
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 the function defined on \(\mathbf{R}^{+}\), \( f(x) = 
\begin{cases} 
\left|\log_3(x) - 1\right| & \text{if } 0 < x \leq 9,\\ 
4 - \sqrt{x} & \text{if } x > 9 
\end{cases} \), where \(a\), \(b\), and \(c\) are three distinct real numbers satisfying \(f(a) = f(b) = f(c)\), find the range of values for \(a \cdot b \cdot c\).

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