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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