Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.  The original answer is in $ae^{b / c}-d$ format, please provide the value of a+b+c+d.

Let's think step by step and output the final answer within \boxed{}.
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