FKMO 2020 Problem 5

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$ . Let $M, L, N$ be the midpoints of segment $BC, AM, AC$ , respectively. The circumcircle of triangle $AMC$ , denoted by $\Omega$ , meets segment $AB$ at $P(\neq A)$ , and the segment $BL$ at $Q$ . Let $O$ be the circumcenter of triangle $BQC$ . Suppose that the lines $AC$ and $PQ$ meet at $X$ , $OB$ and $LN$ meet at $Y$ , and $BQ$ and $CO$ meets at $Z$ . Prove that the points $X, Y, Z$ are collinear.