
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: $\sqrt{2}=2\cos\frac{\pi}{4}$, $\sqrt{2+\sqrt{2}}=2\cos\frac{\pi}{8}$, $\sqrt{2+\sqrt{2+\sqrt{2}}}=2\cos\frac{\pi}{16}$, ${\dots}$ Please deduce the $n$-th equation: $\sqrt{2+\dots+ \sqrt{2+ \sqrt{2} \text{ ($n$ twos)}}}=$ ______.

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