Triangle

A lattice point is an ordered pair $(x, y)$ where both $x$ and $y$ are integers. Given the coordinates of the three vertices of a triangle, all of which are lattice points, count how many lattice points lie strictly inside the triangle. Points that lie on an edge or at a vertex do not count.

The input contains several test cases. Each test case consists of six integers $x_1$, $y_1$, $x_2$, $y_2$, $x_3$, $y_3$, the coordinates of the vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. Every triangle is non-degenerate (it has positive area), and every coordinate satisfies $-15000 \le x_1, y_1, x_2, y_2, x_3, y_3 \le 15000$. The end of the input is marked by a test case with $x_1 = y_1 = x_2 = y_2 = x_3 = y_3 = 0$, which must not be processed.

For each test case, print the number of interior lattice points on its own line.