
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
(a) A function  $f:\mathbb{Z} \rightarrow \mathbb{Z}$  is called  $\mathbb{Z}$ -good if  $f(a^2+b)=f(b^2+a)$  for all  $a, b \in \mathbb{Z}$ . What is the largest possible number of distinct values that can occur among  $f(1), \ldots, f(2023)$ , where  $f$  is a  $\mathbb{Z}$ -good function?

(b) A function  $f:\mathbb{N} \rightarrow \mathbb{N}$  is called  $\mathbb{N}$ -good if  $f(a^2+b)=f(b^2+a)$  for all  $a, b \in \mathbb{N}$ . What is the largest possible number of distinct values that can occur among  $f(1), \ldots, f(2023)$ , where  $f$  is a  $\mathbb{N}$ -good function?

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