
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 defined on the positive real numbers, \( f(x) \), where:

\[ f(x) = \begin{cases} 
\left|\log_{3} x - 1\right| & \text{for } 0 < x \leq 9, \\
4 - \sqrt{x} & \text{for } x > 9,
\end{cases} \]

Let \( a \), \( b \), and \( c \) be three distinct real numbers such that \( f(a) = f(b) = f(c) \). Determine the range of the value of \( a \cdot b \cdot c \).

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