# For Complex Lovers

**Algebra**Level 5

\[\displaystyle{{ Z }_{ 1 },{ Z }_{ 2 },{ Z }_{ 3 }\in C\quad \\ \left| { Z }_{ 1 } \right| =\left| { Z }_{ 2 } \right| =\left| { Z }_{ 3 } \right| =1\\ \sum _{ \text{cyclic} }^{ 1,2,3 }{ \cfrac { { { Z }_{ 1 } }^{ 2 } }{ { { Z }_{ 2 } }{ { Z }_{ 3 } } } } =-1}\]

Let \(\displaystyle{a=\left| { Z }_{ 1 }+{ Z }_{ 2 }+{ Z }_{ 3 } \right| }\)

Let an Set \(A\) contains all possible values of \(a\) . Then find the value of \[\displaystyle{\sum _{ a\in A }^{ }{ a } }\]