Evaluate
$$
\lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}
$$
#*The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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