Eccoci.

Let's call a pair of positive integers $p$, $q$ ($p \lt q$) reciprocal, if there is a positive integer $r\lt p$ such that $r$ equals both the inverse of $p$ modulo $q$ and the inverse of $q$ modulo $p$.

For example, $(3,5)$ is one reciprocal pair for $r=2$.
Let $F(N)$ be the total sum of $p+q$ for all reciprocal pairs $(p,q)$ where $p \le N$.

$F(5)=59$ due to these four reciprocal pairs $(3,5)$, $(4,11)$, $(5,7)$ and $(5,19)$.
You are also given $F(10^2) = 697317$.

Find $F(2\cdot 10^6)$.