
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 \\(f(x) = \left( \frac{x-1}{x+1} \right)^2 (x > 1)\\)  
\\((1)\\) Find the inverse function of \\(f(x)\\) and its domain;  
\\((2)\\) If the inequality \\((1- \sqrt{x})f^{-1}(x) > a(a- \sqrt{x})\\) holds true for the interval \\(x \in \left[ \frac{1}{4}, \frac{1}{2} \right]\\), find the range of the real number \\(a\\).

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