Two non-intersecting circles, \omega and \Omega, have centers C_{\omega} and C_{\Omega} respectively. It is given that the radius of \Omega is strictly larger than the radius of \omega. The two common external tangents of \Omega and \omega intersect at a point P, and an internal tangent of the two circles intersects the common external tangents at X and Y. Suppose that the radius of \omega is 4, the circumradius of \triangle P X Y is 9, and X Y bisects \overline{P C_{\Omega}}. Compute X Y.