IGO 2021 A2

Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$ . A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$ . The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$ . Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$ , and $2\angle AFC=\angle ABC$ . Prove that the tangent at $F$ to $\Gamma_2$ , and lines $BD$ and $CE$ are concurrent.