
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 a non-empty subset \( S \) of the set \(\{1,2, \cdots, 108\}\), such that:
(i) For any two elements \(a\) and \(b\) in \(S\) (where \(a\) and \(b\) can be the same), there exists \(c \in S\) such that \(\gcd(a, c) = 1\) and \(\gcd(b, c) = 1\).
(ii) For any two elements \(a\) and \(b\) in \(S\) (where \(a\) and \(b\) can be the same), there exists \(c' \in S\), different from \(a\) and \(b\), such that \(\gcd(a, c') > 1\) and \(\gcd(b, c') > 1\).

Determine the maximum possible number of elements in \(S\).

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