$0.1+ 0.01+ 0.002+ 0.0003+ 0.00005+ 0.000008+\ldots$

The series above shows the sum of all $n^\text{th}$ Fibonacci number, shifted $n$ places to the right (that is, divided by $10^n$). If its closed form can be represented as $\frac a b$ for coprime positive integers, find $a+b$.

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