
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 \( f: \mathbf{N}_{+} \rightarrow \mathbf{N}_{+} \) such that there exists \( k \in \mathbf{N}_{+} \) and a prime \( p \) such that for any \( n \geqslant k \), \( f(n+p)=f(n) \). Moreover, it must satisfy: if \( m \mid n \), then \( f(m+1) \mid (f(n)+1) \).

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