
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 $A=(a-\frac{{a}^{2}}{a+b})\div \frac{{a}^{2}{b}^{2}}{{a}^{2}-{b}^{2}}$.  
$(1)$ Simplify $A$;  
$(2)$ If point $P\left(a,b\right)$ is the intersection point of the line $y=x-2$ and the inverse proportion function $y=\frac{1}{x}$, find the value of $A$.

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