
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 \( 0 < a < 1 \), a function \( f(t) \) satisfies: for any \( x > 0 \), \( f\left(\log_{a} x\right) = \frac{a\left(x^{2} - 1\right)}{x\left(a^{2} - 1\right)} \), and \( m > n > 0 \). Compare the sizes of \( f\left(\frac{1}{m}\right) \) and \( f\left(\frac{1}{n}\right) \).

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