Olympiad Geometry · Superchallenge problem

A pyramid $VABCD$ has a square base $ABCD$ and all eight of its edges have length $6$. Let $M$ and $N$ be the midpoints of $\overline{VC}$ and $\overline{VD}$. The plane through $A$, $B$, $M$ and $N$ cuts the pyramid into two pieces. Find the volume of the piece that contains $V$, in simplest radical form.