TR-MO 2021 Problem 4

Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$ , projections of $E$ onto $CA$ and $CB$ are $R$ and $S$ . Let $AP \cap CQ=X$ , $AS \cap BR=Y$ and $BX \cap CY=Z$ . Show that $AZ \perp BC$ .