Example 6 Solve the system of equations:
$$
\left\{\begin{array}{l}
x=1+\ln y, \\
y=1+\ln z, \\
z=1+\ln x .
\end{array}\right.
$$
Given that $x, y, z > 0$.
By the monotonicity of $f(x) = \ln x$, if $x > y$, then
$\ln x > \ln y$.
Thus, $z = 1 + \ln x > 1 + \ln y = x$.
Therefore, $y > z$,
which leads to, $x > y > z > x$. Contradiction.
Similarly, if $x < y$, it also leads to a contradiction.
Therefore, $x = y = z$.
Let $g(x) = 1 + \ln x$.
It is easy to see that $g(1) = 1$.
By the monotonicity of the function $g(x)$, $g(x)$ is strictly increasing. Thus, it intersects the $x$-axis at a unique point.
Therefore, the original system of equations has a unique solution
$$
(x, y, z) = (1, 1, 1).
$$