A piece of paper in the shape of a square $FBHD$ with side $a$ is given. Points $G,A$ on $FB$ and $E,C$ on $BH$ are marked so that $FG=GA=AB$ and $BE=EC=CH$. The paper is folded along $DG,DA,DC$ and $AC$ so that $G$ overlaps with $B$, and $F$ and $H$ overlap with $E$. Compute the volume of the obtained tetrahedron $ABCD$.