
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:<br/>$\frac{1}{1×2}=1-\frac{1}{2}$, $\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$, $\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$, adding both sides of these three equations, we get $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=1-\frac{1}{4}=\frac{3}{4}$. Based on the above information, answer the following related questions.<br/>$(1)$ Fill in the blank: $\frac{1}{2022×2023}=\_\_\_\_\_\_$; (written in subtraction form)<br/>$(2)$ Calculate: $1-\frac{1}{2}-\frac{1}{6}-\frac{1}{12}-\frac{1}{20}-\frac{1}{30}$;<br/>$(3)$ Calculate: $\frac{2}{2×4}+\frac{2}{4×6}+\frac{2}{6×8}+⋯+\frac{2}{2022×2024}$.

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