
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 $A=\frac{{x}^{2}-1}{{x}^{2}-2x+1}÷\frac{x+1}{x}+\frac{1}{x-1}$.  
$(1)$ Simplify $A$;  
$(2)$ When $x=(\sqrt{12}-\sqrt{\frac{4}{3}})\times \sqrt{3}$, find the value of $A$.

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