A tangled nest we weave

Calculus Level 5

S=πeπeπeπ876543 \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 1000S\lfloor 1000*S \rfloor.

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

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