
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
Determine all integers  $n\geq 1$  for which the numbers  $1,2,\ldots,n$  may be (re)ordered as  $a_1,a_2,\ldots,a_n$  in such a way that the average  $\dfrac {a_1+a_2+\cdots + a_k} {k}$  is an integer for all values  $1\leq k\leq n$ .

(Dan Schwarz)

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