
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:
\[
\frac{1}{2} + \frac{\sqrt{3}}{2}i = \cos{\frac{\pi}{3}} + i\sin{\frac{\pi}{3}}
\]
\[
\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^2 = \cos{\frac{2\pi}{3}} + i\sin{\frac{2\pi}{3}}
\]
\[
\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^3 = \cos{\pi} + i\sin{\pi}
\]
\[
\left(\frac{1}{2} + \frac{\sqrt{4}}{2}i\right)^4 = \cos{\frac{4\pi}{3}} + i\sin{\frac{4\pi}{3}}
\]
...
According to this pattern, for any $n \in \mathbb{N}^*$, we can infer that $\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^n = $ \_\_\_\_\_\_.

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