
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
(Chebyshev's Inequality) Given \(a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}, b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{n}\), then
$$
n \cdot \sum_{k=1}^{n} a_{k} b_{k} \geqslant \left(\sum_{k=1}^{n} a_{k}\right) \cdot \left(\sum_{k=1}^{n} b_{k}\right) \geqslant n \cdot \sum_{k=1}^{n} a_{k} b_{n-k+1}.
$$

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