Complex Multiplication!

$a$, $b$, $c$ and $d$ are non-zero complex numbers that satisfy

$a+b+c+d=a^3+b^3+c^3+d^3=0$

The sum of all possible values of $(a+b)(a+c)(a+d)$ can be expressed as $p+qi$ where $p$ and $q$ are real numbers and $i$ is the imaginary unit that satisfies $i^2=-1$. What is $p+q$?

This problem is inspired by a problem that appeared in ITT - $1994$.

This problem is from the set "Olympiads and Contests Around the World -1". You can see the rest of the problems here.