MK-MO 2022 Problem 2

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$ . The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$ , which is different from $C$ . If $B'$ is the reflection of $B$ across $F$ , prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$ .

Proposed by Nikola Velov