
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 solutions to the equations $z^2 = 9 + 9\sqrt{7}i$ and $z^2 = 5 + 5\sqrt{2}i$, where $i = \sqrt{-1}$, form the vertices of a parallelogram in the complex plane. Determine the area of this parallelogram, which can be written in the form $p\sqrt{q} - r\sqrt{s}$, where $p, q, r, s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number.
A) $\frac{2}{3}\sqrt{35} - \frac{2}{3}\sqrt{7}$
B) $\frac{3}{4}\sqrt{35} - \frac{3}{4}\sqrt{15}$
C) $\frac{1}{2}\sqrt{21} - 3\sqrt{3}$
D) $\frac{3}{8}\sqrt{15} - \frac{3}{8}\sqrt{7}$

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