They are all related

Algebra Level 3

{a1+b1+c1=λa2+b2+c2=λa3+b3+c3=λ\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 λZ+\lambda \in \mathbb{{Z}^{+}} satisfying the system of equations above with abc=5!abc=5!, determine λ\lambda.

Note: "!!" represents factorial, not exclamatory sign.

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