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To find \(a\) and \(b\) we add the two decomposed fractions using a common denominator. This gives
\begin{equation*}
\frac{1}{(1-x)(1-2x)} = \frac{a(1-2x) + b(1-x)}{(1-x)(1-2x)}\text{.}
\end{equation*}
so
\begin{equation*}
1 = a(1-2x) + b(1-x)\text{.}
\end{equation*}
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