
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
Given non-negative real numbers \(x_{1}, x_{2}, \cdots, x_{n} \ (n \geq 2)\) such that \(x_{1} + x_{2} + \cdots + x_{n} = 1\), find the extremum values of the function
\[ f(x_{1}, x_{2}, \cdots, x_{n}) = \sqrt{x_{1}^{2} + x_{1} x_{2} + x_{2}^{2}} + \sqrt{x_{2}^{2} + x_{2} x_{3} + x_{3}^{2}} + \cdots + \sqrt{x_{n}^{2} + x_{n} x_{1} + x_{1}^{2}}. \]

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