
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 integers  $ n\ge 2$  having the following property: for any  $ k$  integers  $ a_{1},a_{2},\cdots,a_{k}$  which aren't congruent to each other (modulo  $ n$ ), there exists an integer polynomial  $ f(x)$  such that congruence equation  $ f(x)\equiv 0 (mod n)$  exactly has  $ k$  roots  $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$ 

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