
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: $2\sqrt{\frac{2}{3}}=\sqrt{2+\frac{2}{3}}$, verify: $2\sqrt{\frac{2}{3}}=\sqrt{\frac{{{2^3}}}{3}}=\sqrt{\frac{{{2^3}-2+2}}{{{2^2}-1}}}=\sqrt{2+\frac{2}{3}$; $3\sqrt{\frac{3}{8}}=\sqrt{3+\frac{3}{8}}$, verify: $3\sqrt{\frac{3}{8}}=\sqrt{\frac{{{3^3}}}{8}}=\sqrt{\frac{{{3^3}-3+3}}{{{3^2}-1}}}=\sqrt{3+\frac{3}{8}$. <br/> $(1)$ Following the basic idea of the above two equations and their verification processes, guess the result of the transformation of $4\sqrt{\frac{4}{{15}}}$. <br/> $(2)$ Write an equation expressed by $n$ ($n$ is any natural number, and $n\geqslant 2$) to reflect the pattern of the above expressions and provide a proof.

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