
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 sextic polynomial with real coefficients. It has roots $\cos (2\theta) + i \sin (2\theta)$, and $\sin (2\theta) + i \cos (2\theta)$, where $0 < \theta < \frac{\pi}{6}$. Add to $Q(x)$, roots at $x = 1$ and $x = -1$. The polynomial $Q(x)$ also has a quadratic factor $(x^2 + px + q)$ such that the remaining roots have a sum of zero. Find the sum of all roots of $Q(x)$.

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