Algebra and Analysis!

Algebra Level 5

\[\large\ \begin{cases} \displaystyle \sum _{ k=1 }^{ 5 }{ k{ x }_{ k } } = a \\ \displaystyle \sum _{ k=1 }^{ 5 }{ { k }^{ 3 }{ x }_{ k } } = { a }^{ 2 } \\ \displaystyle \sum _{ k=1 }^{ 5 }{ { k }^{ 5 }{ x }_{ k } } = { a }^{ 3 } \end{cases} \]

Find the sum of all possible values of real number \(a\) for which there exists non-negative real numbers \({ { { x }_{ 1 } }, { { x }_{ 2 } },..,{ { x }_{ 5 } }}\) satisfying the system of equations above.

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