
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
The polynomial \( Q(x) \) is a monic, quartic polynomial with real coefficients, having roots at \( e^{i\phi} \) and \( i e^{i\phi} \), where \( 0 < \phi < \frac{\pi}{6} \). When all four roots of \( Q(x) \) are plotted in the complex plane, they form a rectangle. If \( Q(0) \) is twice the area of this rectangle, find the product of the four roots.

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