HMMT 2019 T6

Scalene triangle $A B C$ satisfies $\angle A=60^{\circ}$. Let the circumcenter of $A B C$ be $O$, the orthocenter be $H$, and the incenter be $I$. Let $D, T$ be the points where line $B C$ intersects the internal and external angle bisectors of $\angle A$, respectively. Choose point $X$ on the circumcircle of $\triangle I H O$ such that $H X \| A I$. Prove that $O D \perp T X$.