
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 calculation process:<br/>$\frac{1}{\sqrt{2}+1}=\frac{1×(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\sqrt{2}-1$;<br/>$\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1×(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$;<br/>$\frac{1}{\sqrt{5}+2}=\frac{1×(\sqrt{5}-2)}{(\sqrt{5}+2)(\sqrt{5}-2)}=\sqrt{5}-2$;<br/>Please solve the following problems:<br/>$(1)$ Simplify: $\frac{1}{\sqrt{3}+2}=$______;<br/>$(2)$ According to the above pattern, please directly write $\frac{1}{\sqrt{n+1}+\sqrt{n}}=$______;<br/>$(3)$ Using the above method, simplify: $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots +\frac{1}{\sqrt{2022}+\sqrt{2023}}$.

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