10. (14 points) Given positive real numbers $a, b, c, d$ satisfying $a+b+c+d=abcd$. Find the minimum value of $\sum a^{4}(bcd-1)$, where “$\sum$” denotes the cyclic sum.

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10. (14 points) Given positive real numbers $a, b, c, d$ satisfying $a+b+c+d=abcd$. Find the minimum value of $\sum a^{4}(bcd-1)$, where “$\sum$” denotes the cyclic sum.
10. Since $a, b, c, d > 0$, and $a + b + c + d = abcd$, we have $a(bcd - 1) = b + c + d$.
By the AM-GM inequality,
$$
\begin{array}{l}
abcd = a + b + c + d \geqslant 4 \sqrt[4]{abcd} \\
\Rightarrow abcd \geqslant (\sqrt[3]{4})^4,
\end{array}
$$

where equality holds if and only if $a = b = c = d = \sqrt[3]{4}$. Therefore,
$$
\begin{array}{l}
\sum a^4(bcd - 1) = \sum a^3(b + c + d) \\
\geqslant 12 \sqrt[12]{\prod (a^3 b a^3 c a^3 d)} \\
= 12 \sqrt[12]{a^9 b^9 c^9 d^9 (bcd)(cda)(dab)(abc)} \\
= 12 abcd = 12 (\sqrt[3]{4})^4 = 48 \sqrt[3]{4},
\end{array}
$$

where,