
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 \( a_i, b_i > 0 \) for \( i = 1, 2, \cdots, n \). If \( a_1 \geqslant a_2 \geqslant \cdots \geqslant a_n \) and \( b_1 \leqslant b_2 \leqslant \cdots \leqslant b_n \) or \( a_1 \leqslant a_2 \leqslant \cdots \leqslant a_n \) and \( b_1 \geqslant b_2 \geqslant \cdots \geqslant b_n \), with \( n \geqslant 2 \), \( r, t > 0 \), and \( s = \sum_{i=1}^{n} a_i \), then \( \sum_{i=1}^{n} a_i b_i \leqslant \frac{1}{r + t} \sum_{i=1}^{n} (r s + t a_i) b_i \).

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