RIOPLATENSE 2018 Problem 2

Let $P$ be a point outside a circumference $\Gamma$ , and let $PA$ be one of the tangents from $P$ to $\Gamma$ . Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$ , with $B$ between $P$ and $C$ . Let $D$ be the symmetric of $B$ with respect to $P$ . Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$ . Line $EB$ cuts back to $\omega _1$ in $F$ . Prove that $CF = AB$ .