
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 \(\mathbb{N}\) the set of all positive integers. Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that for all positive integers \(m\) and \(n\), the integer \(f(m) + f(n) - mn\) is nonzero and divides \(mf(m) + nf(n)\).

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