
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
Let \( S \) be a non-empty subset of the set \( \{1, 2, \cdots, 108\} \), satisfying:
(i) For any two distinct numbers \( a \) and \( b \) in \( S \), there exists \( c \in S \) such that \( (a \cdot c) = (b \cdot c) = 1 \);
(ii) For any two numbers \( a \) and \( b \) (whether the same or not) in \( S \), there exists \( c' \in S, c' \) not equal to \( a \) or \( b \), such that \( (a \cdot c') > 1 \) and \( (b \cdot c') > 1 \).

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

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