
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 completely multipiclative functions  $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$  such that for any  $a,b\in \mathbb{Z}$  and  $b\neq 0$ , there exist integers  $q,r$  such that  $$ a=bq+r $$  and  $$ f(r)<f(b) $$  Proposed by Navid Safaei

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