
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
$(1)$ Observation: Fill in the blanks:<br/>①$\frac{2}{3}=\frac{2}{1×3}=1-\frac{1}{3}$,<br/>②$\frac{2}{15}=\frac{2}{3×5}=\frac{1}{3}-\frac{1}{5}$,<br/>③$\frac{2}{35}=\frac{2}{5×7}=\frac{1}{5}-\frac{1}{7}$,$\ldots \frac{2}{(2n-1)(2n+1)}=\_\_\_\_\_\_.$<br/>$ \left(2\right)$ Transfer application: Fill in the blanks:<br/>①$\frac{2}{2021×2023}=$______；<br/>②$\frac{1}{3×5}=\_\_\_\_\_\_\times (\frac{1}{3}-\frac{1}{5})$；<br/>$(3)$ Extended research: Calculate:<br/>$\frac{1}{1×6}+\frac{1}{6×11}+\frac{1}{11×16}+⋯+\frac{1}{10091×10096}+\frac{1}{10096×10101}$.

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