
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 simplifying square roots, we may encounter expressions like $\frac{2}{\sqrt{3}}$, $\sqrt{\frac{2}{5}}$, $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them.<br/>①$\frac{2}{\sqrt{3}}=\frac{2×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{2\sqrt{3}}{3}$; ②$\sqrt{\frac{2}{5}}=\sqrt{\frac{2×5}{5×5}}=\frac{\sqrt{10}}{5}$;<br/>③$\frac{2}{\sqrt{3}+1}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3}+1)×(\sqrt{3}-1)}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3})^{2}-1^{2}}=\frac{2(\sqrt{3}-1)}{2}=\sqrt{3}-1$.<br/>Similar simplification steps are called rationalizing the denominator.<br/>$(1)$ Simplify: $\frac{3}{\sqrt{6}}=$______, $\sqrt{\frac{2}{7}}=$______, $\frac{2}{\sqrt{5}-\sqrt{3}}=$______; <br/>$(2)$ Calculate: $\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{95}+\sqrt{97}}+\frac{1}{\sqrt{97}+\sqrt{99}}$.

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