
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 solution process. For example, simplify $\frac{1}{{\sqrt{2}+\sqrt{1}}}$.
Solution: $\frac{1}{{\sqrt{2}+\sqrt{1}}}=\frac{{\sqrt{2}-\sqrt{1}}}{{({\sqrt{2}+\sqrt{1}})({\sqrt{2}-\sqrt{1}})}}=\frac{{\sqrt{2}-\sqrt{1}}}{{{{({\sqrt{2}})}^2}-{{({\sqrt{1}})}^2}}}=\frac{{\sqrt{2}-\sqrt{1}}}{1}=\sqrt{2}-1$.
Please answer the following questions.
$(1)$ Induction: Please directly write down the results of the following expressions: ① $\frac{1}{{\sqrt{6}+\sqrt{5}}}=$______; ② $\frac{1}{{\sqrt{11}-\sqrt{10}}}=\_\_\_\_\_\_$.
$(2)$ Application: Simplify $\frac{1}{{\sqrt{3}+\sqrt{2}}}+\frac{1}{{\sqrt{4}+\sqrt{3}}}+\frac{1}{{\sqrt{5}+\sqrt{4}}}+⋯+\frac{1}{{\sqrt{2023}+\sqrt{2022}}}$.
$(3)$ Extension: $\frac{1}{{\sqrt{3}+\sqrt{1}}}+\frac{1}{{\sqrt{5}+\sqrt{3}}}+\frac{1}{{\sqrt{7}+\sqrt{5}}}+⋯+\frac{1}{{\sqrt{2n+1}+\sqrt{2n-1}}}=\_\_\_\_\_\_$. (Express using a formula containing $n$, where $n$ is a positive integer)

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