Milk ItAlgebra Level pending
The following inequality holds itself true for all positive reals \(x,y,z\).
\[ xy(x+y) + xz(x+z) + yz(y+z) \geq C \cdot xyz \]
Evaluate the greatest possible value of the constant \( C \).
Be sure to include in your answer the procedure used to check the inequality and its equality case. This question is not original.