CN-TST 2019 Problem 1

$ABCDE$ is a cyclic pentagon, with circumcentre $O$ . $AB=AE=CD$ . $I$ midpoint of $BC$ . $J$ midpoint of $DE$ . $F$ is the orthocentre of $\triangle ABE$ , and $G$ the centroid of $\triangle AIJ$ . $CE$ intersects $BD$ at $H$ , $OG$ intersects $FH$ at $M$ . Show that $AM\perp CD$ .