Example 1 Given the complex number $z_{1}=\mathrm{i}(1-\mathrm{i})^{3}$, when the complex number $z$ satisfies $|z|=1$, find the maximum value of $\left|z-z_{1}\right|$.
(2001 National College Entrance Examination Question)
Solution: Obviously, $z_{1}=2-2 \mathrm{i}$, and $|z|=1$ represents the unit circle centered at the origin in the complex plane (Fig. $7-1$). $\left|z-z_{1}\right|$ represents the distance between the point on the circle and the point corresponding to $z_{1}$. It is easy to see that when $M 、 O 、 Z_{1}$ are collinear, $\left|z-z_{1}\right|$ has a maximum value of $2 \sqrt{2}+1$.