\[\displaystyle \begin{cases} { a }^{ 1 }+{ b }^{ 1 }+{ c }^{ 1 }=\lambda \\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=\lambda \\ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=\lambda \end{cases}\]

If \(\lambda \in \mathbb{{Z}^{+}}\) satisfying the system of equations above with \(abc=5!\), determine \(\lambda\).

**Note:** "\(!\)" represents factorial, not exclamatory sign.

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