Consider the Fibonacci sequence \(F_n\) defined by \(F_0 = 0\), \(F_1 = 1\), and \(F_{n+2} = F_{n+1} + F_n\) for \(n \geq 0\). Evaluate the infinite series
\[
S = \sum_{n=0}^{\infty} \frac{F_n}{10^n}.
\] Let's think step by step and output the final answer within \boxed{}.
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