
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
Let  $A$  be a finite set, and  $A_1,A_2,\cdots, A_n$  are subsets of  $A$  with the following conditions:
(1)  $|A_1|=|A_2|=\cdots=|A_n|=k$ , and  $k>\frac{|A|}{2}$ ;
(2) for any  $a,b\in A$ , there exist  $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$  such that $a,b\in A_r\cap A_s\cap A_t$ ;
(3) for any integer  $i,j\, (1\leq i<j\leq n)$ ,  $|A_i\cap A_j|\leq 3$ .
Find all possible value(s) of  $n$  when  $k$  attains maximum among all possible systems  $(A_1,A_2,\cdots, A_n,A)$ .

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