
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 \( r_1(x_1 - a_1) + r_2(x_2 - a_2) + \cdots + r_n(x_n - a_n) \leq \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} - \sqrt{a_1^2 + a_2^2 + \cdots + a_n^2} \) holds for all real numbers \( x_1, x_2, \cdots, x_n \), determine the values of \( r_1, r_2, \cdots, r_n \).

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