62nd Ukrainian National Mathematical Olympiad

The problem gives us a right triangle $ABC$ with a right angle at $ACB$. Let $W_A$ and $W_B$ be the midpoints of the smaller arcs $BC$ and $AC$ of the circumcircle of $△ABC$, and $N_A$ and $N_B$ be the midpoints of the larger arcs $BC$ and $AC$. Let $P$ and $Q$ be the intersection points of segment $AB$ with lines $N_A{W}_B$ and $N_B{W}_A$, respectively. Prove that $AP=BQ$.

Fig. 7