已知正实数 $x, y, z$ 满足: $x y+y z+z x \neq 1$, 且
$$
\frac{\left(x^2-1\right)\left(y^2-1\right)}{x y}+\frac{\left(y^2-1\right)\left(z^2-1\right)}{y z}+\frac{\left(z^2-1\right)\left(x^2-1\right)}{z x}=4 .
$$
(1) 求 $\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}$ 的值.
(2) 证明: $9(x+y)(y+z)(z+x) \geq 8 x y z(x y+y z+z x)$.