
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 $\omega' = -\frac{1}{4} + \frac{1}{4}i\sqrt{7}.$ Define $S'$ to be the set of all points in the complex plane of the form $a + b\omega' + c\omega'^2$, where $0 \leq a \leq 2$, $0 \leq b \leq 1$, and $0 \leq c \leq 1$. Determine the area of $S'$.
A) $\frac{1}{4}\sqrt{7}$
B) $\frac{1}{2}\sqrt{7}$
C) $\frac{3}{4}\sqrt{7}$
D) $\sqrt{7}$
E) $\frac{1}{2}\sqrt{3}$

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