If

\[ S = {\sum_{n=1}^{\infty}\frac{1}{2n}\left ( \sum_{k=0}^{\infty}\frac{1}{2^{k}k!} \right )^{n}} \]

Suppose \(e^{S}\) may be written as

\[ \frac{a}{b-\sqrt{c \times e}} \]

Then find \(a + b + c\).

**OBS:**

- \(e\) is the base of the natural logarithm.

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