Example 7 Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}$ $=5$, let $s=x^{2}+y^{2}$. Then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}=$ $\qquad$ (1993, National High School Mathematics Competition)
Given:
$$
25 x^{2} y^{2}=\left[5-4\left(x^{2}+y^{2}\right)\right]^{2} \text {. }
$$

Let $s=x^{2}+y^{2}$, and set $x^{2}=\frac{s}{2}+t, y^{2}=\frac{s}{2}-t$. Substituting these into the equation and rearranging, we get:
$$
\frac{25}{4} s^{2}-(5-4 s)^{2}=25 t^{2} \geqslant 0 \text {, }
$$

which simplifies to:
$$
\left(5-\frac{3}{2} s\right)\left(\frac{13}{2} s-5\right) \geqslant 0 \text {. }
$$

Solving this, we get $10 \% s \leqslant \frac{10}{13}$.
$$
\therefore s_{\text {min }}=\frac{10}{13}, s_{\max }=\frac{10}{3} \text {. }
$$

Thus, $\frac{1}{s_{\max }}+\frac{1}{s_{\min }}=\frac{3}{10}+\frac{13}{10}=\frac{8}{5}$.