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Algebra Level 4

If x,y,zR\displaystyle x,y,z\in\mathbb R are positive, solve {x+1yz+z2=3y+1xz=2\displaystyle\begin{cases}x+\frac{1}{yz}+z^2=3\\y+\frac{1}{xz}=2\end{cases}

The solutions are (x1,y1,z1),(x2,y2,z2),,(xn,yn,zn)\displaystyle (x_1,y_1,z_1), (x_2,y_2,z_2),\ldots, (x_n,y_n,z_n).

Find i=1n(xi+yi+zi)\displaystyle\sum_{i=1}^n (x_i+y_i+z_i).

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