
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:  
$1-\frac{1}{2}=\frac{1}{1×2},\frac{1}{2}-\frac{1}{3}=\frac{1}{2×3},\frac{1}{3}-\frac{1}{4}=\frac{1}{3×4} \ldots $,  
$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+…+\frac{1}{19×20}=(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+…+(\frac{1}{19}-\frac{1}{20})=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+…+\frac{1}{19}-\frac{1}{20}=1-\frac{1}{20}=\frac{19}{20}$.  
Based on the above solution process, answer the following questions:  
$(1)$ Calculate: $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+…+\frac{1}{2023×2024}$;  
$(2)$ Calculate: $1+\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}$;  
$(3)$ Calculate: $\frac{1}{1×3}+\frac{1}{3×5}+\frac{1}{5×7}+…+\frac{1}{99×101}$.

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