Cunning Repercussion

\[\large { \begin{cases} { x^5y^4z^3+x^4y^3z^2+x^3y^2z+x^2y+x = 7907799 } \\ { 2x+3y+4z=38} \end{cases} } \]

If \(x, y, z\) are natural numbers that satisfy the system of equations above, evaluate \(x+y+z\).

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