
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For non-negative real numbers $x_{0}$, $x_{1}, \cdots, x_{n}$ such that their sum is 1 and $x_{n+1}=x_{1}$, find the maximum value of
$$
S=\sqrt{x_{0}+\sum_{i=1}^{n} \frac{\left(x_{i}-x_{i+1}\right)^{2}}{2n}} + \sum_{i=1}^{n} \sqrt{x_{i}} \quad (n \geqslant 3).
$$

Present the answer in LaTex format: \boxed{Your answer}
user