6. Let the complex number $z$ satisfy $\left|z+\frac{1}{z}\right| \leqslant 2$. Then the range of $|z|$ is $\qquad$ .
6. $[\sqrt{2}-1, \sqrt{2}+1]$.

Let $|z|=r$. Then
$$
\begin{array}{l}
\left|1-r^{2}\right| \leqslant\left|1+z^{2}\right| \leqslant 2 r \\
\Rightarrow\left(1-r^{2}\right)^{2} \leqslant 4 r^{2} \\
\Rightarrow 3-2 \sqrt{2} \leqslant r^{2} \leqslant 3+2 \sqrt{2} \\
\Rightarrow \sqrt{2}-1 \leqslant r \leqslant \sqrt{2}+1 .
\end{array}
$$