
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
Reading comprehension material: Removing the square root from the denominator is called rationalizing the denominator, for example: <br/>①$\frac{1}{\sqrt{5}}=\frac{1×\sqrt{5}}{\sqrt{5}×\sqrt{5}}=\frac{\sqrt{5}}{5}$; <br/>②$\frac{1}{\sqrt{2}-1}=\frac{1×(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)}=\frac{\sqrt{2}+1}{(\sqrt{2})^{2}-{1}^{2}}=\sqrt{2}+1$ and other operations are all rationalizing the denominator according to the above material. <br/>Based on the above material, <br/>$(1)$ Simplify: $\frac{1}{2\sqrt{3}}$; <br/>$(2)$ Simplify: $\frac{1}{\sqrt{3}-\sqrt{2}}$; <br/>$(3)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\ldots +\frac{1}{\sqrt{2023}+\sqrt{2022}$.

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