
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 $Q(z) = z^6 + (6 - 3\sqrt{2})z^3 + (2\sqrt{2} - 3)$. What is the minimum area enclosed by the hexagon in the complex plane formed by the vertices at the zeros of $Q(z)$?

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