Logic behind cross and dot products

Let $A, B, C,$and $D$ be four distinct points in $3-space$. If $AB×CD$ does not equal $0$ and $AC⋅(AB×CD)=0$, explain why the line through $A$ and $B$ must intersect the line through $C$ and $D$.

Could someone please explain the concept and ideas behind this question?

$$AB × CD \neq 0$$
That means $AB$ and $CD$ are not parallel.
$$AC ⋅ (AB ×CD)=0$$
That means $AC$ is perpendicular to the normal vector $AB ×CD$, so $A,B,C,D$ must be coplanar. But we have shown that $AB$ is not parallel to $CD$, so the two line must intersect at some point on their plane