\[\begin{cases}\begin{align} x_{1} + x_{2} + x_{3} + \cdots + x_{n} &= 2016 \\\\ x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + \cdots + x_{n}^{4} &= 2016\times 512 \end{align}\end{cases}\]

Let \(x_1, x_2, \ldots , x_n \) be real numbers.

Find the smallest possible value of \(n\) such that there exists real solutions to the equations above.

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