IGO 2022 I3

Let $O$ be the circumcenter of triangle $ABC$ . Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$ , respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$ , and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$ .

Proposed by Medeubek Kungozhin, Kazakhstan