A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. Prove that $N$ lies on a constant circle, independent of the choice of $P$.