\[ \large A = \sum_{n=0}^\infty \dfrac{ \binom n0}{n!} + \sum_{n=1}^\infty \dfrac{ \binom n1}{n!} + \sum_{n=2}^\infty \dfrac{ \binom n2}{n!} + \cdots \]

Given that \(A\) has a closed form. Find \( \lfloor 1000A \rfloor \).

\[\]**Notation**: \( \dbinom MN \) denotes the binomial coefficient, \( \dbinom MN = \dfrac{M!}{N!(M-N)!} \).

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