Let $ABCD$ be a square with side length $8$. A second square $A_1B_1C_1D_1$ is formed by joining the midpoints of $AB$, $BC$, $CD$, and $DA$. A third square $A_2B_2C_2D_2$ is formed in the same way from $A_1B_1C_1D_1$, and a fourth square $A_3B_3C_3D_3$ from $A_2B_2C_2D_2$. Find the sum of the areas of these four squares.

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