1. Given $x$, $y$, $z$ are positive real numbers, $x+y+z=1$. Find the range of $x y+y z+z x-2 x y z$.

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Let $0<x \leqslant y \leqslant z<1$. Then $0<x \leqslant \frac{1}{3}$. Using $y z \leqslant\left(\frac{1-x}{2}\right)^{2}$, the original expression can be transformed into finding the extremum of the univariate function
$$
f(x)=x(1-x)+\left(\frac{1-x}{2}\right)^{2}(1-2 x)
$$

Thus, find the range $\left(0, \frac{7}{27}\right]$.