\(x_1, x_2, \ldots, x_k, y_1, y_2, \ldots, y_k\) are nonnegative real numbers. If \[\displaystyle\sum_{n=1}^k x_n^4 = 28561\] and \[\displaystyle\sum_{n=1}^k y_n^4 = 194481,\] find the maximum value of \[\displaystyle \sqrt{\sum_{n=1}^k (x_n+y_n)^4}\] over all \(k\).

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