Four Numbers Game II

In an earlier post a question was left open. I will repeat the question here.

You start with $n$ integers $a_1, a_2, \dots, a_n$ and in one step you perform the following transformation

$$(a_1, a_2, \dots , a_n) \to (|a_1-a_2|, |a_2 - a_3|, \dots, |a_{n-1} - a_n|, |a_n - a_1|)$$

($|x| =$ absolute value of $x$)

You keep performing this operation till all the numbers become zero.

Find all $n \gt 1$ (with proof), such that no matter which integers you start with, the numbers eventually become zero.