
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 the following equations and then answer the questions:
① $\sqrt{1+\frac{1}{{1}^{2}}+\frac{1}{{2}^{2}}}=1+\frac{1}{1}-\frac{1}{1+1}=1\frac{1}{2}$;
② $\sqrt{1+\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}}=1+\frac{1}{2}-\frac{1}{2+1}=1\frac{1}{6}$;
③ $\sqrt{1+\frac{1}{{3}^{2}}+\frac{1}{{4}^{2}}}=1+\frac{1}{3}-\frac{1}{3+1}=1\frac{1}{12}$.
$(1)$ Based on the information provided by the three equations above, please guess $\sqrt{1+\frac{1}{{4}^{2}}+\frac{1}{{5}^{2}}}=\_\_\_\_\_\_=\_\_\_\_\_\_$.
$\sqrt{1+\frac{1}{1{9}^{2}}+\frac{1}{2{0}^{2}}}=\_\_\_\_\_\_=\_\_\_\_\_\_$.
$(2)$ According to the pattern reflected by the above equations, please write an equation expressed with an expression containing $n$ (where $n$ is a positive integer).

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