
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 injective functions  $ f:\mathbb{N} \to \mathbb{N} $  such that   $$  f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2  $$   holds for all  $ a,b \in \mathbb{N} $ . Note that  $ f^{k}\left(n\right) $  means  $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $ 

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