Circle Union

n개의 원을 배치해 한 점이 모든 원에 포함되도록 하면서, 적어도 한 원이 덮는 영역의 넓이를 최대로 구한다.

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문제 설명

예제1

문제

An arrangement of several circles in the plane is interesting if there exists a point that lies inside or on the boundary of each circle. The covered region of an arrangement consists of all points that lie inside or on the boundary of at least one of the circle.

Consider $n$ circles of radii $r\_1, \ldots, r\_n$ respectively. Find the largest possible area of the region covered by these circles in an interesting arrangement.

입력

The first line contains a single integer $n$ ( $1\leq n\leq 10^4$ ).

The second line contains $n$ integers $r\_1, \ldots, r\_n$ ( $1\leq r\_i\leq 10^3$ ).

출력

Print a single real number --- the largest possible covered area. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-6}$ .