VMO 2021 Problem 3

Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$ . a) Prove that $AL$ is perpendicular to $EF$ . b) Let $AL$ intersects $EF$ at $M$ , the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$ , $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.