4. Let the set of positive integers be
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{1000}\right\},
$$

where, $a_{1}<a_{2}<\cdots<a_{1000} \leqslant 2014$, and
$$
B=\left\{a_{i}+a_{j} \mid 1 \leqslant i, j \leqslant 1000, i+j \in A\right\}
$$

is a subset of $A$. Find the number of sets $A$ that satisfy the condition.
(54th Dutch National Team Selection Exam)
Hint: First prove that the set $A$ can be represented as $B \cup C$, where $B \subseteq \{2001,2002, \cdots, 2014\}, C \subseteq \{1, 2, \cdots, 1000\}$. Then prove that when the set $B$ is determined, the set $C$ is unique. Therefore, the number of sets $A$ that satisfy the condition is equal to the number of subsets of the set $\{2001,2002, \cdots, 2014\}$.