
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 following questions: When simplifying algebraic expressions, sometimes we encounter expressions like $\frac{5}{\sqrt{3}}$, $\frac{2}{\sqrt{3}+1}$, in fact, we can further simplify them:<br/>$($1) $\frac{5}{\sqrt{3}}=\frac{5×\sqrt{3}}{\sqrt{3}×\sqrt{3}}=\frac{5}{3}\sqrt{3}$;<br/>$($2) $\frac{2}{\sqrt{3}+1}=\frac{2×(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{2(\sqrt{3}-1)}{(\sqrt{3})^{2}-1}=\sqrt{3}-1$;<br/>$($3) $\frac{2}{\sqrt{3}+1}=\frac{3-1}{\sqrt{3}+1}=\frac{(\sqrt{3})^{2}-{1}^{2}}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)(\sqrt{3}-1)}{\sqrt{3}+1}=\sqrt{3}-1$. The method of simplification mentioned above is called rationalizing the denominator.<br/>$(1)$ Simplify $\frac{2}{\sqrt{5}+\sqrt{3}}$ using different methods:<br/>① Refer to formula (2) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>② Refer to formula (3) to simplify $\frac{2}{\sqrt{5}+\sqrt{3}}=\_\_\_\_\_\_.$<br/>$(2)$ Simplify: $\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{99}+\sqrt{97}}$.

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