
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 equations and complete the following questions: $\sqrt{2+\frac{2}{3}}=2\sqrt{\frac{2}{3}}$, $\sqrt{3+\frac{3}{8}}=3\sqrt{\frac{3}{8}}$, $\sqrt{4+\frac{4}{{15}}}=4\sqrt{\frac{4}{{15}}}$, $\ldots \ldots $.<br/>$(1)$ Based on the above equations, what pattern can you find? Please express the pattern you found with a formula containing a natural number $n$, and indicate the range of values for $n$. Please explain the correctness of the pattern you found;<br/>$(2)$ According to the pattern you found, write a similar equation.

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