
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  $n\geq 1$  such that there exists a permutation  $(a_1,a_2,...,a_n)$  of  $(1,2,...,n)$  such that  $a_1+a_2+...+a_k$  is divisible by  $k$  for  $k=1,2,...,n$ 

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