
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
Determine all integers \( n \geq 1 \) for which there exists a pair of positive integers \( (a, b) \) such that no cube of a prime divides \( a^2 + b + 3 \) and
\[ 
\frac{ab + 3b + 8}{a^2 + b + 3} = n.
\]

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