RU-ALLRUSSIAN 2022 Problem 102

On side $BC$ of an acute triangle $ABC$ are marked points $D$ and $E$ so that $BD = CE$ . On the arc $DE$ of the circumscribed circle of triangle $ADE$ that does not contain the point $A$ , there are points $P$ and $Q$ such that $AB = PC$ and $AC = BQ$ . Prove that $AP=AQ$ .