
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 \( S \) be a subset of the set \(\{1, 2, 3, \ldots, 2015\} \) such that for any two elements \( a, b \in S \), the difference \( a - b \) does not divide the sum \( a + b \). Find the maximum possible size of \( S \).

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