
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 \( n (n > 1) \) be a given positive integer, and let \( x_{1}, x_{2}, \cdots, x_{n} \) be arbitrary real numbers such that \( \sum_{i=1}^{n} x_{i}^{2}=1 \). Find the maximum value of \( \sum_{k=1}^{n} k x_{k}^{2}+\sum_{1 \leqslant i < j \leqslant n}(i+j) x_{i} x_{j} \).

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