Let \omega_{1} and \omega_{2} be intersecting circles in the plane with radii 12 and 15, respectively. Suppose \Gamma is a circle such that \omega_{1} and \omega_{2} are internally tangent to \Gamma at X_{1} and X_{2}, respectively. Similarly, \ell is a line that is tangent to \omega_{1} and \omega_{2} at Y_{1} and Y_{2}, respectively. If X_{1} X_{2}=18 and Y_{1} Y_{2}=9, what is the radius of \Gamma?