Part of a Cube

Geometry Level 5

Cube ABCDEFGHABCDEFGH, labeled as shown above, has edge length 11 and is cut by a plane passing through vertex DD and the midpoints MM and NN of AB\overline{AB} and CG\overline{CG} respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.

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