II. Positive integers $x_{1}, x_{2}, \cdots, x_{n}\left(n \in \mathbf{N}_{+}\right)$ satisfy $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=111$.
Find the maximum possible value of $S=\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}$.
Due to $111 \equiv 7(\bmod 8)$, and $x^{2} \equiv 0,1,4(\bmod 8)(x \in \mathbf{N})$, hence $n \geqslant 4$.
(1) When $n=4$,
$$
S \leqslant \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}{4}}=\frac{2 \sqrt{111}}{4}<\frac{22}{4} \text {. }
$$

Taking $(5,5,5,6)$ as a solution, at this time, $S=\frac{21}{4}$.
(2) When $n \geqslant 5$,
$$
\begin{array}{l}
\frac{x_{1}+x_{2}+\cdots+x_{n}}{n} \leqslant \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}{n}} \\
\leqslant \sqrt{\frac{111}{5}}<\frac{21}{4} .
\end{array}
$$

In summary, the maximum value of $S$ is $\frac{21}{4}$.