HMMT 2007 S7

Three positive reals $x, y$, and $z$ are such that

$$\begin{aligned} x^{2}+2(y-1)(z-1) & =85 \\ y^{2}+2(z-1)(x-1) & =84 \\ z^{2}+2(x-1)(y-1) & =89 \end{aligned}$$

Compute $x+y+z$.