
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
Observing the equations:<br/>$\frac{1}{1×2}=1-\frac{1}{2}$; $\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$; $\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$.<br/>By adding both sides of the above three equations, we get:<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=1-\frac{1}{4}=\frac{3}{4}$.<br/>$(1)$ Make a conjecture and write down: $\frac{1}{n(n+1)}=\_\_\_\_\_\_.$<br/>$(2)$ Calculate:<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\ldots +\frac{1}{2021×2022}$.<br/>$(3)$ Investigate and calculate:<br/>$\frac{1}{1×4}+\frac{1}{4×7}+\frac{1}{7×10}+\ldots +\frac{1}{2020×2023}$.

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