OIM 2021 Problem 2

Consider an acute-angled triangle $ABC$ , with $AC>AB$ , and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$ , respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$ , with $X\neq C$ . The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$ . Let $P$ be the point on segment $AB$ so that $YP = YA$ , with $P\neq A$ , and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$ .