
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 equations and answer the questions below:<br/>The first equation: $(\sqrt{2}+1)(\sqrt{2}-1)=1$;<br/>The second equation: $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=1$;<br/>The third equation: $(2+\sqrt{3})(2-\sqrt{3})=1$;<br/>The fourth equation: $(\sqrt{5}+2)(\sqrt{5}-2)=1$;<br/>$\ldots \ldots $<br/>$(1)$ According to the above pattern, write down the 10th equation: ______;<br/>$(2)$ Using the above pattern, compare the sizes: $\sqrt{18}-\sqrt{17}$______$\sqrt{19}-\sqrt{18}$ (Fill in $ \gt $, $ \lt $, or $=$);<br/>$(3)$ Calculate: $\frac{3}{\sqrt{2}+1}+\frac{3}{\sqrt{3}+\sqrt{2}}+\frac{3}{2+\sqrt{3}}+\ldots +\frac{3}{\sqrt{99}+\sqrt{98}}$.

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