IGO 2018 I3

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$ , respectively. These two circles intersect each other at points $A$ and $B$ . Line $O_1B$ intersects $\omega_2$ for the second time at point $C$ , and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$ . Also $Y$ is the second intersection point of $BD$ and $\omega_2$ . Prove that $CX = DY$ .

Proposed by Alireza Dadgarnia