
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 \( f(x) = \left(\frac{x-1}{x+1}\right)^2 \) for \( x > 1 \), if the inequality \( (1 - \sqrt{x}) f^{-1}(x) > m(m - \sqrt{x}) \) holds for every \( x \) in the interval \(\left[\frac{1}{16}, \frac{1}{4}\right]\), find the range of the real number \( m \).

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