
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
 $p(m)$  is the number of distinct prime divisors of a positive integer  $m>1$  and  $f(m)$  is the  $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$  th smallest prime divisor of  $m$ . Find all positive integers  $n$  satisfying the equation:   $$ f(n^2+2) + f(n^2+5) = 2n-4 $$  

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