Chord Length Between Tangency Points

A circle is inscribed in $\triangle ABC$ with sides $AB = 4$, $BC = 6$, and $AC = 8$. If $P$ and $Q$ are the respective points of tangency of $\overline{AB}$ and $\overline{AC}$ with the circle, determine the length of chord $PQ$. The answer is in the form $\frac{a\sqrt{b}}{c}$ for where $a$, $b$, $c$ are positive integers, $\gcd(a,c) = 1$, and $b$ is square-free. Input the answer as the sum $a+b+c$.