
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
Scenario: In a math activity class, Teacher Wang wrote a series of equations on the blackboard:<br/>$\frac{1}{1×2}=1-\frac{1}{2}$,<br/>$\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$,<br/>$\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$,<br/>$\frac{1}{4×5}=\frac{1}{4}-\frac{1}{5}$,<br/>【Independent Thinking】(1) Find the pattern in the equations and calculate: $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\frac{1}{4×5}+⋯+\frac{1}{2021×2022}$;<br/>【Practical Exploration】(2) After a general study by the math activity group students on the above problem, they changed the difference between the two factors in the denominator to $2$. They proposed a new problem: $\frac{1}{1×3}+\frac{1}{3×5}+\frac{1}{5×7}+\frac{1}{7×9}+⋯+\frac{1}{2021×2023}$, please calculate;<br/>【Problem Extension】(3) Find the value of $\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+⋯+\frac{1}{1+2+3+⋯+2022}$.

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