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Figure E.1. Jewelry
She came up with a new jewelry design. The design uses two parts: a hollow circlet and a convex polygonal component. The design can be customized by specifying the edge lengths of the polygon, which should be multiples of a unit length, so that customers can embed memorial numbers in the jewelry. Note that there can be many different polygons with edges of the specified lengths. Among them, one with a circumscribed circle, that is, a circle that passes through all of its vertices, is chosen so that the polygonal component can be firmly anchored to the circlet
Figure E.2. (a) A pentagon with a circumscribed circle; (b) A pentagon with no circumscribed circle; (c) Another pentagon with no circumscribed circle
For example, Figure E.2(a) has a pentagon with its edge lengths of 3, 1, 6, 1, and 7 units, meaning March 16th and 17th. The radius of the circle is approximately 3.544 units. Figures E.2(b) and E.2(c) show pentagons with the same edge lengths but neither of them has a circumscribed circle.
To commercialize the jewelry, she needs to be able to compute the radius of the circumscribed circle from specified edge lengths. Can you help her by writing a program for this task?
The input consists of a single test case of the following format.
n x1 . . . xn
n is an integer that indicates the number of edges (3 ≤ n ≤ 1000). xk (k = 1, . . . , n) is an integer that indicates the length of the k-th edge (1 ≤ xk ≤ 6000).
You may assume the existence of one or more polygons with the specified edge lengths. You can prove that one of such polygons has a circumscribed circle.
Output the minimum radius of a circumscribed circle of a polygon with the specified edge lengths. Absolute/relative error of the output should be within 10−7.
5 3 1 6 1 7
3 500 300 400
6 2000 3000 4000 2000 3000 4000
10 602 67 67 67 67 67 67 67 67 67
3 6000 6000 1