(1) The quadratic equation in $x$
$$
x^{2}+z_{1} x+z_{2}+m=0
$$

where $z_{1}, z_{2}, m$ are complex numbers, and
$$
z^{2}-4 z^{2}=16+20 \mathrm{i}
$$

Let the two roots of equation (6) be $\alpha, \beta$, and $|\alpha-\beta|=2 \sqrt{7}$, find the maximum and minimum values of $|m|$.
(1994 National Mathematics Competition)
(Answer: $\min |m|=7-\sqrt{41}$ )