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It's a hot summer day, and Bessie the cow is feeling quite lazy. She wants to locate herself at a position in her field so that she can reach as much delicious grass as possible within only a short distance.
There are N patches of grass (1 <= N <= 100,000) in Bessie's field. The ith such patch contains g_i units of grass (1 <= g_i <= 10,000) and is located at a distinct point (x_i, y_i) in the field (0 <= x_i, y_i <= 1,000,000). Bessie would like to choose a point in the field as her initial location (possibly the same point as a patch of grass, and possibly even a point with non-integer coordinates) so that a maximum amount of grass is within a distance of K steps from this location (1 <= K <= 2,000,000).
When Bessie takes a step, she moves 1 unit north, south, east, or west of her current position. For example, to move from (0,0) to (3,2), this requires 5 total steps. Bessie does not need to take integer-sized steps -- for example, 1 total step could be divided up as half a unit north and half a unit east.
Please help Bessie determine the maximum amount of grass she can reach, if she chooses the best possible initial location.
4 3 7 8 6 3 0 0 4 6 0 1 4 2
Bessie is willing to take at most 3 steps from her initial position. There are 4 patches of grass. The first contains 7 units of grass and is located at position (8,6), and so on.
By locating herself at (3,0), the grass at positions (0,0), (6,0), and (4,2) is all within K units of distance.