
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 \( S = \{1, 2, \cdots, n\} \) be a set, and consider \( m \geq 2 \) subsets \( r \)-element subsets \( A_1, A_2, \cdots, A_m \) that have the following property: For each \( A_i \) (\(1 \leq i \leq m\)), there exists a subset \( A_j \) (\(1 \leq j \leq m, j \neq i \)) such that either the maximum element in \( A_i \) is the minimum element in \( A_j \) or the minimum element in \( A_i \) is the maximum element in \( A_j \). Find:
(1) The possible values of \( r \) (when \( n \) is fixed).
(2) The maximum possible value of \( m \) (when both \( n \) and \( r \) are fixed).

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