# Thrown for a loop

Geometry Level 4

Suppose a particle, starting at the origin of a standard $$xy$$-grid, travels in a rectangular spiral, first moving in a straight line east, (i.e., the positive $$x$$ direction), then north, (i.e., the positive $$y$$ direction), then west and then south, repeating this "loop" ad infinitum, such that the $$n$$th move has length $$\displaystyle \tan^{n} \left( \frac{\pi}{10} \right)$$ units. (The argument of the $$\tan$$ function is in radian measure.)

If the (magnitude of the) displacement between the starting and finishing points of the particle is $$\dfrac{\sqrt{a} - b}{c},$$ where $$a,b,c$$ are positive integers with $$a$$ square-free, then find $$a + b + c.$$

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