
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:\mathbb{N} \rightarrow \mathbb{ N }_0 $  satisfy the following conditions:

i)  $ f(ab)=f(a)+f(b)-f(\gcd(a,b)),  \forall a,b \in \mathbb{N} $    

ii) For all primes  $ p $  and natural numbers  $ a $ ,  $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $ 

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