BALTIC-WAY 2025 Problem 12

In an acute triangle $ABC$ with $AB < AC$, the incentre is $I$ and the circumcircle is $\omega$. The line $AI$ intersects the side $BC$ at $D$. Let $T$ be the point on $\omega$ such that $AT \parallel BC$. The line $TI$ intersects $\omega$ again at $P$ and the circumcircle of triangle $APD$ again at $Q$. Prove that $AI = DQ$.