AMO-Bench 2025 Problem 9

Given that $x, y, z$ are all positive real numbers, consider the equation in $w$: $$1121610\sqrt{3270} w^{9} - 407425607 w^{6} + 10360232 = 0$$ which has $k$ real roots $w_{1}, w_{2}, \ldots, w_{k}$, satisfying $$w_{1} < w_{2} < \cdots < w_{k}.$$ If $$m = (w_{1} + w_{k}) \cdot w_{\left\lfloor \tfrac{1 + k}{2} \right\rfloor},$$ (where $\left\lfloor \tfrac{1 + k}{2} \right\rfloor$ denotes the greatest integer not exceeding $\tfrac{1 + k}{2}$), find the minimum value of $$\frac{x^{3} + y^{3} + z^{3}}{xyz} + \frac{63m}{5(x+y+z)} \sqrt[3]{xyz}.$$