Radius of a Circle, Given Tangent

Geometry Level 5

$P$ is a point outside of circle $\Gamma$ . The tangent from $P$ to $\Gamma$ touches at $A$ . A line from $P$ intersects $\Gamma$ at $B$ and $C$ such that $\angle ACP = 120^\circ$ . If $AC = 16$ and $AP = 19$ , then the radius of $\Gamma$ can be expressed as $\frac {a \sqrt{b} }{c}$ , where $b$ is an integer not divisible by the square of any prime and $a, c$ are coprime positive integers. What is $a + b + c$ ?