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 (GCC 5.4.1) (judge verdict: Wrong Answer)

//set many funcs template
//Ver.20181228
#include<stdio.h>
#include<string.h>
#include<stdlib.h>
#include<stdbool.h>
#include<time.h>
#define inf 1072114514
#define llinf 4154118101919364364
#define mod 1000000007
#define pi 3.1415926535897932384

int max(int a,int b){if(a>b){return a;}return b;}
int min(int a,int b){if(a<b){return a;}return b;}
int zt(int a,int b){return max(a,b)-min(a,b);}
int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
int nHr(int a,int b){return nCr(a+b-1,b);}
int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
int sankaku(int x){return ((1+x)*x)/2;}
long long llmax(long long a,long long b){if(a>b){return a;}return b;}
long long llmin(long long a,long long b){if(a<b){return a;}return b;}
long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
long long llsankaku(long long x){return ((1+x)*x)/2;}
double dbmax(double a,double b){if(a>b){return a;}return b;}
double dbmin(double a,double b){if(a<b){return a;}return b;}
double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}

void shuffledget(int x[],int n){
    srand(time(0));
    int i,b[524288],p,c;
    for(i=0;i<n;i++){
        b[i]=i;
    }
    for(i=n;i>=1;i--){
        p=rand()%i;
        c=b[i-1];b[i-1]=b[p];b[p]=c;
    }
    for(i=0;i<n;i++){
        scanf("%d",&x[b[i]]);
    }
}

int dx4[4]={1,-1,0,0};
int dy4[4]={0,0,1,-1};
int dx8[8]={-1,-1,-1,0,0,1,1,1};
int dy8[8]={-1,0,1,-1,1,-1,0,1};

int search(int x,int a[],int n){
    int st=0,fi=n-1,te;
    while(st<=fi){
        te=(st+fi)/2;
        if(a[te]<x){st=te+1;}else{fi=te-1;}
    }
    return st;
}

void prarr(int arr[],int n){
  int i;
  for(i=0;i<n;i++){
    if(i){printf(" ");}
    printf("%d",arr[i]);
  }
  printf("\n");
  return;
}

typedef struct{
int val;
int node;
}sd;

int sdsortfnc(const void *a,const void *b){
if(((sd*)a)->val < ((sd*)b)->val){return -1;}
if(((sd*)a)->val > ((sd*)b)->val){return 1;}
return 0;
}

int n;
double x[131072],y[131072];
char d[131072][2];

double eval(double t){
  int i;
  double sx=llinf,fx=-llinf;
  double sy=llinf,fy=-llinf;
  double nx,ny;
  for(i=0;i<n;i++){
    nx=x[i];ny=y[i];
    if(d[i][0]=='U'){ny+=t;}
    else if(d[i][0]=='D'){ny-=t;}
    else if(d[i][0]=='R'){nx+=t;}
    else{nx-=t;}
    sx=dbmin(nx,sx);
    fx=dbmax(nx,fx);
    sy=dbmin(ny,sy);
    fy=dbmax(ny,fy);
  }
  return (fx-sx)*(fy-sy);
}

int main(void){
    int i,j,p;
    double fc,lc,r,ver;
    scanf("%d",&n);
    for(i=0;i<n;i++){
      scanf("%lf%lf%s",&x[i],&y[i],d[i]);
    }
    r=eval(0);

    p=-1;
    for(i=0;i<n;i++){
      if(d[i][0]!='U'){continue;}
      if(p!=-1){
        if(y[p]<y[i]){p=i;}
      }
      else{p=i;}
    }
    if(p!=-1){
      fc=inf;lc=0;
      for(i=0;i<n;i++){
        if(d[i][0]=='U'){continue;}
        if(y[p]<y[i]){
          ver=y[i]-y[p];
          if(d[i][0]=='D'){
            ver/=2.0;
          }
          fc=dbmin(ver,fc);
          lc=dbmax(ver,lc);
        }
      }
      r=dbmin(eval(fc),r);
      r=dbmin(eval(lc),r);
    }

    p=-1;
    for(i=0;i<n;i++){
      if(d[i][0]!='D'){continue;}
      if(p!=-1){
        if(y[p]>y[i]){p=i;}
      }
      else{p=i;}
    }
    if(p!=-1){
      fc=inf;lc=0;
      for(i=0;i<n;i++){
        if(d[i][0]=='D'){continue;}
        if(y[p]>y[i]){
          ver=y[p]-y[i];
          if(d[i][0]=='U'){
            ver/=2.0;
          }
          fc=dbmin(ver,fc);
          lc=dbmax(ver,lc);
        }
      }
      r=dbmin(eval(fc),r);
      r=dbmin(eval(lc),r);
    }

    p=-1;
    for(i=0;i<n;i++){
      if(d[i][0]!='R'){continue;}
      if(p!=-1){
        if(x[p]<x[i]){p=i;}
      }
      else{p=i;}
    }
    if(p!=-1){
      fc=inf;lc=0;
      for(i=0;i<n;i++){
        if(d[i][0]=='R'){continue;}
        if(x[p]<x[i]){
          ver=x[i]-x[p];
          if(d[i][0]=='L'){
            ver/=2.0;
          }
          fc=dbmin(ver,fc);
          lc=dbmax(ver,lc);
        }
      }
      r=dbmin(eval(fc),r);
      r=dbmin(eval(lc),r);
    }

    p=-1;
    for(i=0;i<n;i++){
      if(d[i][0]!='L'){continue;}
      if(p!=-1){
        if(x[p]>x[i]){p=i;}
      }
      else{p=i;}
    }
    if(p!=-1){
      fc=inf;lc=0;
      for(i=0;i<n;i++){
        if(d[i][0]=='L'){continue;}
        if(x[p]>x[i]){
          ver=x[p]-x[i];
          if(d[i][0]=='R'){
            ver/=2.0;
          }
          fc=dbmin(ver,fc);
          lc=dbmax(ver,lc);
        }
      }
      r=dbmin(eval(fc),r);
      r=dbmin(eval(lc),r);
    }

    printf("%.12lf\n",r);
    return 0;
}