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Bessie the cow has designed what she thinks will be the next big hit video game: "Angry Cows". The premise, which she believes is completely original, is that the player shoots a cow with a slingshot into a one-dimensional scene consisting of a set of hay bales located at various points on a number line; the cow lands on a hay bale with sufficient force to cause the bale to explode, which in turn might set of a chain reaction that causes additional nearby hay bales to explode. The goal is to use a single cow to start a chain reaction that detonates as many hay bales as possible.
There are \(N\) hay bales located at distinct integer positions \(x_1, x_2, \ldots, x_N\) on the number line. If a cow is launched onto a hay bale at position \(x\), this hay bale explodes with a "blast radius" of 1, meaning that any other hay bales within 1 unit of distance are also engulfed by the explosion. These neighboring bales then themselves explode (all simultaneously), each with a blast radius of 2, so these explosions may engulf additional yet-unexploded bales up to 2 units of distance away. In the next time step, these bales also explode (all simultaneously) with blast radius 3. In general, at time \(t\) a set of hay bales will explode, each with blast radius \(t\). Bales engulfed by these explosions will themselves explode at time \(t+1\) with blast radius \(t+1\), and so on.
Please determine the maximum number of hay bales that can explode if a single cow is launched onto the best possible hay bale to start a chain reaction.
The first line of input contains \(N\) (\(1 \leq N \leq 100\)). The remaining \(N\) lines all contain integers \(x_1 \ldots x_N\) (each in the range \(0 \ldots 1,000,000,000\)).
Please output the maximum number of hay bales that a single cow can cause to explode.
6 8 5 6 13 3 4
In this example, launching a cow onto the hay bale at position 5 will cause the bales at positions 4 and 6 to explode, each with blast radius 2. These explosions in turn cause the bales at positions 3 and 8 to explode, each with blast radius 3. However, these final explosions are not strong enough to reach the bale at position 13.