Let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of three-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\) has a \(\mathbf{N}(\mathbf{0}, \mathbf{I})\) distribution. Consider the transformation \(\mathbf{g}(\mathbf{x})=\left[x_{1} x_{2}+x_{3}, x_{1} x_{3}+x_{2}, x_{2} x_{3}+x_{1}ight]\) where \(\mathbf{x}^{\prime}=\left(x_{1}, x_{2}, x_{3}ight)\). Find the asymptotic distribution of \(g\left(\mathbf{X}_{n}ight)\) as \(n ightarrow \infty\).