
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 the interval $(0, +\infty)$ satisfying $f\left(\frac{x_1}{x_2}\right) = f(x_1) - f(x_2)$, and when $x > 1$, $f(x) < 0$.
(1) Find the value of $f(1)$;
(2) Determine the monotonicity of $f(x)$;
(3) If $f(3) = -1$, find the minimum value of $f(x)$ on the interval $[2, 9]$.

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