
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
We know that simplifying $\frac{\sqrt{3}}{\sqrt{5}}$ involves multiplying both the numerator and denominator by $\sqrt{5}$ to eliminate the square root in the denominator, i.e., $\frac{\sqrt{3}}{\sqrt{5}}=\frac{\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}=\frac{\sqrt{15}}{5}$. Similarly, $\frac{1}{\sqrt{2}+1}$ can also be simplified by multiplying both the numerator and denominator by $\sqrt{2}-1$ to eliminate the square root in the denominator, i.e., $\frac{1}{\sqrt{2}+1}=\frac{1 \times (\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\frac{\sqrt{2}-1}{(\sqrt{2})^2-1^2}=\sqrt{2}-1$.  
$(1)$ Simplify the expression $\frac{1}{\sqrt{3}+1}$ using the same method.  
$(2)$ Given $x=\frac{1}{2+\sqrt{3}}$ and $y=\frac{1}{2-\sqrt{3}}$, find the value of $x^{2}-y^{2}$.

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