So Many Solutions!?

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Find the last 3 digits of the number of solutions to \[ \sin x+\cos^2 x+\cos^3 x+\sin^4 x+\sin^5 x+...+\cos^{99}x+\sin^{100} x = \frac{121 \sqrt3}{2} \] for \( x \in [-100 \pi, 100 \pi] \)

Note that the for the power \( n \), we add \( \sin^n x \) if \( x \equiv 1, 4 \pmod 4 \) and we add \( \cos^n x \) if \( x \equiv 2, 3 \pmod 4 \)

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