
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
Given the following equations:

\\(\sqrt{2}=2\cos \frac{\pi }{4}\\),

\\(\sqrt{2+\sqrt{2}}=2\cos \frac{\pi }{8}\\),

\\(\sqrt{2+\sqrt{2+\sqrt{2}}}=2\cos \frac{\pi }{16}\\), ...

Please induce the $n^{th}$ $(n\in \mathbb{N}^{*})$ equation:

\\(\underbrace {\sqrt{2+ \sqrt{2+\cdots+\sqrt{2}}}}_{n\text{ square roots}}=\) \_\_\_\_\_\_\_\_\_\_\_.

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