
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
Find the maximum number of elements in the set \( S \) that satisfies the following conditions:
1. Each element in \( S \) is a positive integer not exceeding 100.
2. For any two distinct numbers \( a \) and \( b \) in \( S \), there exists a number \( c \) in \( S \) such that the greatest common divisor (GCD) of \( a \) and \( c \) is 1, and the GCD of \( b \) and \( c \) is also 1.
3. For any two distinct numbers \( a \) and \( b \) in \( S \), there exists a number \( d \) in \( S \) such that the GCD of \( a \) and \( d \) is greater than 1, and the GCD of \( b \) and \( d \) is also greater than 1.
(2003 CMO problem)

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