Example 6 Solve the equation
\[
\begin{array}{l}
a^{4} \cdot \frac{(x-b)(x-c)}{(a-b)(a-c)}+b^{4} \cdot \frac{(x-c)(x-a)}{(b-c)} \\
+\epsilon^{4} \cdot \frac{(x-a)(x-b)}{(c-a)(c-b)}=x^{4} .
\end{array}
\]
Solution: It is easy to know from the definition that $x_{1}=a$ is a root of the original equation. Since $a, b, c$ in the original equation are cyclically symmetric, by the property of cyclically symmetric expressions, $x_{2}=b$ and $x_{3}=c$ are also roots of the original equation. Let $x_{4}$ be another root of the equation, then
$$
(x-a)(x-b)(x-c)\left(x-x_{4}\right)=0 \text {. }
$$

Expanding and rearranging, we get
$$
x^{4}-\left(a+b+c+x_{4}\right) x^{3}+\cdots=0 .
$$

Since the coefficient of $x^{3}$ in the original equation is 0, we have
$$
\begin{array}{l}
a+b+c+x_{4}=0 . \\
\therefore x_{1}=a, x_{2}=b, x_{3}=c, \\
x_{4}=-(a+b+c) .
\end{array}
$$