The given equation can be written as
\[\frac{z^9 + 1}{z + 1} = 0.\]Then $z^9 + 1 = 0,$ or $z^9 = -1.$  Since $z = e^{i \theta},$
\[e^{9i \theta} = -1.\]This means $9 \theta = \pi + 2 \pi k$ for some integer $k.$  Since $0 \le \theta < 2 \pi,$ the possible values of $k$ are 0, 1, 2, 3, 5, 6, 7, and 8.   (We omit $k = 4,$ because if $k = 4,$ then $\theta = \pi,$ so $z = -1,$ which makes $z + 1 = 0.$)  Therefore, the sum of all possible values of $\theta$ is
\[\frac{\pi}{9} + \frac{3 \pi}{9} + \frac{5 \pi}{9} + \frac{7 \pi}{9} + \frac{11 \pi}{9} + \frac{13 \pi}{9} + \frac{15 \pi}{9} + \frac{17 \pi}{9} = \boxed{8 \pi}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
If $re^{i \theta}$ is a root of
\[z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0,\]where $r > 0$ and $0 \le \theta < 2 \pi,$ then find the sum of all possible values of $\theta.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8 \pi
If $re^{i \theta}$ is a root of
\[z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0,\]where $r > 0$ and $0 \le \theta < 2 \pi,$ then find the sum of all possible values of $\theta.$
test