Who's up to the challenge? 13

$x_1, x_2, \ldots, x_k, y_1, y_2, \ldots, y_k$ 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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