# 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 $\displaystyle{\sum _{ a\in A }^{ }{ a } }$

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