In an acute scalene triangle $ABC$, the incircle $\omega$ touches the sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $P$ be the foot of the perpendicular from $F$ to $DE$. The line $BP$ intersects segment $AC$ at $K$, and the line $AP$ intersects segment $BC$ at $L$. The altitude through vertex $C$ in $\triangle ABC$ intersects the circumcircle of $\triangle CKL$ at a point $Q$. Prove that line $PQ$ passes through the center of $\omega$.