Logarithms?

Algebra Level 4

\[\large \begin {cases} x^{xyz} =y^2 \\ y^{xyz+1}=z^3 \\ z^{xyz+2}=x^4 \end {cases} \]

Let \(x\), \(y\) and \(z\) be positive integers that satisfy the system of equations above. Find the sum of all possible \(x,y,z\).

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