This Spiral is So Radical Dude!

Calculus Level 3

A particle begins at the origin and moves lengths of 11+5\frac{1}{\sqrt1+\sqrt5} units east, 12+6\frac{1}{\sqrt2+\sqrt6} units north, 13+7\frac{1}{\sqrt3+\sqrt7} units west, 14+8\frac{1}{\sqrt4+\sqrt8} units south... ad infinitum.

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

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