Suppose $a$ is a real number such that the equation

$a\cdot (2\cos x+1)=2\cos (2x)-4\cos x-1$

has more than one solution in the interval $(0,\pi )$. The set of all such $a$ can be written in the form

$(p,q)\cup (q,r)$,

where $p,q,r$ are real numbers with $p<q<r$. What is $p+q+r$?