I need the following estimate for something I am working on, but I don't immediately see how to establish it.

For $x, y, z \in \mathbb{R}_{\ge 0}$, show that $$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz + zx),$$ and I suspect the only point of equality is (1,1,1).

It feels like the sort of thing that ought to have a simple Olympiad-style proof using standard inequalities, but I haven't had any luck thus far; of course, I'll take any proof that I can get.

Also, if this inequality is reminiscent of any others, I would be grateful for references!