
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 problem-solving process:<br/>The first equation: $\sqrt{1-\frac{3}{4}}=\sqrt{\frac{1}{4}}=\sqrt{(\frac{1}{2})^2}=\frac{1}{2}$.<br/>The second equation: $\sqrt{1-\frac{5}{9}}=\sqrt{\frac{4}{9}}=\sqrt{(\frac{2}{3})^2}=\frac{2}{3}$;<br/>The third equation: $\sqrt{1-\frac{7}{16}}=\sqrt{\frac{9}{16}}=\sqrt{(\frac{3}{4})^2}=\frac{3}{4}$;<br/>$\ldots $<br/>$(1)$ According to the pattern you discovered, please write down the fourth equation: ______<br/>$(2)$ According to the pattern you discovered, please write down the nth equation (n is a positive integer): ______;<br/>$(3)$ Using this pattern, calculate: $\sqrt{(1-\frac{3}{4})×(1-\frac{5}{9})×(1-\frac{7}{16})×⋯×(1-\frac{21}{121})}$;

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