Find \(x,y,z\) satisfying the equations

\[\large{ \begin{cases} (x+y)(x+y+z)=66 \\ (y+z)(x+y+z)=99 \\ (z+x)(x+y+z)=77 \end{cases}} \]

Submit your answer as the sum of all possible values of \((x+y+z).\)

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