
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
Find all natural numbers  $n (n \geq 2)$  such that there exists reals  $a_1, a_2, \dots, a_n$  which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]

Let  $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$ .  $A_i(i=1,2,\dots,20)$  contains eight numbers, three of which are chosen from  $A$  and the other five numbers from  $B$ .  $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$ . Find the minimum possible value of  $n$ .

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