
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
Determine the area of the rhombus formed by the roots of the quartic equation
\[
z^4 + 4iz^3 + (5 + 5i)z^2 + (10 - i)z + (1 - 4i) = 0.
\]

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