
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
When simplifying square roots, we sometimes encounter expressions like $\frac{2}{\sqrt{3}+1}$, which need to be further simplified as follows: $\frac{2}{\sqrt{3}+1}=\frac{2\sqrt{3}-1}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{2\sqrt{3}-1}{(\sqrt{3})^{2}-1^{2}}=\frac{2\sqrt{3}-1}{2}=\sqrt{3}-1$. The steps involved in this simplification process are called rationalizing the denominator. It can also be simplified using the following method: $\frac{2}{\sqrt{3}+1}=\frac{3-1}{\sqrt{3}+1}=\frac{(\sqrt{3})^{2}-1^{2}}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)(\sqrt{3}-1)}{\sqrt{3}+1}=\sqrt{3}-1$.

$(1)$ Simplify: $\frac{2}{\sqrt{7}+\sqrt{6}}$;  
$(2)$ Choose the appropriate method to simplify: $\frac{1}{\sqrt{n+1}+\sqrt{n}}$ (where $n$ is a positive integer);  
$(3)$ Find the value of $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\cdots+\frac{1}{\sqrt{2022}+\sqrt{2023}$.

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