
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 an integer \( n \geq 2 \), and real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) that satisfy \( \sum_{i=1}^{n} i x_{i} = 1 \),

(1) Find the minimum value of \( \sum_{k=1}^{n} x_{k}^{2} + \sum_{1 \leq i < j \leq n} x_{i} x_{j} \).

(2) Find the minimum value of \( \sum_{k=1}^{n} x_{k}^{2} + \sum_{1 \leq i < j \leq n} (x_{i} - x_{j})^{2} \).

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