
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
Consider the following equations:  
$\sqrt{1+\frac{1}{{1}^{2}}+\frac{1}{{2}^{2}}}=1+\frac{1}{1}-\frac{1}{2}=1\frac{1}{2}=\frac{3}{2}$;  
$\sqrt{1+\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}}=1+\frac{1}{2}-\frac{1}{3}=1\frac{1}{6}=\frac{7}{6}$;  
$\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}=1+\frac{1}{3}-\frac{1}{4}=1\frac{1}{12}=\frac{13}{12}$;  
Based on the patterns reflected in the above three equations, please make a conjecture:  
$(1)\sqrt{1+\frac{1}{5^2}+\frac{1}{6^2}}=$______;  
$(2)\sqrt{1+\frac{1}{{n}^{2}}+\frac{1}{(n+1)^{2}}}=\_\_\_\_\_\_;(n$ is a positive integer);  
$(3)$ Calculate using the above pattern: $\sqrt{\frac{37}{36}+\frac{1}{49}}$.

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