
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
We call a permutation  $(a_1, a_2,\cdots , a_n)$  of the set  $\{ 1,2,\cdots, n\}$  "good" if for any three natural numbers  $i <j <k$ ,  $n\nmid a_i+a_k-2a_j$  find all natural numbers  $n\ge 3$  such that there exist a "good" permutation of a set  $\{1,2,\cdots, n\}$ .

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