Score : 1000 points
There are N cities in a two-dimensional plane. The coordinates of the i-th city is (x_i, y_i). Initially, the amount of water stored in the i-th city is a_i liters.
Snuke can carry any amount of water from a city to another city. However, water leaks out a bit while he carries it. If he carries l liters of water from the s-th city to the t-th city, only max(l-d_{s,t}, 0) liters of water remains when he arrives at the destination. Here d_{s,t} denotes the (Euclidean) distance between the s-th city and the t-th city. He can perform arbitrary number of operations of this type.
Snuke wants to maximize the minimum amount of water among the N cities. Find the maximum X such that he can distribute at least X liters of water to each city.
The input is given from Standard Input in the following format:
N x_1 y_1 a_1 : x_N y_N a_N
Print the maximum of the minimum amount of water among the N cities. The absolute error or the relative error must be at most 10^{-9}.
3 0 0 10 2 0 5 0 5 8
6.500000000000
The optimal solution is to carry 3.5 liters of water from the 1st city to the 2nd city. After the operation, both the 1st and the 2nd cities will have 6.5 liters of water, and the 3rd city will have 8 liters of water.
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