MathNet 2023-part-06 — 191 problems (EN–VI) Question 167

$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If the point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\frac{AM}{AO}$.


Figure 21