# Complex Multiplication!

Algebra Level 5

$$a$$, $$b$$, $$c$$ 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.

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