\[2z^{8}-3z^{7}-12z^{6}+12z^{5}+22z^{4}-12z^{3}-12z^{2}+3z+2=0\]

Two of the roots of the equation above can be written in the form \(\frac{-A \pm \sqrt{B}}{C}\) while another two in the form \(\frac{-A \pm \sqrt{D}}{E}\), where \(A,B,C,D,E\) are distinct positive integers with \(B\) and \(D\) squarefree. Find \(A+B+C+D+E\).

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