If \(x+y+z=2196\)

\(\sqrt [ 3 ]{ x } +y+z=2076\)

\(x+\sqrt [ 3 ]{ y } +z=1860\)

\(x+y+\sqrt [ 3 ]{ z } =480\)

and if \(x\), \(y\) and \(z\) are positive integers,

Find \({ x }^{ \frac { 2 }{ 3 } }+{ y }^{ \frac { 2 }{ 3 } }+{ z }^{ \frac { 2 }{ 3 } }\)

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