FKMO 2019 Problem 2

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$ . Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$ . $BF$ and $DE$ meets at $G$ , and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$ . $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$ . $XY$ and $ML$ meets at $P$ , $YZ$ and $MN$ meets at $Q$ . Prove that $BP$ and $DQ$ are parallel.