
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
First observe: $1-\frac{1}{{2}^{2}}=\frac{1}{2}\times \frac{3}{2}$, $1-\frac{1}{{3}^{2}}=\frac{2}{3}\times \frac{4}{3}$, $1-\frac{1}{{4}^{2}}=\frac{3}{4}\times \frac{5}{4}$, $\ldots$  
$(1)$ Based on the above formulas, please write: $1-\frac{1}{{8}^{2}}=( $______).  
$(2)$ Investigate the pattern and fill in the blanks: $1-\frac{1}{{n}^{2}}=\_\_\_\_\_\_\times \_\_\_\_\_\_$.  
$(3)$ Calculate: $(1-\frac{1}{{2}^{2}})\cdot (1-\frac{1}{{3}^{2}})\cdot (1-\frac{1}{{4}^{2}})\ldots \cdot (1-\frac{1}{201{9}^{2}})$.

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