Let $\mathcal{K}$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt{3}$ along a common hypotenuse. Eight copies of $\mathcal{K}$ are used to form the polygon shown below. Let $A$ be the leftmost point of the polygon and $D$ be the top-rightmost point of the polygon. What is the distance $AD$?
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