
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  $\mathbb{Z}_{>0}$  denote the set of positive integers. For any positive integer  $k$ , a function  $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$  is called *$k$ -good* if  $\gcd(f(m) + n, f(n) + m) \le k$  for all  $m \neq n$ . Find all  $k$  such that there exists a  $k$ -good function.

*Proposed by James Rickards, Canada*

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