
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 $\theta = \frac{1}{3} + \frac{1}{3}i\sqrt{2}$. Define $T$ as the set of all points in the complex plane represented by $x+y\theta+z\theta^2$, where $0 \leq x \leq 1$, $0 \leq y \leq 1$, and $0 \leq z \leq 1$. Determine the area of $T$.
A) $\frac{\sqrt{2}}{9}$  
B) $\frac{2\sqrt{2}}{9}$  
C) $\frac{4}{9}\sqrt{2}$  
D) $\frac{8}{9}\sqrt{2}$  
E) $\frac{1}{3}$

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