Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of 
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms.

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