HMMT November 2022 T9

Call an ordered pair $(a, b)$ of positive integers fantastic if and only if $a, b \leq 10^{4}$ and

$$\operatorname{gcd}(a \cdot n!-1, a \cdot(n+1)!+b)>1$$

for infinitely many positive integers $n$. Find the sum of $a+b$ across all fantastic pairs $(a, b)$.