Three. (Full marks 20 points) Let the complex numbers $z_{1}, z_{2}, \cdots, z_{10}$ form a geometric sequence. Given that $z_{1} \neq 1, z_{2}=z_{10}=1$. Find the modulus and argument of $z_{1}$.

保留源文本的换行和格式，翻译结果如下：

Three. (Full marks 20 points) Let the complex numbers $z_{1}, z_{2}, \cdots, z_{10}$ form a geometric sequence. Given that $z_{1} \neq 1, z_{2}=z_{10}=1$. Find the modulus and argument of $z_{1}$.
$\begin{array}{l}\text { Three, } \because z_{1} \neq 1 \text {, let } z_{1}=r(\cos \theta+i \sin \theta) \text {. } \\ \text { Also, } z_{2}=1, \therefore q=\frac{1}{z_{1}}=\frac{1}{r}(\cos \theta-i \sin \theta) \text {. } \\ \because z_{10}=z_{2} \cdot q^{9}=\frac{1}{r^{8}}(\cos \theta-i \sin \theta)^{8} \\ =\frac{1}{r^{8}}(\cos 8 \theta-i \sin 8 \theta)=1 \text {, } \\ \therefore \frac{1}{r^{8}}=1 \Rightarrow r=1 \text {. } \\ \text { From }\left\{\begin{array}{l}\cos 8 \theta=1, \\ \sin 8 \theta=0\end{array} \Rightarrow 8 \theta=2 k \pi \text {, }\right. \\ \therefore \theta=\frac{k \pi}{4}, k \in \mathbb{Z}, k \neq 8 m, m \in \mathbb{Z} . \\\end{array}$