KöMaL 2023 B5357

Let $ABC$ be a triangle. The tangents at $B$ and $C$ to the circumcircle of $ABC$ intersect at $P$. Let $PB\cap AC=D$ and $PC\cap AB=E$. The perpendicular bisector of the line segment $BC$ intersects $AC$ and $AB$ at $F$ and $G$, respectively. The circumcircles of the triangles $PDF$ and $PEG$ intersect at $M$ for the second time. Let $A'$ be the reflection of $A$ over $FG$ and let $O$ be the circumcenter of the triangle $AFG$. Prove that the line $OA'$ passes through one of the intersections of the circles $MFG$ and $ADE$.