Robot Lawn Mowers#2723
A landscaping company is designing an automated lawn maintenance system for a large property. They have two advanced robot lawn mowers that need to be deployed efficiently. Each robot is tethered to a charging post with a power cable, allowing it to operate within a circular area around its post.
The property has several trees and obstacles scattered throughout. Each robot must be installed at one of these anchor points. However, there are important constraints: a robot's cable must not reach any other obstacle (otherwise the cable would get tangled around it), and the two robots must not reach each other (otherwise they might collide or interfere with each other's operation).
The company wants to maximize the total lawn area that both robots can cover. Each robot can be tethered with a cable length between and meters from its anchor point.
The first line contains an integer (2 \le N \le 1000) — the number of potential anchor points (trees and obstacles) on the property. The next lines contain the coordinates of each anchor point. The origin is at the center of the property. Coordinates are given in meters with centimeter precision. The absolute values of coordinates do not exceed meters. The robots must stay within the property boundaries. The physical size of the robots can be neglected. It is guaranteed that a valid configuration exists.
Print the maximum total area (in square meters) that both robot lawn mowers can cover, with a precision of at least decimal places.
Example 1