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

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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