
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
Observation and Thinking:  
① $2\sqrt{\frac{2}{3}}=\sqrt{2\frac{2}{3}}$  
② $3\sqrt{\frac{3}{8}}=\sqrt{3\frac{3}{8}}$  
③ $4\sqrt{\frac{4}{15}}=\sqrt{4\frac{4}{15}}$  
Verification of Equation ①:  
$2\sqrt{\frac{2}{3}}=\sqrt{\frac{2^3}{3}}=\sqrt{\frac{{({{2^3}-2})+2}}{{{2^2}-1}}}=\sqrt{\frac{{2({{2^2}-1})+2}}{{{2^2}-1}}}=\sqrt{2\frac{2}{3}$  
Verification of Equation ②:  
$3\sqrt{\frac{3}{8}}=\sqrt{\frac{3^3}{8}}=\sqrt{\frac{{({{3^3}-3})+3}}{{{3^2}-1}}}=\sqrt{\frac{{3({{3^2}-1})+3}}{{{3^2}-1}}}=\sqrt{3\frac{3}{8}$  
$(1)$ Following the verification process of Equation ① and Equation ② above, please write out the verification process of Equation ③.  
$(2)$ Conjecture: $\sqrt{5\frac{5}{24}}=$______  
$(3)$ Try to express and verify this pattern using an equation containing $n$ ($n$ is a natural number, and $n\geqslant 2$).

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