
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 that $x_{i} \geqslant 0 \ (i=1,2, \cdots, n)$ and $\sum_{i=1}^{n} x_{i}^{2} + 2 \sum_{1 \leqslant k < j \leqslant n} \sqrt{\frac{k}{j}} x_{k} x_{j} = 1$, find the maximum and minimum values of $\sum_{i=1}^{n} x_{i}$.

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