
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 expressions:<br/>The $1$st expression: $\frac{1}{1×2}=1-\frac{1}{2}$,<br/>The $2$nd expression: $\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$,<br/>The $3$rd expression: $\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$.<br/>$\ldots $<br/>Based on this pattern, answer the following questions:<br/>$(1)$ $\frac{1}{99×100}=\_\_\_\_\_\_.$<br/>$(2)$ The $n$th expression is ______.<br/>$(3)$ Using the above pattern, calculate: $\frac{1}{2×4}+\frac{1}{4×6}+\frac{1}{6×8}+…+\frac{1}{2022×2024}$.

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