
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 that \( a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n} > 0 \) and \( b_{1} \geqslant b_{2} \geqslant \cdots \geqslant b_{n} > 0 \), then:

\[ a_{1}^{b_{1}} a_{2}^{b_{2}} \cdots a_{n}^{b_{1}} \geqslant a_{1}^{b_{1}} a_{2}^{b_{d}} \cdots a_{n}^{b_{n}} \geqslant a_{1}^{b_{1}} a_{2}^{b_{1}} \cdots \cdots a_{n}^{b_{1}} \]

where \( b_{i 1}, b_{i 2}, \cdots, b_{i 1} \) is any permutation of \( b_{1}, b_{2}, \cdots, b_{n} \).

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