Points, Lines & Planes · Superchallenge problem

Lines $\ell$ and $m$ are skew. Five points are marked on $\ell$, four points are marked on $m$, and one more point $Q$ is marked that is on neither line. Point $Q$ is in general position: it does not lie in any plane that contains $\ell$ and a marked point of $m$, nor in any plane that contains $m$ and a marked point of $\ell$. How many distinct planes pass through at least three of the ten marked points?