
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
If \(0 < x_{1} < x_{2} < \cdots < x_{n-1} < a\), \( b_{i} \in \mathbf{R}^{+}(i=1,2,\cdots, n)\), and \( k \in \mathbf{N}^{*} \), then
\[ 
\frac{b_{1}^{k+1}}{x_{1}^{k}} + \frac{b_{2}^{k+1}}{\left(x_{2}-x_{1}\right)^{k}} + \cdots + \frac{b_{n}^{k+1}}{\left(a-x_{n-1}\right)^{k}} \geqslant \frac{\left(b_{1}+b_{2}+\cdots+b_{n}\right)^{k+1}}{a^{k}}, 
\]
with equality if and only if \( x_{i} = \frac{a b_{i}}{\sum_{i=1}^{n} b_{i}} \).

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