Example 10 Given $\left(x^{2}-x+1\right)^{6}=a_{12} x^{12}+$
$$
\begin{array}{l}
a_{11} x^{11}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} \text {. Then } a_{12}+a_{10} \\
+\cdots+a_{2}+a_{0}=\ldots
\end{array}
$$
(Fifth National Partial Provinces Junior High School Mathematics Correspondence Competition)
Solution: In the given equation, let $x=1$, we get
$$
a_{12}+a_{11}+a_{10}+\cdots+a_{2}+a_{1}+a_{0}=1 \text {; }
$$

Let $x=-1$, we get
$$
a_{12}-a_{11}+a_{10}-\cdots+a_{2}-a_{1}+a_{0}=3^{6} \text {. }
$$

Adding the above two equations, we get
$$
\begin{array}{l}
2\left(a_{12}+a_{10}+\cdots+a_{2}+a_{0}\right)=1+3^{6} . \\
\text { Therefore, the original expression }=\frac{1}{2}\left(1+3^{6}\right)=365 .
\end{array}
$$