Let I be the incenter of a triangle A B C with A B=20, B C=15, and B I=12. Let C I intersect the circumcircle \omega_{1} of A B C at D \neq A. Alice draws a line l through D that intersects \omega_{1} on the minor arc A C at X and the circumcircle \omega_{2} of A I C at Y outside \omega_{1}. She notices that she can construct a right triangle with side lengths I D, D X, and X Y. What is the length of I Y?