
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  $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$ .

*Proposed by Bojan Bašić, Serbia*

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