
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
Let \( A = \{ a_{1}, a_{2}, \cdots, a_{1000} \} \) be a set of positive integers, where \( a_{1} < a_{2} < \cdots < a_{1000} \leq 2014 \), and \( B = \{ a_{i} + a_{j} \mid 1 \leq i, \, j \leq 1000, \, i + j \in A \} \) is a subset of \( A \). Find the number of such sets \( A \) that satisfy the given condition.

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