Consider the Fibonacci sequence $F_0, F_1, F_2, \ldots$ defined by $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for $n \geq 0$. Find the sum of the infinite series
\[
\sum_{n=0}^{\infty} \frac{F_n}{2^{n+1}}.
\] Let's think step by step and output the final answer within \boxed{}.
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