Let A B C be a triangle with sides A B=34, B C=15, A C=35 and let \Gamma be the circle of smallest possible radius passing through A tangent to B C. Let the second intersections of \Gamma and sides A B, A C be the points X, Y. Let the ray X Y intersect the circumcircle of the triangle A B C at Z. If A Z=\frac{p}{q} for relatively prime integers p and q, find p+q.