This Spiral is So Radical Dude!

Calculus Level 4

A particle begins at the origin and moves lengths of \(\frac{1}{\sqrt1+\sqrt5}\) units east, \(\frac{1}{\sqrt2+\sqrt6}\) units north, \(\frac{1}{\sqrt3+\sqrt7}\) units west, \(\frac{1}{\sqrt4+\sqrt8}\) units south... ad infinitum.

If the magnitude of the displacement of the particle from the origin is \(P\), enter your answer as \(\lfloor 1000P\rfloor\).

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