
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 positive integers  $n\geq1$  such that there exists a pair  $(a,b)$  of positive integers, such that  $a^2+b+3$  is not divisible by the cube of any prime, and   $$ n=\frac{ab+3b+8}{a^2+b+3}. $$  

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