
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 \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).

Find the smallest value of \( n \).

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