
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:①$\frac{1}{1×2}=1-\frac{1}{2}$；②$\frac{1}{2}-\frac{1}{3}$；③$\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$. Adding both sides of the above three equations, we get $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=1-\frac{1}{4}=\frac{3}{4}$.<br/>$(1)$ Please write down the $n$th equation:______；<br/>$(2)$ Given $|ab-2|$ and $\left(b-1\right)^{2}$ are opposite in sign, find the value of $\frac{1}{ab}+\frac{1}{(a+1)(b+1)}+\frac{1}{(a+2)(b+2)}+⋯+\frac{1}{(a+2023)(b+2023)}$；<br/>$(3)$ Investigate and 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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