Radius of a Circle, Given Tangent

Geometry Level 5

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

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