
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
Let $Q(x)$ be a monic, quartic polynomial with real coefficients. Two of its roots are $\cos \phi + i \sin 2\phi$ and $\cos 2\phi + i \sin \phi,$ where $0 < \phi < \frac{\pi}{6}$. When the four roots of $Q(x)$ are plotted in the complex plane, they form a parallelogram whose area is $Q(0)$. Find the sum of the four roots.

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