
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 equations and answer the following questions:<br/>①$\sqrt{1+\frac{1}{{1}^{2}}+\frac{1}{{2}^{2}}}=1+\frac{1}{1}-\frac{1}{1+1}=1\frac{1}{2}$;<br/>②$\sqrt{1+\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}}=1+\frac{1}{2}-\frac{1}{2+1}=1\frac{1}{6}$;<br/>③$\sqrt{1+\frac{1}{{3}^{2}}+\frac{1}{{4}^{2}}}=1+\frac{1}{3}-\frac{1}{3+1}=1\frac{1}{12}$.<br/>$(1)$ Based on the information provided by the above three equations, guess the result of $\sqrt{1+\frac{1}{{4}^{2}}+\frac{1}{{5}^{2}}}$ and verify it;<br/>$(2)$ According to the pattern reflected by the above equations, try to write an equation represented by $n$ ($n$ is a positive integer);<br/>$(3)$ Use the above pattern to calculate $\sqrt{\frac{101}{100}+\frac{1}{121}$ (write out the process following the example).

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