A cubed pairwise ratio sum bounds cyclic quadratic ratios

Let $x,y,z>0$ ,prove that;

${\frac { \left( y+z \right) ^{3}}{yz}}+{\frac { \left( z+x \right) ^{3}}{xz}}+{\frac { \left( x+y \right) ^{3}}{xy}}\geq 16(\,{\frac {{x}^{2}}{x+y}}+\,{\frac {{y}^{2}}{y+z}}+\,{\frac {{z}^{2}}{z+x}})$ Please do it by C-S or A-G

Source: InternLM Lean-Workbook, record lean_workbook_24254 (Apache-2.0). Complete source proposition preserved; proof developed independently.