SHARYGIN 2021 Problem 17

Let $ABC$ be an acute-angled triangle. Points $A_0$ and $C_0$ are the midpoints of minor arcs $BC$ and $AB$ respectively. A circle passing though $A_0$ and $C_0$ meet $AB$ and $BC$ at points $P$ and $S$ , $Q$ and $R$ respectively (all these points are distinct). It is known that $PQ\parallel AC$ . Prove that $A_0P+C_0S=C_0Q+A_0R$ .