Can you guess what I guessed!

Algebra Level 5

\[x^{4}-22x^{2}-12x+22=0\] The above equation has roots of the form \(\pm \sqrt{a} \pm \sqrt{b \pm \sqrt{c}}\), where \(a, b\) and \(c\) are constant positive integers, and \(a\) and \(\sqrt[3]{b}\) are relatively prime.

Find \(a+b+c\).

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