
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
$(1)$ Calculate: $(\sqrt{3}+1)(\sqrt{3}-1)+\sqrt{6}\div \sqrt{2}$.  
$(2)$ Simplify first, then evaluate: $(\frac{1}{x-1}-\frac{1}{x+1})\div \frac{x+2}{{x}^{2}-1}$, where $x=\sqrt{2}-2$.

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