
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 \( P \) be the set of the first 2009 prime numbers and let \( X \) be the set of all natural numbers which have only prime factors from \( P \). Determine all natural numbers \( k \) for which a function \( f: X \rightarrow X \) exists that satisfies the following equation for all \( m, n \in X \):

\[
f(m f(n))=f(m) n^{k}
\]

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