EGMO 2024 Problem 2

Let $ABC$ be a triangle with $AC > AB$, and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC, CA, AB$ at $D, E, F$ respectively. Let $X$ and $Y$ be two points on minor arcs $\widehat{DF}$ and $\widehat{DE}$ of the incircle, respectively, such that $\angle BXD = \angle DYC$. Let line $XY$ meet line $BC$ at $K$. Let $T$ be the point on $\Omega$ such that $KT$ is tangent to $\Omega$ and $T$ is on the same side of line $BC$ as $A$. Prove that lines $TD$ and $AI$ meet on $\Omega$.