Circle in the Middle

Three mutually externally tangent circles have radii $3$, $4$, and $3$. A fourth, smaller circle sits in the curved gap between them, tangent to all three. Find its radius. The answer is in the form $\frac{-a+\sqrt{b}}{c}$ for where $a$, $b$, $c$ are positive integers and $\gcd(a,c) = 1$. Input the answer as the sum $a+b+c$.