
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 \( n \in \mathbf{Z}_{+} \) and any sequence \(\{x_{k}\}\) satisfying \(0 = x_{0} < x_{1} < x_{2} < \cdots < x_{n} = 1\), the following inequality holds:
\[ 
\sum_{k=1}^{n} x_{k}^{2} \left(x_{k} - x_{k-1}\right) > C.
\]

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