
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 that $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3$, find the area of $S$, where $S$ denotes all points in the complex plane of the form $a+b\omega+c\omega^2$, where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$

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