
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
Determine the largest real number \( a \) such that for all \( n \geqslant 1 \) and for all real numbers \( x_{0}, x_{1}, \ldots, x_{n} \) satisfying \( 0 = x_{0} < x_{1} < x_{2} < \cdots < x_{n} \), we have
\[
\frac{1}{x_{1} - x_{0}} + \frac{1}{x_{2} - x_{1}} + \cdots + \frac{1}{x_{n} - x_{n-1}} \geqslant a \left( \frac{2}{x_{1}} + \frac{3}{x_{2}} + \cdots + \frac{n+1}{x_{n}} \right).
\]

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