ISL 2020 G3

Let $A B C D$ be a convex quadrilateral with $\angle A B C>90^{\circ}, \angle C D A>90^{\circ}$, and $\angle D A B=\angle B C D$. Denote by $E$ and $F$ the reflections of $A$ in lines $B C$ and $C D$, respectively. Suppose that the segments $A E$ and $A F$ meet the line $B D$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $B E K$ and $D F L$ are tangent to each other. (Slovakia)