In an acute-angled triangle $ ABC$, $ \omega$ is the circumcircle and $ O$ its center, $ \omega _1$ the circumcircle of triangle $ AOC$, and $ OQ$ the diameter of $ \omega _1$. Let $ M$ and $ N$ be points on the lines $ AQ$ and $ AC$ respectively such that the quadrilateral $ AMBN$ is a parallelogram. Prove that the lines $ MN$ and $ BQ$ intersect on $ \omega _1$.