
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
A line of six equilateral triangles, each with a side length of $3\sqrt{3}$, are placed in such a way that the midpoint of the base of one triangle is a vertex of the next. What is the area of the region covered by the union of these six triangular regions?
A) $\frac{513\sqrt{3}}{16}$
B) $\frac{405\sqrt{3}}{8}$
C) $\frac{243\sqrt{3}}{8}$
D) $\frac{162\sqrt{3}}{4}$
E) $\frac{729\sqrt{3}}{16}$

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