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The base of a solid is an ellipse bounded by the curve $2x^2 + y^2 = 8$.

Find the volume of the solid formed by cross-sections perpendicular to the $x$-axis that are isosceles right triangles with a leg in the base.

$\cfrac{32}{3}$

$\cfrac{64}{3}$

$\cfrac{128}{3}$

$\cfrac{80\sqrt{2}}{3}$