
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 by $\mathbb{N}=\{1,2,3, \ldots\}$ the set of natural numbers excluding zero and by $\mathbb{N}^{*}=\{0,1,2,3, \ldots\}$ the set of natural numbers including zero. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}^{*}$ that are increasing, i.e., $f(n) \geq f(m)$ if $n>m$, and such that $f(nm)=f(n)+f(m)$ for all $n, m \in \mathbb{N}$.

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