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>
using namespace std;
using ll = long long;
#define rep(i, a, b) for(int i = a; i < b; i++)
#define rrep(i, a, b) for(int i = b - 1; i >= a; i--)
#define ALL(a) a.begin(), a.end()
#define pii pair<int,int>
#pragma GCC optimize("Ofast")
#define pcnt __builtin_popcount
#define buli(x) __builtin_popcountll(x)
#define pb push_back
#define mp make_pair
#define UNIQUE(v) v.erase( unique(v.begin(), v.end()), v.end() );
template<class T>bool chmax(T &a, const T &b) {if(a<b){a = b; return 1;} return 0; };
template<class T>bool chmin(T &a, const T &b) {if(a>b){a = b; return 1;} return 0; };
inline void IN(void){return;}
template <typename First, typename... Rest> void IN(First& first, Rest&... rest){cin >> first;IN(rest...);return;}
inline void OUT(void){cout << "\n";return;}
template <typename First, typename... Rest> void OUT(First first, Rest... rest){cout << first << " ";OUT(rest...);return;}
const double EPS = 1e-9;
const int mod = 1e9 + 7;
const int INF = 1e9;
const long long LLINF = 1e18;
long long lcm(ll a, ll b){return a * b / __gcd(a,b);}
struct IoSetup { IoSetup() {
    cin.tie(nullptr);ios::sync_with_stdio(false);
    cout << fixed << setprecision(10);
    cerr << fixed << setprecision(10);
} } iosetup;
template< int mod > struct ModInt {
    int x; ModInt() : x(0) {}
    ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}
    ModInt &operator+=(const ModInt &p) {if((x += p.x) >= mod) x -= mod;return *this;}
    ModInt &operator-=(const ModInt &p) {if((x += mod - p.x) >= mod) x -= mod;return *this;}
    ModInt &operator*=(const ModInt &p) {x = (int) (1LL * x * p.x % mod);return *this;}
    ModInt &operator/=(const ModInt &p) {*this *= p.inverse();return *this;}
    ModInt operator-() const { return ModInt(-x); }
    ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }
    ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }
    ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }
    ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }
    bool operator==(const ModInt &p) const { return x == p.x; }
    bool operator!=(const ModInt &p) const { return x != p.x; }
    ModInt inverse() const {int a = x, b = mod, u = 1, v = 0, t;
    while(b > 0) { t = a / b; swap(a -= t * b, b); swap(u -= t * v, v); }return ModInt(u);}
    ModInt pow(int64_t n) const {ModInt ret(1), mul(x); while(n > 0) {if(n & 1) ret *= mul;mul *= mul;n >>= 1;}return ret;}
    friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x;}
    friend istream &operator>>(istream &is, ModInt &a) { int64_t t; is >> t; a = ModInt< mod >(t); return (is); }
    static int get_mod() { return mod; }
}; using modint = ModInt< mod >;
const int dx[4] = {1, 0, -1, 0};
const int dy[4] = {0, 1, 0, -1};

int main(){
    iosetup;
    int N;cin >> N;
    set<int> X[3], Y[3]; // 0: const, 1: plus, 2: minus
    //set<int> Xm, Xp, Ym, Yp;
    vector<pair<pii, char> > P(N);
    rep(i, 0, N){
        int x, y; cin >> x >> y;
        char d; cin >> d;
        P[i] = mp(mp(x, y), d);
        if(d == 'U' || d == 'D'){
            X[0].insert(x);
            if(d == 'U') Y[1].insert(y);
            else Y[2].insert(y);
        }else{
            Y[0].insert(y);
            if(d == 'L') X[2].insert(x);
            else X[1].insert(x);
        }
    }
    vector<long double> times;
    //rep(i, 0, 3) if(!Y[i].empty()) cerr << i << " " << *Y[i].rbegin() << endl;
    if(!X[0].empty() && !X[1].empty()) times.push_back(*X[0].rbegin() - *X[1].rbegin());
    if(!X[2].empty() && !X[0].empty()) times.push_back(*X[2].rbegin() - *X[0].rbegin());
    if(!X[2].empty() && !X[1].empty()) times.push_back((double)(*X[2].rbegin() - *X[1].rbegin()) / 2.0);
    if(!X[0].empty() && !X[1].empty()) times.push_back(*X[0].begin() - *X[1].begin());
    if(!X[0].empty() && !X[2].empty()) times.push_back(*X[2].begin() - *X[0].begin());
    if(!X[2].empty() && !X[1].empty()) times.push_back((double)(*X[2].begin() - *X[1].begin()) / 2.0);
    if(!Y[0].empty() && !Y[1].empty()) times.push_back(*Y[0].rbegin() - *Y[1].rbegin());
    if(!Y[0].empty() && !Y[2].empty()) times.push_back(*Y[2].rbegin() - *Y[0].rbegin());
    if(!Y[2].empty() && !Y[1].empty()) times.push_back((double)(*Y[2].rbegin() - *Y[1].rbegin()) / 2.0);
    if(!Y[0].empty() && !Y[1].empty()) times.push_back(*Y[0].begin() - *Y[1].begin());
    if(!Y[2].empty() && !Y[0].empty()) times.push_back(*Y[2].begin() - *Y[0].begin());
    if(!Y[2].empty() && !Y[1].empty()) times.push_back((double)(*Y[2].begin() - *Y[1].begin()) / 2.0);
    sort(ALL(times));
    UNIQUE(times);
    long double ans = 1e20;
    for(auto time: times){
        if(time < 0) continue;
        long double xM = -LLINF, xm = LLINF, yM = -LLINF, ym = LLINF;
        rep(i, 0, N){
            long double tx = P[i].first.first, ty = P[i].first.second;
            char d = P[i].second;
            if(d == 'R') tx += time;
            else if(d == 'L') tx -= time;
            else if(d == 'U') ty += time;
            else ty -= time;
            chmax(xM, tx);
            chmin(xm, tx);
            chmax(yM, ty);
            chmin(ym, ty);
        }
        //cerr << time << " " << (xM-xm)*(yM-ym) << endl;
        chmin(ans, (xM-xm)*(yM-ym));
    }
    cout << ans << endl;

    return 0;
}