
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 positive integers \(a, b, n\) such that \(\gcd(a, b) = 1\) and \(a + b\) divides \(n\). Let \(S\) be a subset of \(\{1, 2, \ldots, n\}\) such that the difference between any two elements in \(S\) is not \(a\) or \(b\). What is the maximum number of elements that \(S\) can contain?

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