Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$.

Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.