
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 \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) which satisfies
\[
f(a) + f(b) + f(c) = \frac{f(a-b) + f(b-c) + f(c-a)}{k}
\]
for any integers \( a, b, c \) with \( a + b + c = 0 \).

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