1. Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$, if $S=x^{2}+y^{2}$, let the maximum value of $S$ be $p$, and the minimum value of $S$ be $q$, then $\frac{1}{p}+\frac{1}{q}=$ $\qquad$ .
Let $x=\sqrt{S} \cos \theta, y=\sqrt{S} \sin \theta$, then $4-\frac{5}{2} \sin 2 \theta=\frac{5}{S} \in\left[\frac{3}{2}, \frac{13}{2}\right] \Rightarrow \frac{1}{S} \in\left[\frac{3}{10}, \frac{13}{10}\right]$, so $\frac{1}{p}+\frac{1}{q}=\frac{3}{10}+\frac{13}{10}=\frac{8}{5}$.