HMMT 2017 T5

Let $A B C$ be an acute triangle. The altitudes $B E$ and $C F$ intersect at the orthocenter $H$, and point $O$ denotes the circumcenter. Point $P$ is chosen so that $\angle A P H=\angle O P E=90^{\circ}$, and point $Q$ is chosen so that $\angle A Q H=\angle O Q F=90^{\circ}$. Lines $E P$ and $F Q$ meet at point $T$. Prove that points $A, T, O$ are collinear.