1. Given $x_{i} \in \mathbf{R}_{+}(i=1,2,3,4)$, and $x_{1}+x_{2}+$ $x_{3}+x_{4}=\pi$. Then the minimum value of $\prod_{i=1}^{4}\left(\sin x_{i}+\frac{1}{\sin x_{i}}\right)$ is $\qquad$ _.
$=1 . \frac{81}{4}$.
Notice that
$$
\sin x_{1}+\frac{1}{2 \sin x_{1}}+\frac{1}{2 \sin x_{1}} \geqslant 3 \sqrt[3]{\frac{1}{4 \sin x_{1}}},
$$

which means $\sin x_{1}+\frac{1}{\sin x_{1}} \geqslant 3 \sqrt[3]{\frac{1}{4 \sin x_{1}}}$.
Thus, the original expression
$$
\geqslant 3^{4} \sqrt[3]{\frac{1}{4^{4} \sin x_{1} \cdot \sin x_{2} \cdot \sin x_{3} \cdot \sin x_{4}}}.
$$

For equation (1), equality holds if and only if $\sin x_{1}=\frac{1}{2 \sin x_{1}}$, i.e., $x_{1}=\frac{\pi}{4}$.

Therefore, equality holds in the original expression if and only if $x_{1}=x_{2}=x_{3}=x_{4}=\frac{\pi}{4}$.
Thus, the minimum value of the original expression is $\frac{81}{4}$.