6. (1 mark) Points $A$ and $B$ lie on a plane. A straight line passing through $A$ will divide the plane into 2 regions. A further straight line through $B$ will altogether divide the plane into 4 regions, and so on. If 1002 and 1000 straight lines are drawn passing through $A$ and $B$ respectively, what is the maximum number of regions formed?
(1 分) $A$ 和 $B$ 是平面上的兩個點。經過 $A$ 點的一條直線將平面分為兩部分, 經過 $B$ 點的另一條直線再將平面共分為四部分。現各有 1002 及 1000 條直線, 分別經過 $A$ 點和 $B$ 點, 問這些直線最多可將平面分為多少部分？
6. The 1000 lines through B divide the plane into 2000 regions. For the first line drawn through A, the number of new regions formed is 1001 . For each new line drawn through $\mathrm{A}$, the number of new regions formed is 1002 . Therefore the total number of regions formed is $1002 \times 1002-1+2000=\underline{\underline{1006003}}$.