There's A Maximum?

{xy+yz+xz+2(x+y+z)=53x(yz1)=y+z2\begin{cases} xy+yz+xz+2(x+y+z)=53 \\ x(yz-1)=y+z-2 \end{cases}

Given that x,yx,y and zz are positive integers satisfying the system of equations above, find the maximum value of x+yx+y.

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