Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. What is the area of the region of the plane that is covered by the union of the five triangular regions? Express your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of a + b.

Let's think step by step and output the final answer within \boxed{}.
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