
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
Let  $0=x_0<x_1<\cdots<x_n=1$  .Find the largest real number $ C$  such  that for any positive integer  $ n $  ,  we have   $$ \sum_{k=1}^n  x^2_k (x_k - x_{k-1})>C $$  

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