
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
Suppose $\alpha = \frac{1}{2} + \frac{1}{2}i\sqrt{3}$. Let $T$ denote all points in the complex plane of the form $x + y\alpha + z\alpha^2$, where $0 \leq x \leq 2, 0 \leq y \leq 2,$ and $0 \leq z \leq 2$. Calculate the area of $T$.

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