$a_{n}=\left(n-\frac{1}{2}\right)\left(\frac{9}{10}\right)^{n}, b_{n}=a_{n+1}-a_{n}$. Find the minimum value of $\left\{\frac{b_{n+1}}{b_{n}}\right\}$ and multiply it by 10. What is the result?

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