
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 the set of positive integers
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{1000}\right\},
$$
where $ a_{1} < a_{2} < \cdots < a_{1000} \leq 2017 $. If for any $ 1 \leq i, j \leq 1000$, $ i+j \in A $ implies $ a_{i} + a_{j} \in A $, determine the number of sets $ A $ that satisfy these conditions.

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