Please reason step by step, and put your final answer within \boxed{}.
From 4 integers, any 3 are selected to find their average. Then, the sum of this average and the remaining integer is calculated. This yields 4 numbers: $8$, $12$, $10 \frac{2}{3}$, and $9 \frac{1}{3}$. Find the sum of the original 4 integers.
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