Distance for 3D Objects

Geometry Level 4

A sphere is inscribed in a regular tetrahedron with side length of \(2\) such that it is tangent to each of the tetrahedron's four faces. Let points \(P\) and \(Q\) be two of the points of tangency. If \(d\) is the distance between \(P\) and \(Q\), then \(d^6\) can be expressed as \(\dfrac{m}{n}\), where \(m\) and \(n\) are positive, coprime integers. Find \(m+n\).

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