### Problem Statement

Given a set S of points in the plane we can generate a new set T in the following fashion:
• 1) For all distinct points p and q in S, the midpoint of the line segment from p to q should be added to T.
• 2) Remove all duplicate points in T so that there is only one copy of each point in T.
The process used to convert S into T is called a round.

You will be given int[]s xs and ys denoting the points contained in S. Point i will have x-coordinate xs[i] and y-coordinate ys[i]. After applying rnds rounds to S, you will return the best point. One point is better than another if it has a larger y-coordinate. In case of a tie, the point with the larger x-coordinate is then better. The returned point should have the form (quotes for clarity) "x y" where x and y denote the x and y coordinates respectively. Each coordinate has the format (quotes for clarity) "####.####". In other words, there should be exactly 4 digits before the decimal place and 4 digits afterward. When necessary, round down to the nearest ten-thousandth.

### Definition

 Class: PointLifeGame Method: simulate Parameters: int[], int[], int Returns: String Method signature: String simulate(int[] xs, int[] ys, int rnds) (be sure your method is public)

### Constraints

-xs must contain between 3 and 50 elements inclusive.
-ys must contain the same number of elements as xs.
-Each element of xs will be between 0 and 5000 inclusive.
-Each element of ys will be between 0 and 5000 inclusive.
-rnds must be between 1 and 10 inclusive.
-Each given point will be distinct.

### Examples

0)

 `{0,0,10,10}` `{0,10,0,10}` `1`
`Returns: "0005.0000 0010.0000"`
 The given points are arranged in a 10 by 10 square. After 1 round the best point lies on the middle of the top edge of the original square.
1)

 `{0,0,10,10}` `{0,10,0,10}` `10`
`Returns: "0005.0097 0007.5000"`
 Same as before, but now there are 10 rounds.
2)

 `{0,10,20}` `{0,10,0}` `1`
`Returns: "0015.0000 0005.0000"`
 Here we have a triangular arrangement. After 1 round the best point lies on the upper right edge of the original triangle.
3)

 ```{1,2,3,4,5,6,7,8,9,10, 1,2,3,4,5,6,7,8,9,10, 1,2,3,4,5,6,7,8,9,10, 1,2,3,4,5,6,7,8,9,10, 1,2,3,4,5,6,7,8,9,10}``` ```{1,1,1,1,1,1,1,1,1,1, 2,2,2,2,2,2,2,2,2,2, 3,3,3,3,3,3,3,3,3,3, 4,4,4,4,4,4,4,4,4,4, 5,5,5,5,5,5,5,5,5,5}``` `10`
`Returns: "0009.0009 0005.0000"`
4)

 `{3,3,8,0}` `{2,1,1,3}` `10`
`Returns: "0002.4658 0002.1875"`

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=3447

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=5867&pm=3447