Imagine a cube with a positive integer written on each of its six faces. Now imagine cutting the vertices of this cube; the resulting solid is known as a truncated cube, or truncated hexahedron, and is shown in the image. The six original squares are now octagons, but the integers remain unchanged, and the eight vertices have become triangles. Let the number on each triangular face be the product of the three numbers written on the octagons that share an edge with it.

The sum of the numbers on all the triangles is equal to 6006. Find the smallest possible sum of the original six numbers written on the faces of the cube.

[This question has been adapted from 104 Number Theory Problems: From the Training of the USA IMO Team]

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