
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

Suppose \(a_{1}, a_{2}, \cdots, a_{n}\) are given real numbers, not all zero, and \(r_{1}, r_{2}, \cdots, r_{n}\) are real numbers. If the inequality

\[
r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leq \sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}-\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}
\]

holds for any real numbers \(x_{1}, x_{2}, \cdots, x_{n}\), find the values of \(r_{1}, r_{2}, \cdots, r_{n}\).

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