Find all positive integers $n \geq 1$ such that there exists a pair $(a, b)$ of positive integers, satisfying the following conditions:
1. $a^2 + b + 3$ is not divisible by the cube of any prime.
2. \( n = \frac{ab + 3b + 8}{a^2 + b + 3} \)

Let's think step by step and output the final answer within \boxed{}.
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