HMMT 2006 U12

For each positive integer $n$ let $S_{n}$ denote the set $\{1,2,3, \ldots, n\}$. Compute the number of triples of subsets $A, B, C$ of $S_{2006}$ (not necessarily nonempty or proper) such that $A$ is a subset of $B$ and $S_{2006}-A$ is a subset of $C$.