
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]


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Let \( a_{1}, a_{2}, \cdots, a_{n} \) be given real numbers, not all zero. If the real numbers \( r_{1}, r_{2}, \cdots, r_{n} \) satisfy the inequality
$$
\sum_{k=1}^{n} r_{k}\left(x_{k}-a_{k}\right) \leqslant\left(\sum_{k=1}^{n} x_{k}^{2}\right)^{\frac{1}{2}}-\left(\sum_{k=1}^{n} a_{k}^{2}\right)^{\frac{1}{2}}
$$
for any 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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