
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  $n=2k+1>1$  so that there exists a permutation  $a_0, a_1,\ldots,a_{k}$  of  $0, 1, \ldots, k$  such that 
\[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\]

*Proposed by usjl*

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