
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 positive integers  $k$  for which there exists a nonlinear function  $f:\mathbb{Z} \rightarrow\mathbb{Z}$  such that the equation   $$ f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k} $$   holds for any integers  $a,b,c$  satisfying  $a+b+c=0$  (not necessarily distinct).

*Evan Chen*

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