
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 \( n \geq 2, x_{1}, x_{2}, \cdots, x_{n} \) are all real numbers, and \( \sum_{i=1}^{n} x_{i}^{2} + \sum_{i=1}^{n-1} x_{i} x_{i+1} = 1 \). For each fixed \( k \) (\( k \in \mathbf{Z}, 1 \leq k \leq n \)), find the maximum value of \( \left|x_{k}\right| \).

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