RU-SPB 2020 Problem 113

$BB_1$ is the angle bisector of $\triangle ABC$ , and $I$ is its incenter. The perpendicular bisector of segment $AC$ intersects the circumcircle of $\triangle AIC$ at $D$ and $E$ . Point $F$ is on the segment $B_1C$ such that $AB_1=CF$ .Prove that the four points $B, D, E$ and $F$ are concyclic.