If \( n(2 n+1) \int_{0}^{1}\left(1-x^{n}\right)^{2 n} d x=1177 \int_{0}^{1}\left(1-x^{n}\right)^...

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If \( n(2 n+1) \int_{0}^{1}\left(1-x^{n}\right)^{2 n} d x=1177 \int_{0}^{1}\left(1-x^{n}\right)^{2 n+1} d x \), then \( n \) \( \in N \) is equal to
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