Let $ABC$ be an acute triangle with circumcenter $O$ and incenter $I$. Points $E$, $M$
lie on $AC$ and $F$, $N$ on $AB$ so that $BE \perp AC$,
$CF\perp AB$, $\angle ABM = \angle CBM$ and
$\angle ACN = \angle BCN$. Prove that $I$ lies on $EF$ if and only if $O$ lies on $MN$.

I honestly can't make any progress. I tried playing around on geogebra but it seems like when I get a configuration where $O$ and $I$ are on the desired lines, $O$ and $I$ stay on these lines as I move $A$ around the circumcenter...