CGMO 2021 Problem 7

In an acute triangle $ABC$ , $AB \neq AC$ , $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$ . $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$ . Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$ .

Proof that $B, C, Y, O, Z$ lie on a circle.