
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 \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt3\). Define \(T\) as the set of all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 2\), and \(0 \leq c \leq 1\). Determine the area of \(T\).
A) \(4\sqrt{3}\)
B) \(6\sqrt{3}\)
C) \(8\sqrt{3}\)
D) \(12\sqrt{3}\)
E) \(16\sqrt{3}\)

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