
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
Observe the following set of equations and then answer the questions: $(\sqrt{2}+1)(\sqrt{2}-1)=1$, $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=1$, $(\sqrt{4}+\sqrt{3})(\sqrt{4}-\sqrt{3})=1$, $({\sqrt{5}+\sqrt{4}})({\sqrt{5}-\sqrt{4}})=1$, $\ldots \ldots $.  
$(1)$ Using the above pattern, calculate: $\frac{1}{{\sqrt{2}+1}}+\frac{1}{{\sqrt{3}+\sqrt{2}}}+\frac{1}{{\sqrt{4}+\sqrt{3}}}+\ldots+\frac{1}{{\sqrt{2023}+\sqrt{2022}}}$;  
$(2)$ Using the above pattern, compare the sizes of $\sqrt{99}-\sqrt{98}$ and $\sqrt{98}-\sqrt{97}$.

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