
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 $f(x)=\left( \frac {x+1}{x}\right)^{2}$ for $x > 0$.  
$(1)$ Find the inverse function $f^{-1}(x)$;  
$(2)$ If for $x\geqslant 2$, the inequality $(x-1)f^{-1}(x) > a(a- \sqrt {x})$ always holds, find the range of the real number $a$.

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