
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 two ordered sequences

$$
a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n} \quad \text{and} \quad b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{n},
$$

then

$$\frac{1}{n} \sum_{i=1}^{n} a_{i} b_{n+1-i} \leqslant \left(\frac{1}{n} \sum_{i=1}^{n} a_{i}\right) \left(\frac{1}{n} \sum_{i=1}^{n} b_{i}\right) \leqslant \frac{1}{n} \sum_{i=1}^{n} a_{i} b_{i}.$$

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