March Break 2024 - Problem 8

Consider a sphere tangent to four faces of a unit octahedron that share a vertex. Suppose this sphere passes through the opposite vertex. If the radius of this sphere can be expressed as \frac{\sqrt{a}-\sqrt{b}}{c}, where a, b, and c are positive integers, and a, b are not divisible by the square of any prime, find 100a+10b+c.