Amazing Nested not-quite-radical 4

Calculus Level 4

Let \[\Large R = \sqrt[2 \cdot 3]{\frac{2}{\sqrt[4 \cdot 5]{\frac{2}{\sqrt[6 \cdot 7]{\frac{2}{\ddots}}}}}} .\]

Find \( \left \lfloor{1000 R}\right \rfloor .\)

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