3D Marion's Theorem

Geometry Level 3

This problem can be viewed as the 3D analog of Marion's theorem.

Imagine that each edge of a tetrahedron is trisected. Then, through each of these 12 points and its two opposite vertices, a plane is constructed for a total of 12 planes.

Now, let \(V\) denote the volume of the tetrahedron, and \(V_M\) the volume of the 3D figure carved out by the 12 planes inside the tetrahedron. If \(V_M=\frac{A}{B}V,\) where \(A\) and \(B\) are coprime positive integers, find \(A+B.\)

The 3D figure in question is shown below:


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