A tangled nest we weave

Calculus Level 5

\[ \Large S = \displaystyle\sqrt{\pi \sqrt[3]{e \sqrt[4]{\pi \sqrt[5]{e \sqrt[6]{\pi \sqrt[7]{e \sqrt[8]{\pi \cdots}}}}}}} \]

Find the value of \(\lfloor 1000*S \rfloor\).

Clarification: The small numbers \(3, 4,5,\ldots \) indicate the \(n^\text{th} \) roots and are not powers.

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