
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

Let \( k \) be any positive integer. If for all positive integers \( m \) and \( n \) (with \( m \neq n \)), it holds that
$$
(f(m)+n, f(n)+m) \leqslant k,
$$
then the function \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) is called " \( k \)-good". Find all positive integers \( k \) such that there exists a \( k \)-good function.

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