69 If $11 z^{10}+10 i z^{9}+10 i z-11=0$, then $|z|=$

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However, the translation is already provided in the response, so there's no need for further action. The text is a mathematical problem asking for the magnitude of $z$ given a specific polynomial equation. If you need further assistance with solving the problem or have any other requests, feel free to let me know!
69 1. From the problem, we have
$$
z^{9}=\frac{11-10 i z}{11 z+10 i} .
$$

Let $z=a+b i ，(a, b \in \mathbf{R})$, then
$$
\begin{aligned}
\left|z^{9}\right| & =\left|\frac{11-10 i z}{11+10 i}\right|=\frac{\sqrt{11^{2}+220 b+10^{2}\left(a^{2}+b^{2}\right)}}{\sqrt{11^{2}\left(a^{2}+b^{2}\right)+220 b+10^{2}}} \\
& =\frac{f(a, b)}{g(a, b)} .
\end{aligned}
$$

If $a^{2}+b^{2}>1$, then
$$
g(a, b)>f(a, b),
$$

thus $\left|z^{9}\right|<1$. On the other hand, if $a^{2}+b^{2}<1$, then
$$
g(a, b)<f(a, b),
$$

thus $\left|z^{9}\right|>1$, which also leads to a contradiction. Therefore, $|z|=1$.