
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
On each side of a unit square, an equilateral triangle with side length 2 is constructed. At one end of each triangle, a new equilateral triangle of side length 1 is added. Another triangle of side length 1 is added at the opposite end of each initial triangle. Similar to the original, their interiors and the square’s interior do not overlap. Let $R'$ be the region comprising the square and attached triangles, and $S'$ the minimal convex polygon encompassing $R'$. Compute the area that lies within $S'$ but outside $R'$.
A) $4.5\sqrt{3} - 0.5$
B) $6\sqrt{3} + 1$
C) $7.5\sqrt{3} - 1$
D) $9\sqrt{3} - 2$
E) $10.5\sqrt{3} - 3$

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