Problem:
Score: 600 points

Problem Statement

E869120 is initially standing at the origin (0, 0) in a two-dimensional plane.

He has N engines, which can be used as follows:

When E869120 uses the i-th engine, his X- and Y-coordinate change by x_i and y_i, respectively. In other words, if E869120 uses the i-th engine from coordinates (X, Y), he will move to the coordinates (X + x_i, Y + y_i).

E869120 can use these engines in any order, but each engine can be used at most once. He may also choose not to use some of the engines.

He wants to go as far as possible from the origin.
Let (X, Y) be his final coordinates. Find the maximum possible value of \sqrt{X^2 + Y^2}, the distance from the origin.

Constraints

1 \leq N \leq 100

-1 \ 000 \ 000 \leq x_i \leq 1 \ 000 \ 000

-1 \ 000 \ 000 \leq y_i \leq 1 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
: :
x_N y_N

Output

Print the maximum possible final distance from the origin, as a real value.
Your output is considered correct when the relative or absolute error from the true answer is at most 10^{-10}.

Sample Input 1

3
0 10
5 -5
-5 -5

Sample Output 1

10.000000000000000000000000000000000000000000000000

The final distance from the origin can be 10 if we use the engines in one of the following three ways:

Use Engine 1 to move to (0, 10).

Use Engine 2 to move to (5, -5), and then use Engine 3 to move to (0, -10).

Use Engine 3 to move to (-5, -5), and then use Engine 2 to move to (0, -10).

The distance cannot be greater than 10, so the maximum possible distance is 10.

Sample Input 2

5
1 1
1 0
0 1
-1 0
0 -1

Sample Output 2

2.828427124746190097603377448419396157139343750753

The maximum possible final distance is 2 \sqrt{2} = 2.82842....
One of the ways to achieve it is:

Use Engine 1 to move to (1, 1), and then use Engine 2 to move to (2, 1), and finally use Engine 3 to move to (2, 2).

Sample Input 3

5
1 1
2 2
3 3
4 4
5 5

Sample Output 3

21.213203435596425732025330863145471178545078130654

If we use all the engines in the order 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5, we will end up at (15, 15), with the distance 15 \sqrt{2} = 21.2132... from the origin.

Sample Input 4

3
0 0
0 1
1 0

Sample Output 4

1.414213562373095048801688724209698078569671875376

There can be useless engines with (x_i, y_i) = (0, 0).

Sample Input 5

1
90447 91000

Sample Output 5

128303.000000000000000000000000000000000000000000000000

Note that there can be only one engine.

Sample Input 6

2
96000 -72000
-72000 54000

Sample Output 6

120000.000000000000000000000000000000000000000000000000

There can be only two engines, too.

Sample Input 7

10
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20

Sample Output 7

148.660687473185055226120082139313966514489855137208

Solution in Fortran (gfortran v4.8.4) (judge verdict: Wrong Answer)

module ABC139

  ! <module>s to import
  use, intrinsic :: iso_fortran_env

  ! require all variables to be explicitly declared
  implicit none

  type coordinate
    real(REAL64), public :: x
    real(REAL64), public :: y
    real(REAL64), public :: distance
  end type coordinate

  ! accessibility of <subroutine>s and <function>s in this <module>
  public :: task_F

  ! contained <subroutine>s and <function>s are below
  contains

  subroutine task_F

    ! variables for this <subroutine>
    integer(INT32)   :: num_engines
    real(REAL64)     :: x_max, x_read, x_trial
    real(REAL64)     :: y_max, y_read, y_trial
    real(REAL64)     :: distance_max
    real(REAL64)     :: distance_trial

    ! support variables for this <subroutine>
    integer(INT32) :: itr

    ! STEP.01
    ! read out the given data (the number of engines)
    read(unit=INPUT_UNIT, fmt=*) num_engines

    ! STEP.02
    ! initialize the variables for this <subroutine>
    distance_max = 0.0_REAL64
    x_max        = 0.0_REAL64
    y_max        = 0.0_REAL64
    x_trial      = 0.0_REAL64
    y_trial      = 0.0_REAL64

    ! STEP.03
    ! calculate the answer of this task
    do itr = 1, num_engines, 1

      ! read out the given data (the movement by the each engine)
      read(unit=INPUT_UNIT, fmt=*) x_read, y_read

      x_trial        = x_trial + x_read
      y_trial        = y_trial + y_read
      distance_trial = sqrt(x_trial * x_trial + y_trial * y_trial)

      if (distance_trial .gt. distance_max) then
        x_max        = x_trial
        y_max        = y_trial
        distance_max = distance_trial
      end if

    end do

    ! STEP.04
    ! output the answer of this task
    write(unit=OUTPUT_UNIT, fmt='(ES23.15e3)', advance='yes') distance_max

    ! STEP.END
    return

  end subroutine task_F

end module ABC139

program main

  ! <module>s to import
  use, non_intrinsic :: ABC139

  ! require all variables to be explicitly declared
  implicit none

  call task_F

end program main