
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 \( S = \{1, 2, \cdots, n\} \). Suppose the 3-element subsets \( A_1, A_2, \cdots, A_m \) (\( m > 1 \)) satisfy:
(1) Every 2-element subset of \( S \) is contained in exactly one \( A_i \) (\( 1 \leq i \leq m \));
(2) Each \( A_i \) and \( A_j \) (\( i \neq j, 1 \leq i, j \leq m \)) have exactly one common element.
What is \( m \)?

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