
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  $\mathbb{N}$  be the set of positive integers. Determine all positive integers  $k$  for which there exist functions  $f:\mathbb{N} \to \mathbb{N}$  and  $g: \mathbb{N}\to \mathbb{N}$  such that  $g$  assumes infinitely many values  and such that   $$  f^{g(n)}(n)=f(n)+k $$   holds for every positive integer  $n$ .

(*Remark.* Here,  $f^{i}$  denotes the function  $f$  applied  $i$  times i.e  $f^{i}(j)=f(f(\dots f(j)\dots ))$ .)

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