
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
Find the greatest real number \( a \) such that for all positive integers \( n \) and all real numbers \( x_{0}, x_{1}, \cdots, x_{n} \) where \( 0=x_{0}<x_{1}<\cdots<x_{n} \), the following inequality holds:
\[ 
\sum_{i=1}^{n} \frac{1}{x_{i}-x_{i-1}} \geqslant a \sum_{i=1}^{n} \frac{i+1}{x_{i}}. 
\]

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