
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 maximum real number $\lambda$ such that the inequality $\sum_{k=1}^{n} x_{k}^{3}(x_{k} - x_{k-1}) \geq \frac{1}{4} + \frac{\lambda}{n}$ holds for any positive integer $n$ and any real numbers $0 = x_0 \leq x_1 \leq \ldots \leq x_n = 1$.

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