
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
We denote the set of nonzero integers and the set of non-negative integers with  $\mathbb Z^*$  and  $\mathbb N_0$ , respectively. Find all functions  $f:\mathbb Z^* \to \mathbb N_0$  such that: $a)$   $f(a+b)\geq min(f(a), f(b))$  for all  $a,b$  in  $\mathbb Z^*$  for which  $a+b$  is in  $\mathbb Z^*$ . $b)$   $f(ab)=f(a)+f(b)$  for all  $a,b$  in  $\mathbb Z^*$ .

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