
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a function \( f(x) \) defined on \( \mathbf{R}^{+} \) as
\[ 
f(x) = \begin{cases} 
\left|\log_3 x - 1\right|, & 0 < x \leq 9, \\ 
4 - \sqrt{x}, & x > 9 
\end{cases} 
\]
and let \( a, b, c \) be three distinct real numbers that satisfy \( f(a) = f(b) = f(c) \). Determine the range of \( abc \).

Present the answer in LaTex format: \boxed{Your answer}
user