
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
Let \( f(x) \) be an increasing function defined on \( (0, +\infty) \) such that \( f\left(\frac{x}{y}\right) = f(x) - f(y) \), and \( f(3) = 1 \). Solve the inequality
$$
f(x) - f\left(\frac{1}{x-5}\right) \geqslant 2.
$$

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