
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
Determine all pairs \((f, g)\) of functions from the set of positive integers to itself that satisfy
$$
f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1
$$
for every positive integer \(n\). Here, \(f^{k}(n)\) means \(\underbrace{f(f(\ldots f)}_{k}(n) \ldots)\).

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