Evaluate
$$\lim_{n \to \infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^2 \{\frac{\pi}{2n}(x_{1}+x_{2}+\cdots  x_{n})\}dx_{1}dx_{2}\cdots dx_{n}$$.
The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\).

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