Algebra Level 5

\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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