
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Given a set of positive integers
$$
A = \{a_1, a_2, \ldots, a_{1000}\}
$$
such that $a_1 < a_2 < \cdots < a_{1000} \leq 2014$, and
$$
S_A = \{a_i + a_j \mid 1 \leq i, j \leq 1000, i+j \in A\}
$$
is a subset of $A$. Determine the number of such sets $A$.

Present the answer in LaTex format: \boxed{Your answer}
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