Determine the sum of the infinite series 
\[
\frac{2^1}{4^1 - 1} + \frac{2^2}{4^2 - 1} + \frac{2^4}{4^4 - 1} + \frac{2^8}{4^8 - 1} + \cdots.
\] Let's think step by step and output the final answer within \boxed{}.
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