IGO 2019 A3

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$ , respectively. These two circles intersect at points $X$ and $Y$ . $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$ . Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$ , respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$ . Prove that lines $AM, BN$ and $O_1O_2$ concur.

Proposed by Dominik Burek - Poland