Ukrainian Mathematical Olympiad

Two circles ${\omega }_1$ and ${\omega }_2$ intersect each other at two distinct points $A$ and $B$. The tangent line of the circle ${\omega }_1$ at the point $A$ and the tangent line of the circle ${\omega }_2$ at the point $B$ meet at point $C$. The first one of these two lines intersects the circle ${\omega }_2$ for the second time at point $T$ (distinct from $A$). A point $X$ (distinct from $A$ and $B$) belongs to the circle ${\omega }_1$, and the straight line $XA$ intersects the circle ${\omega }_2$ for the second time at point $Y$ (distinct from $A$). The straight lines $YB$ and $XC$ meet at point $Z$. Prove that $TZ$ is parallel to $XY$.