
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
We know that $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=1$. Therefore, by multiplying both the numerator and denominator of $\frac{1}{\sqrt{3}+\sqrt{2}}$ by "$\sqrt{3}-\sqrt{2}$", the denominator becomes $1$. The original expression can be simplified to $\sqrt{3}-\sqrt{2}$, so we have $\frac{1}{\sqrt{3}+\sqrt{2}}=\sqrt{3}-\sqrt{2}$. Please solve the following questions following the method above:
$(1)$ Simplify: $\frac{1}{\sqrt{5}+2}=$______ $\frac{1}{\sqrt{6}-\sqrt{5}}=$______;
$(2)$ If $x=\frac{1}{3+2\sqrt{2}}$ and $y=\frac{1}{3-2\sqrt{2}}$, find the value of $\left(x-y\right)^{2}-xy$;
$(3)$ Calculate the value of the following expression based on the above rule: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{2022}+\sqrt{2021}}$.

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