AIME 2026b Problem 11

Find the greatest integer $n$ such that the cubic polynomial$$x^{3} -\frac{n}{6}x^{2} + (n - 11)x - 400$$has roots $\alpha^{2}$, $\beta^{2}$, and $\gamma^{2}$, where $\alpha$, $\beta$, and $\gamma$ are complex numbers, and there are exactly seven different possible values for $\alpha + \beta + \gamma$.