Can you spot the trick?

Algebra Level 5

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

Let \(a,b,c\) and \(d\) be four real numbers satisfying the system of equations above.

What is the sum of all possible different values of \(a+2b+3c+4d\)?

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