
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 calculations and complete the following questions:  
$\sqrt{\frac{1}{2}-\frac{1}{3}}=\frac{1}{2}\sqrt{\frac{2}{3}}$ Verification: $\sqrt{\frac{1}{2}-\frac{1}{3}}=\sqrt{\frac{1}{2\times3}}=\sqrt{\frac{2}{{2}^{2}\times3}}=\frac{1}{2}\sqrt{\frac{2}{3}$;  
$\sqrt{\frac{1}{2}(\frac{1}{3}-\frac{1}{4})}=\frac{1}{3}\sqrt{\frac{3}{8}}$ Verification: $\sqrt{\frac{1}{2}(\frac{1}{3}-\frac{1}{4})}=\sqrt{\frac{1}{2\times3\times4}}=\sqrt{\frac{2}{2\times3^{2}\times4}}=\frac{1}{3}\sqrt{\frac{3}{8}$;  
$\sqrt{\frac{1}{3}(\frac{1}{4}-\frac{1}{5})}$ Verification: $\sqrt{\frac{1}{3}(\frac{1}{4}-\frac{1}{5})}=\sqrt{\frac{1}{3\times4\times5}}=\sqrt{\frac{4}{3\times4^{2}\times5}}=\frac{1}{4}\sqrt{\frac{4}{15}$;  
$(1)$ According to the basic idea of the above three equations and their verification process, guess the transformation result of $\sqrt{\frac{1}{4}(\frac{1}{5}-\frac{1}{6})}$ and verify it;  
$(2)$ Based on the rules reflected in the above equations, express it in the form of an equation containing $n(n\geqslant 1$ is a natural number).

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