
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
Denote by \(\mathbb{Q}^{+}\) the set of positive rational numbers. A function \(f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}\) satisfies
- \(f(p)=1\) for all primes \(p\), and
- \(f(ab)=a f(b) + b f(a)\) for all \(a, b \in \mathbb{Q}^{+}\).

For which positive integers \(n\) does the equation \(n f(c) = c\) have at least one solution \(c\) in \(\mathbb{Q}^{+}\)?

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