
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:

(1) For all positive integers \( n \), we have
\[ f(n!) = f(n)! \]

(2) For all distinct integers \( m \) and \( n \), \((m-n) \) divides \( f(m)-f(n) \).

(2012, USA Mathematical Olympiad)

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