Determine all real numbers $a, b, c, d$ that satisfy the following system of equations.

$$
\left\{\begin{array}{r}
a b c+a b+b c+c a+a+b+c=1 \\
b c d+b c+c d+d b+b+c+d=9 \\
c d a+c d+d a+a c+c+d+a=9 \\
d a b+d a+a b+b d+d+a+b=9
\end{array}\right.
$$
Answer: $a=b=c=\sqrt[3]{2}-1, d=5 \sqrt[3]{2}-1$.

Substituting $A=a+1, B=b+1, C=c+1, D=d+1$, we obtain

$$
\begin{aligned}
& A B C=2 \\
& B C D=10 \\
& C D A=10 \\
& D A B=10
\end{aligned}
$$

Multiplying (1), (2), (3) gives $C^{3}(A B D)^{2}=200$, which together with (4) implies $C^{3}=2$. Similarly we find $A^{3}=B^{3}=2$ and $D^{3}=250$. Therefore the only solution is $a=b=c=\sqrt[3]{2}-1, d=5 \sqrt[3]{2}-1$.