\[ \large{ \begin{cases}xy+x+y=23 \\ yz+y+z=31 \\ zx + z+x=47 \end{cases}} \]

Let \(x,y\) and \(z\) be numbers satisfying the system of equations above, enter your answer as the sum of all possible values of \(x,y\) and \(z\).

**This is part of the set My Problems and THRILLER**

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