
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
First, observe the following calculations, then complete:<br/>$\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$;<br/>$\frac{1}{\sqrt{4}+\sqrt{3}}=\frac{\sqrt{4}-\sqrt{3}}{(\sqrt{4}+\sqrt{3})(\sqrt{4}-\sqrt{3})}=\sqrt{4}-\sqrt{3}$;<br/>$(1)$ Calculate: $\frac{1}{\sqrt{5}+\sqrt{4}$;<br/>$(2)$ Observe the above solution process, please write directly the result of $\frac{1}{\sqrt{n}+\sqrt{n-1}}(n≥2)$ as ______;<br/>$(3)$ Based on your conjecture and induction, apply the rule to calculate: Find the value of $(\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+…+\frac{1}{\sqrt{2023}+\sqrt{2022}})(\sqrt{2023}+1)$.

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