IGO 2021 I3

Given a convex quadrilateral $ABCD$ with $AB = BC$ and $\angle ABD = \angle BCD = 90$ .Let point $E$ be the intersection of diagonals $AC$ and $BD$ . Point $F$ lies on the side $AD$ such that $\frac{AF}{F D}=\frac{CE}{EA}$ .. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$ . Point $L$ is the second intersection of $EF$ and $\omega$ . Prove that the line $KL$ passes through the midpoint of $CE$ . Proposed by Mahdi Etesamifard and Amir Parsa Hosseini - Iran