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 \( a + b + c\)?

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