DE-BUNDESWETTBEWERB 2020 Problem 7

Two lines $m$ and $n$ intersect in a unique point $P$ . A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$ , but not at the same time.

Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.