Let $A$ be a regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. The ratio of the area of $B$ to the area of $A$ can be written in simplest form as $\frac{a + \sqrt{b}}{c}$, where $a$, $b$, and $c$ are integers. Find $a + b + c$.

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