For a natural number $n \geq 2$, let $C _ { n }$ be the circle obtained by translating the circle $C$ with center at the origin and radius 1 by $\frac { 2 } { n }$ in the $x$-direction. Let $l _ { n }$ be the length of the common chord of circles $C$ and $C _ { n }$. When $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \left( n l _ { n } \right) ^ { 2 } } = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]