
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 pairs of positive integer   $\alpha$  and function  $f : N \to N_0$  that satisfies
(i)  $f(mn^2) = f(mn) + \alpha f(n)$  for all positive integers  $m, n$ .
(ii) If  $n$  is a positive integer and  $p$  is a prime number with  $p|n$ , then  $f(p) \ne 0$  and  $f(p)|f(n)$ .

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