CN-TST 2022 Problem 4

Let $ABC$ be an acute triangle with $\angle ACB>2 \angle ABC$ . Let $I$ be the incenter of $ABC$ , $K$ is the reflection of $I$ in line $BC$ . Let line $BA$ and $KC$ intersect at $D$ . The line through $B$ parallel to $CI$ intersects the minor arc $BC$ on the circumcircle of $ABC$ at $E(E \neq B)$ . The line through $A$ parallel to $BC$ intersects the line $BE$ at $F$ . Prove that if $BF=CE$ , then $FK=AD$ .