KöMaL 2018 A733

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\ne X$ and $B\ne X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.