The 37th Korean Mathematical Olympiad Final Round

Let $O$ be the circumcircle of a triangle $ABC$. Choose a point $D(\ne A)$ on the extension of segment $BA$ towards $A$. Points $E$ and $F$ are two distinct points on circle $O$ such that lines $DE$ and $DF$ are both tangent to circle $O$. Segment $EF$ intersects side $CA$ at point $T(\ne C)$. Choose a point $P(\ne B,C)$ on the arc $BC$ of circle $O$ that doesn't contain point $A$. Let the intersection point of line $DP$ and circle $O$ be $Q(\ne P)$. Lines $BQ$ and $DT$ intersect at a point $X$ (other than $Q$). Let $Y(\ne P)$ be the intersection point of line $PT$ and circle $O$. Prove that the points $C,X,\text{and }Y$ are collinear.