
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:  
$$\frac {1}{2} + \frac { \sqrt {3}}{2}i = \cos \frac {\pi}{3} + i\sin \frac {\pi}{3}$$  
$$(\frac {1}{2} + \frac { \sqrt {3}}{2}i)^2 = \cos \frac {2\pi}{3} + i\sin \frac {2\pi}{3}$$  
$$(\frac {1}{2} + \frac { \sqrt {3}}{2}i)^3 = \cos\pi + i\sin\pi$$  
$$(\frac {1}{2} + \frac { \sqrt {4}}{2}i)^4 = \cos \frac {4\pi}{3} + i\sin \frac {4\pi}{3}$$  
...  
Following this pattern, we can infer that for any $n \in \mathbb{N}^*$, $(\frac {1}{2} + \frac { \sqrt {3}}{2}i)^n = $ ______.

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