
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 functions \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for all positive integers \( m \) and \( n \), \( f(m) + f(n) - mn \) is non-zero and divides \( mf(m) + nf(n) \).

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