
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 positive numbers \(a_{1}, a_{2}, \cdots, a_{n}\) and \(m > 1\), find real numbers \(r_{1}, r_{2}, \cdots, r_{n}\) such that for any non-negative real numbers \(x_{1}, x_{2}, \cdots, x_{n}\), the following inequality holds:
$$
\sum_{i=1}^{n} r_{i}\left(x_{i}-a_{i}\right) \leqslant\left(\sum_{i=1}^{n} x_{i}^{m}\right)^{\frac{1}{m}}-\left(\sum_{i=1}^{n} a_{i}^{m}\right)^{\frac{1}{m}} .
$$

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