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

The numbers \(x\), \(y\), and \(z\) satisfy the system of equations above, where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).

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