
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  $\mathbb{N}$  denote the set of positive integers. Find all functions  $f : \mathbb{N} \rightarrow \mathbb{N}$  such that for positive integers  $a$  and  $b,$  \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]

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