NMC 2021 Problem 4

Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $A B C D$ is a convex quadrilateral. Suppose that $A B$ and $C D$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $B D$ and $A C$ intersect at a point $F$. Let $X \neq D$ be the point on $\omega$ such that $D X$ and $E F$ are parallel. Let $Y$ be the reflection of $D$ through $E F$ and suppose that $Y$ is inside the circle $\omega$.

Show that $A, X$, and $Y$ are collinear.