APMO 2016 - Asian Pacific Mathematics Olympiad Question 3

Let $AB,AC$ be distinct noncollinear rays. A circle $\omega$ with center $O$ is tangent to ray $AC$ at $E$ and ray $AB$ at $F$. Let $R$ lie on segment $EF$. The line through $O$ parallel to $EF$ meets line $AB$ at $P$. Let $N=PR\cap AC$, and let $M$ be the intersection of $AB$ with the line through $R$ parallel to $AC$. Prove that $MN$ is tangent to $\omega$.

Newly drawn configuration: