IGO 2019 I5

Let $ABC$ be a triangle with $\angle A = 60^\circ$ . Points $E$ and $F$ are the foot of angle bisectors of vertices $B$ and $C$ respectively. Points $P$ and $Q$ are considered such that quadrilaterals $BFPE$ and $CEQF$ are parallelograms. Prove that $\angle PAQ > 150^\circ$ . (Consider the angle $PAQ$ that does not contain side $AB$ of the triangle.)

Proposed by Alireza Dadgarnia