Triangle $ABC$ has circumcircle $\Omega$ and incircle $\omega$, where $\omega$ is tangent to $BC, CA, AB$ at $D,E,F$, respectively. $T$ is an arbitrary point on $\omega$. $EF$ meets $BC$ at $K$, $AT$ meets $\Omega$ again at $P$, $PK$ meets $\Omega$ again at $S$. $X$ is a point on $\Omega$ such that $S, D, X$ are colinear. Let $Y$ be the intersection of $AX$ and $EF$, prove that $YT$ is tangent to $\omega$. Proposed by chengbilly