Consider the sets $\begin{aligned} A &= \{15, 22, 31, 45, 56, 67, 79 \} \\ B &= \{2, 3, 5, 6, 7, 9,12, 24 \} \\ C &= \{14, 21, 31, 40, 53, 62, 78, 81, 90 \} \\ D &= \{11,12, 17,23,54,55,62,98,110,111 \}. \end{aligned}$

How many ways are there to choose a number out of each set such that the sum of the four numbers is a multiple of $8$?

This problem is posed by Daniel W.

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