\[\large \begin{align} x+\sqrt{y}&=11 \\ y+\sqrt{x}&=7 \end{align}\]

Let \(n\) be the number of real pairs \((x, y)\) to the above system of equations. Define these solution pairs as \((x_1, y_1), \ldots, (x_n, y_n)\). Let

\[S=\displaystyle \sum_{i=1}^n \left(x_i + y_i\right).\]

Find \(n+S\).

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