KMO 2019 Problem 2

Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$ -excenter of triangle $ABC$ . Let $\Gamma$ the circle centered at $E$ and passes $A$ . $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$ , and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$ . $L$ is the perpendicular foot from $I$ to $AC$ . Now if $AE$ and $DK$ intersects at $F$ , prove that $BE\cdot CI=2\cdot CF\cdot CL$ .