
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 patterns: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}}\ldots $<br/>$(1)$ Guess: The $(n-1)$th equation is ______；<br/>$(2)$ Explain the correctness of your guess；<br/>$(3)$ Application: If $a\sqrt{\frac{7}{b}}=\sqrt{a+\frac{7}{b}}$, then $a+b=\_\_\_\_\_\_$.

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