
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 positive integer \( n \geq 3 \), each side and diagonal of a regular \( n \)-gon is labeled with a positive integer not exceeding \( r \), satisfying the following conditions: 
(1) Every positive integer \( 1, 2, 3, \cdots, r \) appears on at least one side or diagonal; 
(2) For every triangle \( \triangle P_{i} P_{j} P_{k} \), there are two sides with the same labeled number, and this number is greater than the number on the third side. Determine: 
(1) The maximum positive integer \( r \) that satisfies the above conditions;
(2) For this maximum positive integer \( r \), find the number of different labeling methods that meet the requirements of the problem.

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