
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:<br/>$\frac{1}{1×2}=1-\frac{1}{2}=\frac{1}{2}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}=\frac{2}{3}$;<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=\frac{3}{4}$;<br/>$\ldots $<br/>Based on the pattern you discovered, answer the following questions:<br/>$(1)\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}×\frac{1}{4×5}=$______;<br/>$(2)$If $n$ is a positive integer, then $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\frac{1}{4×5}+…+\frac{1}{n(n+1)}=$______;<br/>$(3)$Calculate $1-\frac{1}{1×2}-\frac{1}{2×3}-\frac{1}{3×4}-…-\frac{1}{99×100}$.

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