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.