13. Given a complex number $z$ satisfies $\left|z+\frac{1}{z}\right|=1$, find the maximum and minimum values of $|z|$.

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13. Let $z=r e^{j 6}$. (where: $r>0.0 \leqslant \theta<2 \pi$)

So $z+\frac{1}{z}=r e^{j \theta}+\frac{1}{r} \cdot e_{i(-\theta)}=\left(r+\frac{1}{r}\right) \cos \theta+i \cdot\left(r-\frac{1}{r}\right) \cdot \sin \theta$
Also, $1=\left|z+\frac{1}{z}\right| \Rightarrow 1=\left(r+\frac{1}{r}\right)^{2} \cos ^{2} \theta+\left(r-\frac{1}{r}\right) \sin ^{2} \theta$
Given $0 \leqslant \sin ^{2} \theta \leqslant 1$. So $\left(r+\frac{1}{r}\right)^{2} \leqslant 5 \Rightarrow r+\frac{1}{r} \leqslant \sqrt{5}$.
Therefore,
$$
\frac{\sqrt{5}-1}{2} \leqslant r \leqslant \frac{\sqrt{5}+1}{2}
$$
The value is $\frac{1}{2}$, and the minimum value is $\frac{\sqrt{5}-1}{2}$.