BALTIC-WAY 2018 Problem 13

The bisector of the angle $A$ of a triangle $A B C$ intersects $B C$ in a point $D$ and intersects the circumcircle of the triangle $A B C$ in a point $E$. Let $K, L, M$ and $N$ be the midpoints of the segments $A B, B D, C D$ and $A C$, respectively. Let $P$ be the circumcenter of the triangle $E K L$, and $Q$ be the circumcenter of the triangle $E M N$. Prove that $\angle P E Q=\angle B A C$.