SILK-ROAD 2022 Problem 1

Convex quadrilateral $ABCD$ is inscribed in circle $w.$ Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$ (Kungozhin M.)