
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 \( n \) (\( n \geq 2 \)) such that there exist real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying \( |a_{i} - a_{j}| \) for \( 1 \leq i < j \leq n \) equals \( \{1,2, \cdots, \frac{n(n-1)}{2}\} \).
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 set \( A_{i} \) with five elements for \( i=1, 2, \cdots, 20 \), such that \( |A_{i} \cap A_{j}| \leq 2 \) for \( 1 \leq i < j \leq 20 \). Find the minimum value of \( n \).

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