
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
(1) Find all natural numbers \( n (n \geqslant 2) \) such that there exist real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying \( \left\{\left|a_{i}-a_{j}\right|\right. | 1 \leqslant i<j \leqslant n\}=\left\{1,2, \cdots, \frac{n(n-1)}{2}\right\} \).

(2) Let \( A=\{1,2,3,4,5,6\} \) and \( B=\{7,8,9, \cdots, n\} \). Select three numbers from \( A \) and two numbers from \( B \) to form a five-element set \( A_{i}, i=1,2, \cdots, 20 \) such that \( \left|A_{i} \cap A_{j}\right| \leqslant 2 \) for \( 1 \leqslant i<j \leqslant 20 \). Find the minimum value of \( n \).

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