Consider two geometric sequences $16$, $a_1$, $a_2$, $\ldots$ and $56$, $b_1$, $b_2$, $\ldots$ with the same common nonzero ratio. Given that $a_{2023} = b_{2020}$, compute $b_6 - a_6$.

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