
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
Read the following material and then answer the questions. When performing square root operations, sometimes we encounter expressions like $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify it as: $\frac{2}{\sqrt{3}+1}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3}+1)×(\sqrt{3}-1)}=\frac{2×(\sqrt{3}-1)}{{(\sqrt{3})}^{2}-{1}^{2}}=\sqrt{3}-1$. The steps of this simplification are called rationalizing the denominator. $(1)$ Simplify $\frac{2}{\sqrt{5}+\sqrt{3}$; $(2)$ Given that the integer part of $\frac{1}{2-\sqrt{3}}$ is $a$ and the decimal part is $b$, find the value of $a^{2}+b^{2}$.

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