A interesting number game in a lecture note about coxeter group

Question

Suppose n>1 integers with positive sum are arranged in a circle. If at least one number is negative, then the player may pick a negative number, add it to its neighbors, and reverse its sign. The game terminates when all the numbers are nonnegative. Prove that the game terminates in the same number of steps and in the same final position no matter how it is played.