
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 deduce the $n^{th}$ ($n\in \mathbb{N}^*$) equation: $\sqrt{2+\cdots +\sqrt{2+\sqrt{2}}}= \quad .$

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