VN-TST 2022 Problem 4

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$ . A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$ , respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$ . $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$ , $IN\perp AB$ . The midperpendiculars of $IK$ intersects $MN$ at $H$ . Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$ .