$\omega$ is a circle with diameter $AB$. Points $C, D$ lie on $\omega$ such that $C, D$ are on different sides of $AB$. A line passing through $C$ and parallel to $AD$ cuts $AB$ at $F$, and a line passing through $D$ and parallel to $AC$ cuts $AB$ at $E$. $A, B, C$ and $D$ are in a way that $E, F$ are inside $\omega$. The line passing through $E$ and perpendicular to $AB$ cuts $BD$ at $X$ and the line passing through $F$ and perpendicular to $AB$ cuts $BC$ at $Y$. Prove that the perimeter of triangle $\triangle AXY$ equals to $2CD$.