9. (16 points) Given the function defined on $\mathbf{R}_{+}$
$$
f(x)=\left\{\begin{array}{ll}
\left|\log _{3} x-1\right|, & 09 .
\end{array}\right.
$$

Let $a, b, c$ be three distinct real numbers, satisfying $f(a)=f(b)=f(c)$. Find the range of $a b c$.
II. 9. Let's assume $a<b<c$.
Since $f(x)$ is strictly decreasing on the interval $(0,3]$, strictly increasing on the interval $[3,9]$, and strictly decreasing on the interval $[9,+\infty)$, and given $f(3)=0, f(9)=1$, we can deduce from the graph that
$$
a \in(0,3), b \in(3,9), c \in(9,+\infty),
$$

and $f(a)=f(b)=f(c) \in(0,1)$.
From $f(a)=f(b) \Rightarrow 1-\log _{3} a=\log _{3} b-1$
$\Rightarrow \log _{3} a+\log _{3} b=2$
$\Rightarrow a b=3^{2}=9 \Rightarrow a b c=9 c$.
Also, $0<f(c)=4-\sqrt{c}<1$, so $c \in(9,16)$.
Thus, $a b c=9 c \in(81,144)$.
Therefore, the range of $a b c$ is $(81,144)$.
[Note] For any $r \in(81,144)$, let $c_{0}=\frac{r}{9}$, then $c_{0} \in(9,16)$. Thus, $f\left(c_{0}\right) \in(0,1)$. Draw a horizontal line $l$ through the point $\left(c_{0}, f\left(c_{0}\right)\right)$, then the line $l$ intersects the graph of $f(x)$ at two other points $(a, f(a))$, $(b, f(b))(a \in(0,3), b \in(3,9))$, satisfying:
$f(a)=f(b)=f(c)$, and $a b=9$.
Thus, $a b c=r$.