
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
Find the vertices formed by the solutions of the equations \( z^2 = 5 + 5\sqrt{7}i \) and \( z^2 = 3 + 3\sqrt{2}i \) in the complex plane. Compute the area of the parallelogram formed by these vertices.
A) \( 5\sqrt{7} - 2\sqrt{2} \)
B) \( 4\sqrt{7} - 3\sqrt{2} \)
C) \( 5\sqrt{7} - 3\sqrt{2} \)
D) \( 6\sqrt{7} - 3\sqrt{2} \)

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