10. (20 points) Given complex numbers $z$ and $k$ satisfy $|z|=|1+k z|(|k|<1)$.
Find the range of $|z|$ (expressed in terms of $|k|$).
10. Let the positive direction of $k$ be the positive direction of the real axis.

Thus, if $0 \leqslant k0,(k-1)|z|-1<0$, then
$$
\left\{\begin{array}{l}
(k-1)|z|+1 \geqslant 0, \\
(k+1)|z|-1 \geqslant 0 .
\end{array}\right.
$$

Therefore, $\frac{1}{k+1} \leqslant|z| \leqslant \frac{1}{1-k}$, and the equality holds if and only if $|z|=$ $\frac{1}{1-k}$ or $z=\frac{-1}{k+1}$.