In the additive group of ordered pairs of integers ( $m, n$ ) [with addition defined componentwise: $\left.(m, n)+\left(m^{\prime}, n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements
$$
(3,8), \quad(4,-1)
$$
Then $H$ has another set of generators of the form
$$
(1, b), \quad(0, a)
$$
for some integers $a, b$ with $a>0$. Find $a$.
[Elements $g_{i}, \ldots, g_{k}$ are said to generate a subgroup $H$ if (i) each $g_{i} \in H$, and (ii) every $h \in H$ can be written as a sum $h=n_{1} g_{1}+\cdots+n_{k} g_{k}$ where the $n_{i}$ are integers (and where, for example, $3 g_{1}-2 g_{2}$ means $\left.\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\right)\right]$


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