
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 quadratic polynomial $Q(x) = x^2 + px + q$ whose roots are $\sin \frac{\pi}{6}$ and $\sin \frac{5\pi}{6}$. Find the value of $pq$.
A) $-\frac{1}{2}$
B) $-\frac{1}{4}$
C) $\frac{1}{4}$
D) $-\frac{1}{8}$

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