XK-MO 2009 Problem 5

In a circle four distinct points are fixed and each of them is assigned with a real number. Let those numbers be $x_1,x_2,x_3,x_4$ such that $x_1+x_2+x_3+x_4>0$ . Now we define a game with these numbers: If one of them, i.e. $x_i$ , is a negative number, the player makes a move by adding the number $x_i$ to his neighbors and changes the sign of the chosen number. The game ends when all the numbers are negative. Prove that this game ends in a finite number of steps.