
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 expressions:<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/>$\ldots \ldots $<br/>$(1)$ Fill in the blanks according to the above expressions:<br/>①$\frac{1}{8×9}=$______；<br/>②$\frac{1}{n×(n+1)}=\_\_\_\_\_\_$(n$ is a positive integer).<br/>$(2)$ Calculate based on the above expressions and the pattern you discovered:<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+⋯+\frac{1}{2021×2022}+\frac{1}{2022×2023}$.

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