
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
Consider a monic quartic polynomial $Q(x)$ with real coefficients, having four complex roots: $\cos \alpha + i \sin \alpha$, $\cos \alpha - i \sin \alpha$, $\cos(\alpha + \frac{\pi}{6}) + i \sin(\alpha + \frac{\pi}{6})$, and $\cos(\alpha + \frac{\pi}{6}) - i \sin(\alpha + \frac{\pi}{6})$, where $0 < \alpha < \frac{\pi}{6}$. The area of the quadrilateral formed by plotting these roots in the complex plane is equal to a quarter of $Q(0)$. Find $Q(0)$ and the sum of the four roots of this polynomial.

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