Infinite Factorial Summation with a Twist

Calculus Level 4

\[\large 1 + \dfrac{1 + \frac{1}{1!}}{2} + \dfrac{1 + \frac{1}{1!} + \frac{1}{2!}}{2^2} + \dfrac{1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!}}{2^3} + \ldots \]

If the above series can be expressed as \(S\), find \(\big \lfloor 100S \rfloor\).

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