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Given an integer $n>1$, determine the real numbers $x_1, x_2, \ldots, x_n \geq 1$ and $x_{n+1}>$ 0 , such that the following conditions are simultaneously fulfilled:

(i) $\sqrt{x_1}+\sqrt[3]{x_2}+\cdots+\sqrt[n+1]{x_n}=n \sqrt{x_{n+1}}$,

(ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1+}$
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