
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 $z = -\frac{1}{2} + \frac{\sqrt{3}i}{2}$.
(1) If $\overline{z}$ is the complex conjugate of $z$, find the value of $\overline{z}^2 + \overline{z} + 1$;
(2) By analogy with the knowledge of series, find the value of $S_{2016} = 1 + z + z^2 + \ldots + z^{2015}$.

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