
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 $\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}$, define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2$, where $0 \leq a \leq 0.5$, $0 \leq b \leq 1$, and $0 \leq c \leq 1$. Calculate the area of $T$.

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