
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 positive integers $n$ and $k$ (with $n > k$), and $x_{1}, x_{2}, \cdots, x_{k} > 0$. Find $x_{k+1}, x_{k+2}, \cdots, x_{n} > 0$ such that $\sum_{1 \leqslant i, j \leq n} \frac{x_{i}}{x_{j}}$ is minimized.

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