Unknown powers

Algebra Level 4

\[ \large{\begin{cases} a+b+c=5 \\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=83 \\ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=245\\ { a }^{ 4 }+{ b }^{ 4 }+{ c }^{ 4 }=3107 \end{cases} }\]

Let \(a,b\) and \(c\) be integers satisfying the system of equations above. Find the value of \({ a }^{ 5 }+{ b }^{ 5 }+{ c }^{ 5 }\).

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