Problem:
Score : 600 points

Problem Statement

There are N points in a two-dimensional plane. The initial coordinates of the i-th point are (x_i, y_i). Now, each point starts moving at a speed of 1 per second, in a direction parallel to the x- or y- axis. You are given a character d_i that represents the specific direction in which the i-th point moves, as follows:

If d_i = R, the i-th point moves in the positive x direction;

If d_i = L, the i-th point moves in the negative x direction;

If d_i = U, the i-th point moves in the positive y direction;

If d_i = D, the i-th point moves in the negative y direction.

You can stop all the points at some moment of your choice after they start moving (including the moment they start moving).
Then, let x_{max} and x_{min} be the maximum and minimum among the x-coordinates of the N points, respectively. Similarly, let y_{max} and y_{min} be the maximum and minimum among the y-coordinates of the N points, respectively.

Find the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}) and print it.

Constraints

1 \leq N \leq 10^5

-10^8 \leq x_i,\ y_i \leq 10^8

x_i and y_i are integers.

d_i is R, L, U, or D.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 d_1
x_2 y_2 d_2
.
.
.
x_N y_N d_N

Output

Print the minimum possible value of (x_{max} - x_{min}) \times (y_{max} - y_{min}).

The output will be considered correct when its absolute or relative error from the judge's output is at most 10^{-9}.

Sample Input 1

2
0 3 D
3 0 L

Sample Output 1

0

After three seconds, the two points will meet at the origin. The value in question will be 0 at that moment.

Sample Input 2

5
-7 -10 U
7 -6 U
-8 7 D
-3 3 D
0 -6 R

Sample Output 2

97.5

The answer may not be an integer.

Sample Input 3

20
6 -10 R
-4 -9 U
9 6 D
-3 -2 R
0 7 D
4 5 D
10 -10 U
-1 -8 U
10 -6 D
8 -5 U
6 4 D
0 3 D
7 9 R
9 -4 R
3 10 D
1 9 U
1 -6 U
9 -8 R
6 7 D
7 -3 D

Sample Output 3

273

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#define rep(i,n) for (int i = 0; i < (n); ++i)
#define mins(x,y) x = min(x,y)
#define maxs(x,y) x = max(x,y)
using namespace std;

const double inf = 1e10;

struct D{
  vector<double>l,r;
  D():l(3,inf),r(3,-inf) {}
  void add(double x,int v){
    ++v;
    mins(l[v],x);
    maxs(r[v],x);
  }
  double get(double t){
    double nl = inf, nr = -inf;
    rep(i,3){
      mins(nl,l[i]+i*t);
      maxs(nr,r[i]+i*t);
    }
    return nr-nl;
  }
  vector <double> events(){
    vector<double>ts;
    rep(i,3)rep(j,i){
      {
        double t = (l[j]-l[i])/i-j;
        if(t>0)ts.push_back(t);
      }
      {
        double t = (l[j]-l[i])/i-j;
        if(t>0)ts.push_back(t);
      }
    }
    return ts;
  }
};

int main(){
  int n;
  cin >> n;
  D xd,yd;
  rep(i,n){
    int x,y;
    char d;
    cin >> x >> y >> d;
    if (d == 'L') xd.add(x,-1), yd.add(y,0);
    if (d == 'R') xd.add(x,1), yd.add(y,0);
    if (d == 'U') xd.add(x,0), yd.add(y,1);
    if (d == 'D') xd.add(x,0), yd.add(y,-1);
  }
  vector<double> ts;
  ts.push_back(0);
  ts.push_back(inf);
  {
    auto nts =xd.events();
    ts.insert(ts.end(),nts.begin(),nts.end());
  }
  {
    auto nts = yd.events();
    ts.insert(ts.end(),nts.begin(),nts.end());
  }
  sort(ts.begin(),ts.end());
  double ans = 1e18;
  for(double t:ts){
    double now = xd.get(t)*yd.get(t);
    mins(ans,now);
  }
  printf("%.10f\n", ans);
}