
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 \( Q(z) = z^8 + \left(3\sqrt{3} + 5\right)z^4 - \left(3\sqrt{3} + 8\right) \). What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of \( Q(z) \)?

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