
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 smallest real number \(a\) such that for all positive integers \(n\) and real numbers \(0 = x_0 < x_1 < \cdots < x_n\), the following inequality holds:
\[ 
a \sum_{k=1}^{n} \frac{\sqrt{(k+1)^{3}}}{\sqrt{x_{k}^{2}-x_{k-1}^{2}}} \geqslant \sum_{k=1}^{n} \frac{k^{2}+3k+3}{x_{k}}.
\]

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