What an Amusing Nested Radical!

Calculus Level 3

2222543\Large \sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\cdots}}}}

Let the value of the above expression be denoted by II. Find 1000I\lfloor1000I\rfloor.

Details and Assumptions:

You may use a scientific calculator to evaluate the final result.

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