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Algebra Level 4

The complex roots of :

\[ (x+2)^{2}+(x+3)^{3}=(x+4)^{4}\]

are \(\frac{-a\pm\sqrt{b}i}{c}\) for positive coprime integers \(a,b,c\).

Find \(a+b+c\).

Details and Assumptions

\( i^2 = -1 \)

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