MathNet 2013 Question 4

Let $ABC$ be an acute triangle. Circle $\omega_1$, with diameter $AC$, intersects side $BC$ at $F$ (other than $C$). Circle $\omega_2$, with diameter $BC$, intersects side $AC$ at $E$ (other than $C$). Ray $AF$ intersects $\omega_2$ at $K$ and $M$ with $AK \lt AM$. Ray $BE$ intersects $\omega_1$ at $L$ and $N$ with $BL \lt BN$. Prove that lines $AB$, $ML$, $NK$ are concurrent.

(This problem was suggested by Steve Dinh.)