GAIC Math 2024 Problem 43

Divide the set $S=\{1,2,\cdots,2006\}$ into two disjoint subsets $A$ and $B$ such that:

(1) $B \in A$;

(2) If $a\in A$ and $b\in B$ with $a+b\in S$, then $a+b\in B$;

(3) If $a\in A$, $b\in B$, and $a b\in S$, then $a b\in A$.

The number of elements in set $A$ is \underline{\hspace{2cm}}.