
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 \( k>1 \) be a real number. Calculate:

(a) \( L = \lim _{n \rightarrow \infty} \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x \).

(b) \( \lim _{n \rightarrow \infty} n\left[L - \int_{0}^{1}\left(\frac{k}{\sqrt[n]{x} + k - 1}\right)^{n} \, \mathrm{d}x\right] \).

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