
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 \(a_{1}, a_{2}, \cdots, a_{n}\) be given non-zero real numbers. If the inequality \(r_{1}\left(x_{1}-a_{1}\right)+ \cdots + r_{n}\left(x_{n}-a_{n}\right) \geq 0\) holds for all 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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