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AIME II 2026 (Problem 11)

Find the greatest integer nn such that the cubic polynomial
x3−n6x2+(n−11)x−400x^3-\frac n6x^2+(n-11)x-400
has roots α2,β2,\alpha^2,\beta^2, and γ2\gamma^2, where α,β,\alpha,\beta, and γ\gamma are complex numbers, and there are exactly seven different possible values for α+β+γ\alpha+\beta+\gamma.