IGO 2021 I5

Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$ . Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies, prove that all such lines $XY$ pass through a fixed point, or they are all parallel. Proposed by Josef Tkadlec - Czech Republic