
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
The sequence  $\{c_{n}\}$  is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let  $\lambda$  be the limit value  $\lim_{n\to\infty}c_{n}.$   Find  $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ 

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