Example 10 Find $A^{2}$, where $A$ is the sum of the absolute values of all roots of the following equation:
$$
x=\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}
$$
(9th American Invitational Mathematics Examination)
Solution: According to the structural characteristics of the original equation, set
$$
\begin{array}{ll}
x=\sqrt{19}+\frac{91}{y}, & y=\sqrt{19}+\frac{91}{z}, \\
z=\sqrt{19}+\frac{91}{u}, & u=\sqrt{19}+\frac{91}{v}, \\
v=\sqrt{19}+\frac{91}{x} . &
\end{array}
$$

Assume $x>y$ and $x<y$ respectively, then
$$
\begin{array}{l}
x>y \Rightarrow y>z \Rightarrow z>u \Rightarrow u>v \Rightarrow v>x, \\
x<y \Rightarrow y<z \Rightarrow z<u \Rightarrow u<v \Rightarrow v<y .
\end{array}
$$

Both conclusions contradict the assumptions. Therefore, only $x=y$. Substituting
$$
x=y \text { into } x=\sqrt{19}+\frac{91}{y} \text {, we get } x=\sqrt{19}+\frac{91}{x} \text {. }
$$

Solving, we get $x_{1}=\frac{\sqrt{19}+\sqrt{383}}{2}, x_{2}=\frac{\sqrt{19}-\sqrt{383}}{2}$.
$$
\begin{aligned}
\therefore A^{2}= & \left(\left|\frac{\sqrt{19}+\sqrt{383}}{2}\right|\right. \\
& \left.+\left|\frac{\sqrt{19}-\sqrt{383}}{2}\right|\right)^{2} \\
= & (\sqrt{383})^{2}=383 .
\end{aligned}
$$