
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
Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).

Determine the maximum value of \( |S| \).

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