
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 set \( S \) that satisfy the following conditions:
1. Every element in \( S \) is a positive integer not exceeding 100;
2. For any two different elements \( a \) and \( b \) in \( S \), there exists an element \( c \) in \( S \) such that the greatest common divisor of \( a \) and \( c \) is 1, and the greatest common divisor of \( b \) and \( c \) is also 1;
3. For any two different elements \( a \) and \( b \) in \( S \), there exists an element \( d \) in \( S \) (distinct from \( a \) and \( b \)) such that the greatest common divisor of \( a \) and \( d \) is greater than 1, and the greatest common divisor of \( b \) and \( d \) is also greater than 1.

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