Strange Factorial Sum

Calculus Level 3

Let \[S=\dfrac{1}{4!}+\dfrac{1}{5!}+\dfrac{1}{8!}+\dfrac{1}{9!}+\cdots +\dfrac{1}{(4n)!}+\dfrac{1}{(4n+1)!}+\cdots\]

Find the value of \[\lfloor 1000S\rfloor\]

Details and Assumptions

Please provide the exact value in your solution.

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