Example 1 Given that $a$ and $b$ are real numbers, and $a^{2} + ab + b^{2} = 3$. If the maximum value of $a^{2} - ab + b^{2}$ is $m$, and the minimum value is $n$, find the value of $m + n$. ${ }^{\text {[2] }}$
Let $a^{2}-a b+b^{2}=t$.
Combining this with the given equation, we get
$$
a b=\frac{3-t}{2}, a+b= \pm \sqrt{\frac{9-t}{2}} .
$$

Thus, $a$ and $b$ are the two real roots of the quadratic equation in $x$:
$$
x^{2} \pm \sqrt{\frac{9-t}{2}} x+\frac{3-t}{2}=0
$$

Therefore, $\Delta=\left( \pm \sqrt{\frac{9-t}{2}}\right)^{2}-4 \times \frac{3-t}{2} \geqslant 0$.
Solving this, we get $t \geqslant 1$.

Also, $\frac{9-t}{2} \geqslant 0$, which implies $t \leqslant 9$.
Thus, $1 \leqslant t \leqslant 9$.
Hence, $m=9, n=1$
$$
\Rightarrow m+n=10 \text {. }
$$