
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
Let  $f : N \to R$  be a function, satisfying the following condition:
for every integer  $n > 1$ , there exists a prime divisor  $p$  of  $n$  such that  $f(n) = f \big(\frac{n}{p}\big)-f(p)$ .
If  $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$ ,  determine the value of  $f(2007^2) + f(2008^3) + f(2009^5)$ 

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