Let \Gamma be a circle with center A, radius 1 and diameter B X. Let \Omega be a circle with center C, radius 1 and diameter D Y, where X and Y are on the same side of A C. \Gamma meets \Omega at two points, one of which is Z. The lines tangent to \Gamma and \Omega that pass through Z cut out a sector of the plane containing no part of either circle and with angle 60^{\circ}. If \angle X Y C=\angle C A B and \angle X C D=90^{\circ}, then the length of X Y can be written in the form \frac{\sqrt{a}+\sqrt{b}}{c} for integers a, b, c where \operatorname{gcd}(a, b, c)=1. Find a+b+c.