Example 22 (Mathematical Problem 1434 from "Mathematical Bulletin") Let positive numbers $x, y, z$ be not equal to 1, and $xy + yz + zx = 1$. Try to find the maximum value of $f(x, y, z) = x(1-y^2)(1-z^2) + y(1-z^2)(1-x^2) + z(1-x^2)(1-y^2)$.