
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 an integer \( n > 1 \), for a positive integer \( m \), let \( S_{m} = \{1, 2, \cdots, mn\} \). Suppose there exists a collection \( \mathscr{F} \) with \( |\mathscr{F}| = 2n \) satisfying the following conditions:
1. Each set in the collection \( \mathscr{F} \) is an \( m \)-element subset of \( S_{m} \);
2. Any two sets in the collection \( \mathscr{F} \) share at most one common element;
3. Each element of \( S_{m} \) appears in exactly two sets of the collection \( \mathscr{F} \).

Find the maximum value of \( m \).
(2005, USA Mathematical Olympiad Selection Test)

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