
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 non-zero real numbers \(a_{1}, a_{2}, \cdots, a_{n}\), find the values of real numbers \(r_{1}, r_{2}, \cdots, r_{n}\) such that the inequality
$$
\sum_{k=1}^{n} r_{k}\left(x_{k}-a_{k}\right) \leqslant \left(\sum_{k=1}^{n} x_{k}^{2}\right)^{\frac{1}{2}} - \left(\sum_{k=1}^{n} a_{k}^{2}\right)^{\frac{1}{2}}
$$
holds for any real numbers \(x_{1}, x_{2}, \cdots, x_{n}\).

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