IL-GILLIS 1996 Problem 6

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$ . What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.