One, (40 points) For non-negative real numbers $x_{0}$, $x_{1}, \cdots, x_{n}$ that sum to 1, and with the convention that $x_{n+1}=x_{1}$, find
$$
S=\sqrt{x_{0}+\sum_{i=1}^{n} \frac{\left(x_{i}-x_{i+1}\right)^{2}}{2 n}}+\sum_{i=1}^{n} \sqrt{x_{i}}(n \geqslant 3)
$$

the maximum value.