KöMaL 2018 B4936

Let $AB$ be a fixed chord that is not a diameter in a circle $k$. The midpoint of $AB$ is $F$. Let $P$ be a point on the circle $k$, different from $A$ and $B$. Let the line $PF$ intersect circle $k$ again at $X$, and let $Y$ be the reflection of $X$ in the perpendicular bisector of $AB$. Prove that there exists a point in the plane that lies on the line $PY$ for all $P$.