So Many Solutions!?

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 $x \equiv 2, 3 \pmod 4$