FKMO 2021 Problem 1

An acute triangle $\triangle ABC$ and its incenter $I$ , circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$ , $AC$ in $D$ , $E$ . The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$ . Denote the circumcircle of $DEF$ as $\omega$ , and its center as $K$ . $\omega$ and $FI$ intersect in $P$ ( $\neq F$ ). Prove that $O,K,P$ is collinear.