
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
Given the equations $z^{2}=4+4 \sqrt{15} i$ and $z^{2}=2+2 \sqrt{3} i$, the roots are the coordinates of the vertices of a parallelogram in the complex plane. Find the value of the area $S$ of the parallelogram.

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