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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using static System.Math;

namespace Program {
    public class Solver {
        Random rnd = new Random(0);

        public void Solve() {
            var n = ri;
            var LX = new List<long>();
            var LY = new List<long>();
            var RX = new List<long>();
            var RY = new List<long>();
            var FX = new List<long>();
            var FY = new List<long>();
            for (int i = 0; i < n; i++) {
                var x = rl;
                var y = rl;
                var d = rs;
                if (d == "L" || d == "R") {
                    if (d == "R") RX.Add(x);
                    else LX.Add(x);
                    FY.Add(y);
                }
                else {
                    if (d == "U") RY.Add(y);
                    else LY.Add(y);
                    FX.Add(x);
                }
            }
            var ans = 1e100;
            var P = Enumerate(2, x => new List<KeyValuePair<long, int>>());

            var T = new List<double>() { 0 };
            for (int k = 0; k < 2; k++) {
                if (LX.Count > 0) {
                    P[k].Add(new KeyValuePair<long, int>(LX.Min(), -1));
                    P[k].Add(new KeyValuePair<long, int>(LX.Max(), -1));
                }
                if (RX.Count > 0) {
                    P[k].Add(new KeyValuePair<long, int>(RX.Min(), 1));
                    P[k].Add(new KeyValuePair<long, int>(RX.Max(), 1));
                }
                if (FX.Count > 0) {
                    P[k].Add(new KeyValuePair<long, int>(FX.Min(), 0));
                    P[k].Add(new KeyValuePair<long, int>(FX.Max(), 0));
                }
                Swap(ref LX, ref LY);
                Swap(ref RX, ref RY);
                Swap(ref FX, ref FY);
                for (int _ = 0; _ < P[k].Count; _++)
                    for (int __ = _ + 1; __ < P[k].Count; __++) {
                        var p = P[k][_];
                        var q = P[k][__];
                        if (p.Value == q.Value) continue;
                        var d = q.Key - p.Key;
                        var e = -q.Value + p.Value;
                        if (d * e >= 0) T.Add(d * 1.0 / e);
                    }
            }
            Debug.WriteLine(T.AsJoinedString());
            foreach (var t in T) {
                double xmi = INF;
                double xma = -INF;
                double ymi = INF;
                double yma = -INF;
                foreach (var lx in LX) { xmi = Min(xmi, lx - t); xma = Max(xma, lx - t); }
                foreach (var lx in RX) { xmi = Min(xmi, lx + t); xma = Max(xma, lx + t); }
                foreach (var lx in FX) { xmi = Min(xmi, lx); xma = Max(xma, lx); }
                foreach (var lx in LY) { ymi = Min(ymi, lx - t); yma = Max(yma, lx - t); }
                foreach (var lx in RY) { ymi = Min(ymi, lx + t); yma = Max(yma, lx + t); }
                foreach (var lx in FY) { ymi = Min(ymi, lx); yma = Max(yma, lx); }
                ans = Min(ans, (xma - xmi) * (yma - ymi));
            }

            Console.WriteLine(ans);
        }

        const long INF = 1L << 60;
        int ri => sc.Integer();
        long rl => sc.Long();
        double rd => sc.Double();
        string rs => sc.Scan();
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static IEnumerable<int> Rep(int n) => Enumerable.Range(0, n);
        static IEnumerable<int> RRep(int n) => Enumerable.Range(0, n).Reverse();
        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < a.Length; ++i) a[i] = f(i);
            return a;
        }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex {
    static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
        return string.Join(st, ie);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        solver.Solve();
        Console.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO {
    using System.IO;
    using System.Text;
    using System.Globalization;

    public class Printer : StreamWriter {
        public override IFormatProvider FormatProvider => CultureInfo.InvariantCulture;
        public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
    }

    public class StreamScanner {
        public StreamScanner(Stream stream) { str = stream; }

        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;

        private byte read() {
            if (isEof) return 0;
            if (ptr >= len) {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0) {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }

        public char Char() {
            byte b = 0;
            do b = read(); while ((b < 33 || 126 < b) && !isEof);
            return (char)b;
        }
        public string Scan() {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine() {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
        public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
        public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
}

#endregion