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 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp> 
#include <ext/pb_ds/tree_policy.hpp> 
using namespace __gnu_pbds;
using namespace std;
typedef tree<pair<int,int>,null_type,less<pair<int,int>>,rb_tree_tag,tree_order_statistics_node_update> oset;
#define sim template < class c
#define ris return * this
#define dor > debug & operator <<
#define eni(x) sim > typename \
  enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) {
sim > struct rge { c b, e; };
sim > rge<c> range(c i, c j) { return rge<c>{i, j}; }
sim > auto dud(c* x) -> decltype(cerr << *x, 0);
sim > char dud(...);
struct debug {
#ifdef LOCAL
~debug() { cerr << endl; }
eni(!=) cerr << boolalpha << i; ris; }
eni(==) ris << range(begin(i), end(i)); }
sim, class b dor(pair < b, c > d) {
  ris << "(" << d.first << ", " << d.second << ")";
}
sim dor(rge<c> d) {
  *this << "[";
  for (auto it = d.b; it != d.e; ++it)
	*this << ", " + 2 * (it == d.b) << *it;
  ris << "]";
}
#else
sim dor(const c&) { ris; }
#endif
};
#define imie(...) " [" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "] "
using ll = long long;
#define mins(x,y) x = min(x,y)
#define maxs(x,y) x = max(x,y)
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;
			for(int i = 0 ;  i < 3 ; i++){
				mins(nl,l[i]+i*t);
				maxs(nr,r[i]+i*t);
			}
			return nr-nl;
		}
		vector<double> events(){
			vector<double> ts;
			for(int i = 0 ; i < 3 ; i++){
				for(int j = 0 ; j < i ; j++){
				{
					double t = (l[j] - l[i])/(i-j);
					if( t > 0)
						ts.push_back(t);
				}
				{
					double t = (r[j] - r[i])/(i-j);
					if( t > 0)
						ts.push_back(t);
				}
			}
		}
		return ts;
	}
};

void test_case(){
	int n; cin >> n;
	D xd , yd;
	for(int i = 0 ; i < n ; i++){
		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);
	}
	cout<<ans<<"\n";
}
 
int main() {
	ios_base::sync_with_stdio(false);
	cin.tie(NULL);
	cout.tie(NULL);
	cout<<setprecision(10);
	int T=1;
	//~ cin >> T;
	while(T--){
		test_case();
	}
	return 0;
}