Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_i b_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_i b_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?

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