
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 largest real number \( C \) such that for any positive integer \( n \) and any sequence of numbers \(\{x_k\}\) satisfying \(0 = x_0 < x_1 < x_2 < \cdots < x_n = 1\), the following inequality holds:
\[ \sum_{k=1}^{n} x_k^2 (x_k - x_{k-1}) > C. \]

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