Radius of a Sphere Inside a Cube

A cube has edge length 1. A sphere is tangent to three faces of the cube and to a space diagonal of the cube. What is the radius of the sphere? The answer is in the form $\frac{a-\sqrt{b}}{c}$ where $a$, $b$, $c$ are positive integers, and $\gcd(a,c) = 1$. Input the value of $a+b+c$.