\[\begin{cases} a+b=8 \\ ab+c+d = 23 \\ ad+bc = 28 \\ cd = 12 \end{cases}\]

Let the above system of equations have \(n\) sets of real solutions. Let these solutions be \((a_1, b_1, c_1, d_1), \ldots, (a_n, b_n, c_n, d_n)\). Find the value of

\[n + \displaystyle \sum_{i=1}^n (a_i + b_i + c_i + d_i ). \]

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