
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
Find all functions  $f:\mathbb{N} \to \mathbb{N}$  such that for any two positive integers  $a$  and  $b$  we have   $$  f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1) $$  Where  $f^n(m)$  is defined in the standard iterative manner. 

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