Prof. Du and Prof. Pang plan to build a sky garden near the city of Allin. In the garden, there will be a plant maze consisting of straight and circular roads. On the blueprint of the plant maze, Prof. Du draws n{n}n circles indicating the circular roads. All of them have center (0,0){}(0,0). The radius of the i{i}i-th circle is i{i}i. Meanwhile, Prof. Pang draws m{m}m lines on the blueprint indicating the straight roads. All of the lines pass through (0,0){}(0,0). Each circle is divided into 2m{2m}2m parts with equal lengths by these lines. Let Q{Q}Q be the set of the n+m{n+m}n+m roads. Let P{P}P be the set of all intersections of two different roads in Q{Q}Q. Note that each circular road and each straight road have two intersections. For two different points a∈Pain Pa∈P and b∈Pbin Pb∈P, we define dis{dis}dis to be the shortest distance one needs to walk from a{a}a to b{b}b along the roads. Please calculate the sum of dis{dis}dis for all {a,b}P{a, b}subseteq P{a,b}P.
Prof. Du and Prof. Pang plan to build a sky garden near the city of Allin. In the garden, there will be a plant maze consisting of straight and circular roads. On the blueprint of the plant maze, Prof. Du draws n{n}n circles indicating the circular roads. All of them have center (0,0){(0, 0)}(0,0). The radius of the i{i}i-th circle is i{i}i. Meanwhile, Prof. Pang draws m{m}m lines on the blueprint indicating the straight roads. All of the lines pass through (0,0){(0, 0)}(0,0). Each circle is divided into 2m{2m}2m parts with equal lengths by these lines. Let Q{Q}Q be the set of the n+m{n+m}n+m roads. Let P{P}P be the set of all intersections of two different roads in Q{Q}Q. Note that each circular road and each straight road have two intersections. For two different points a∈Pain Pa∈P and b∈Pbin Pb∈P, we define dis({a,b}){dis({a, b})}dis({a,b}) to be the shortest distance one needs to walk from a{a}a to b{b}b along the roads. Please calculate the sum of dis({a,b}){dis({a, b})}dis({a,b}) for all {a,b}⊆P{a, b}subseteq P{a,b}⊆P.
