A Carpenter's Tetradecahedron

Geometry Level 3

The artistic carpenter forms a tetradecahedron from a cube with side length 3". For each of the cube's vertices, he measures an inch out on the edges, and cuts out the resulting right tetrahedron from each corner, leaving 6 octagonal faces, and 8 triangular faces. The volume is \(\frac{a}{b}\) where \(a\) and \(b\) are relatively prime positive integers and the surface area is \(c+d\sqrt{e}\), and \(e\) is prime. What is the sum \(a+b+c+d+e\)?

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