4. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$, let $S=x^{2}+y^{2}$. Then the value of $\frac{1}{S_{\text {min }}}+\frac{1}{S_{\text {max }}}$ is $\qquad$
(1993, National High School Mathematics Competition)
(Let $x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta$, substituting into the known conditions yields $\sin 2 \theta=\frac{8 S-10}{5 S}$. Then $\left|\frac{8 S-10}{5 S}\right| \leqslant 1$. Solving this, we get $\frac{10}{13} \leqslant S$ $\leqslant \frac{10}{3}$. Therefore, $\frac{1}{S_{\max }}+\frac{1}{S_{\max }}=\frac{8}{5}.$)