Suppose that $x, y,z$ are positive integers satisfying $x \le y \le z$, and such that the product of all three numbers is twice their sum. What is the sum of all possible values of $z$?

Asoftball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. they lost by one run in exactly five games. in each of their other games, they scored twice as many runs as their opponent. how many total runs did their opponents score?