Given a unit cube, extend 3 of its edges into non-intersecting, non-parallel lines (in blue) in 3D space. Then five and **only** five spheres of the same radius are each tangent to all 3 of the lines.

What is the radius of each of these spheres?

The figure at right illustrates a sphere tangent to the 3 blue lines, but there can only be **one** such sphere with that radius. (Remember, we need **five** identical spheres satisfying the tangency condition!)

Suppose the radius of each of the five spheres can be expressed as \(\frac{a}{b}\sqrt{c},\) where integers \(a\) and \(b\) are coprime and integer \(c\) is square-free.

What is the product \(a \times b \times c?\)

**Note:** Spheres may intersect one another.

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