Example 10 Let real numbers $x, y, z, w$ satisfy
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Try to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
Let $1^{2}=a_{1}, 3^{2}=a_{2}, 5^{2}=a_{3}, 7^{2}=a_{4}, 2^{2}=\lambda_{1}, 4^{2}=\lambda_{2}, 6^{2}=\lambda_{3}, 8^{2}=\lambda_{4}$. Construct $f(x)=\prod_{i=1}^{4}\left(x-a_{i}\right)-\prod_{i=1}^{4}\left(x-\lambda_{i}\right)=\left[\sum_{i=1}^{4}\left(\lambda_{i}-a_{i}\right)\right] x^{3}+\cdots$
Then $f\left(\lambda_{k}\right)=\prod_{i=1}^{4}\left(\lambda_{i}-a_{i}\right)$.
By Lagrange's formula, we get $f(x)=\sum_{j=1}^{4} \prod_{\substack{i \neq j \\ 1 \leqslant i \leqslant 4}} \frac{x-a_{i}}{a_{j}-a_{i}} \cdot f\left(a_{j}\right)$.
Substituting $x=\lambda_{k}$ in the above equation and dividing both sides by $f\left(\lambda_{k}\right)$, we get
$$
\frac{A_{1}}{\lambda_{k}-a_{1}}+\frac{A_{2}}{\lambda_{k}-a_{2}}+\frac{A_{3}}{\lambda_{k}-a_{3}}+\frac{A_{4}}{\lambda_{k}-a_{4}}=1,
$$

where $A_{j}=\frac{f\left(a_{j}\right)}{\prod_{\substack{i \leqslant j \\ 1 \leqslant 4}}\left(a_{j}-a_{i}\right)}$, and $j, k=1,2,3,4$.
The system of equations (2) has a unique solution. Comparing the coefficients of $x^{3}$ in (1) and (2), we get $\sum_{j=1}^{4} A_{j}=\sum_{i=1}^{4}\left(\lambda_{i}-a_{i}\right)$, and $x^{2}, y^{2}, z^{2}$, $w^{2}$ are the solutions of the known system of equations, then
$$
\begin{aligned}
x^{2}+y^{2}+z^{2}+w^{2} & =\sum_{j=1}^{4} A_{j}=\sum_{i=1}^{4} \lambda_{i}-\sum_{i=1}^{4} a_{i} \\
& =\left(2^{2}+4^{2}+6^{2}+8^{2}\right)-\left(1^{2}+3^{2}+5^{2}+7^{2}\right)=36
\end{aligned}
$$