
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
Observing the following expressions:<br/>$1-\frac{1}{{2}^{2}}=1-\frac{1}{4}=\frac{3}{4}=\frac{1}{2}\times \frac{3}{2}$;<br/>$1-\frac{1}{{3}^{2}}=1-\frac{1}{9}=\frac{8}{9}=\frac{2}{3}\times \frac{4}{3}$;<br/>$1-\frac{1}{{4}^{2}}=1-\frac{1}{16}=\frac{15}{16}=\frac{3}{4}\times \frac{5}{4}$;<br/>$1-\frac{1}{{5}^{2}}=1-\frac{1}{25}=\frac{24}{25}=\frac{4}{5}\times \frac{6}{5}$;<br/>$\ldots $<br/>$(1)$ Fill in the blanks according to the pattern you discovered: $1-\frac{1}{{6}^{2}}=\_\_\_\_\_\_\times \_\_\_\_\_\_$, $1-\frac{1}{1{0}^{2}}=\_\_\_\_\_\_\times \_\_\_\_\_\_$;<br/>$(2)$ Calculate using the pattern you discovered:<br/>$(1-\frac{1}{{2}^{2}})\times (1-\frac{1}{{3}^{2}})\times (1-\frac{1}{{4}^{2}})\times \ldots \times (1-\frac{1}{202{0}^{2}})\times (1-\frac{1}{202{1}^{2}})$.

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