An algebra problem by Benny Sampson Uche

Algebra Level pending

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

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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