Question 10
One thousand non-zero numbers $x_1,x_2,x_3,x_4,\ldots,x_{1000}$ are written in a cyclic order around a circle. It turns out that each number that is written in an odd position is equal to the sum of its two neighbors and that each number that is written in an even position is equal to the product of its two neighbors. What are the possible values for the sum of these one thousand numbers?