
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 expressions and their verification processes:  
$\sqrt{2+\frac{2}{3}}=2\sqrt{\frac{2}{3}}$,  
Verification: $\sqrt{2+\frac{2}{3}}=\sqrt{\frac{2×3+2}{3}}=\sqrt{\frac{2^3}{3}}=2\sqrt{\frac{2}{3}}$.  
$\sqrt{3+\frac{3}{8}}=3\sqrt{\frac{3}{8}}$,  
Verification: $\sqrt{3+\frac{3}{8}}=\sqrt{\frac{3×8+3}{8}}=\sqrt{\frac{3^3}{8}}=3\sqrt{\frac{3}{8}}$.  
$(1)$ Following the basic idea of the above two equations and their verification processes, guess the transformation result of $\sqrt{4+\frac{4}{15}}$ and verify it;  
$(2)$ Based on the rules reflected in the above expressions, directly write out the equation represented by $n(n\geqslant 2$, and $a$ is an integer).

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