PMO 2021 Problem 8

In right triangle $ABC$ , $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$ . $M$ is the midpoint of $AB$ , $P$ is the foot of the altitude from $C$ , and $N$ is the midpoint of $CP$ . Line $AB$ meets the circumcircle of $CNB$ again at $Q$ . $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$ , and $V$ lies on segment $SP$ such that $AV = VP$ . Line $SP$ meets the circumcircle of $CPB$ again at $T$ . $W$ lies on ray $VA$ past $A$ such that $2AW = ST$ , and $O$ is the circumcenter of $SPM$ . Prove that lines $OM$ and $BW$ are perpendicular.