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

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