
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
Determine the maximum number of elements in the set \( S \) that satisfy the following conditions:
1. Each element in \( S \) is a positive integer not exceeding 100;
2. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is 1;
3. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is greater than 1.

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