
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
\(X\) is the unit \(n\)-cube, \([0, 1]^n\). Let 
\[k_n = \int_X \cos^2 \left( \frac{\pi(x_1 + x_2 + \cdots + x_n)}{2n} \right) dx_1 \, dx_2 \, \cdots \, dx_n.\]
What is \(\lim_{n \to \infty} k_n\)?

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