
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 $(0, +\infty)$ that satisfies the functional equation $f\left( \frac{x}{y} \right) = f(x) - f(y)$, and $f(x) < 0$ when $x > 1$,
1. Find the value of $f(1)$;  
2. Determine the monotonicity of $f(x)$ and provide the reasoning;  
3. If $f(3) = -1$, solve the inequality $f(|x|) < -2.$

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