MK-MO 2017 Problem 4

Let $O$ be the circumcenter of the acute triangle $ABC$ ( $AB < AC$ ). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$ , respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$ , respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$ . Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$ .