
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 \( x_{i} \geq 0 \) for \( i = 1, 2, \ldots, n \) and let \(\sum_{i=1}^{n} x_{i}^{2} + 2 \sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}} x_{k} x_{j} = 1\). Find the maximum and minimum values of \(\sum_{i=1}^{n} x_{i}\).

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