Fibonacci Fractions

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

The series above shows the sum of all nthn^\text{th} Fibonacci number, shifted nn places to the right (that is, divided by 10n10^n). If its closed form can be represented as ab \frac a b for coprime positive integers, find a+ba+b.

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