Circles \omega_{1} and \omega_{2} are externally tangent to each other. Circle \Omega is placed such that \omega_{1} is internally tangent to \Omega at X while \omega_{2} is internally tangent to \Omega at Y. Line \ell is tangent to \omega_{1} at P and \omega_{2} at Q and furthermore intersects \Omega at points A and B with A P<A Q. Suppose that A P=2, P Q=4, and Q B=3. Compute the length of line segment \overline{X Y}.