Problem:
Score : 700 points

Problem Statement

We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.

The i-th edge connects Vertex U_i and Vertex V_i.
The time it takes to traverse this edge is D_i minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).

Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 10^9 + 7.

Constraints

1 \leq N \leq 100 000

1 \leq M \leq 200 000

1 \leq S, T \leq N

S \neq T

1 \leq U_i, V_i \leq N (1 \leq i \leq M)

1 \leq D_i \leq 10^9 (1 \leq i \leq M)

If i \neq j, then (U_i, V_i) \neq (U_j, V_j) and (U_i, V_i) \neq (V_j, U_j).

U_i \neq V_i (1 \leq i \leq M)

D_i are integers.

The given graph is connected.

Input

Input is given from Standard Input in the following format:

N M
S T
U_1 V_1 D_1
U_2 V_2 D_2
:
U_M V_M D_M

Output

Print the answer.

Sample Input 1

4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1

Sample Output 1

2

There are two ways to choose shortest paths that satisfies the condition:

Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.

Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.

Sample Input 2

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 2

2

Sample Input 3

3 3
1 3
1 2 1
2 3 1
3 1 2

Sample Output 3

2

Sample Input 4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Sample Output 4

6

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Numerics;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
using Number = System.Int64;
using static System.Math;
using ll = System.Int64;
namespace Program {
    public class Solver {
        Random rnd = new Random();
        public void Solve() {
            var n = ri;
            var m = ri;
            var s = ri - 1; var t = ri - 1;
            var G = Enumerate(n, x => new List<KeyValuePair<int, long>>());
            for (int i = 0; i < m; i++)
            {
                var u = ri - 1;
                var v = ri - 1;
                var d = ri;
                G[u].Add(new KeyValuePair<int, Number>(v, d));
                G[v].Add(new KeyValuePair<int, Number>(u, d));
            }
            long[] da, db;
            ModInt[] pa, pb;
            solve(n, s, G, out da, out pa);
            solve(n, t, G, out db, out pb);
            Debug.WriteLine(da.AsJoinedString());
            Debug.WriteLine(db.AsJoinedString());
            var all = pa[t] * pb[s];
            for (int i = 0; i < n; i++)
            {
                if (da[i] == db[i] && 2 * da[i] == da[t]) all -= pa[i] * pb[i];
                foreach (var e in G[i])
                {
                    var j = e.Key;
                    var dd = e.Value;
                    if (da[i] + dd != da[j]) continue;
                    if (db[j] + dd != db[i]) continue;
                    var A = da[i]; var B = da[i] + dd;
                    var C = db[j]; var D = db[j] + dd;
                    var ma = Max(A, C);
                    var mi = Min(B, D);
                    if (ma < mi)
                    {
                        all -= pa[i] * pb[j];
                    }

                }
            }
            Console.WriteLine(all);
        }

        void solve(int n, int src, List<KeyValuePair<int, long>>[] G, out long[] dist, out ModInt[] pat) {
            dist = Enumerate(n, x => 1L << 60);
            pat = new ModInt[n];

            dist[src] = 0;
            pat[src] = 1;
            var pq = new PriorityQueue<KeyValuePair<int, long>>((l, r) => l.Value.CompareTo(r.Value));
            pq.Enqueue(new KeyValuePair<int, Number>(src, 0));
            while (pq.Any())
            {
                var p = pq.Dequeue();
                var u = p.Key; var d = p.Value;
                foreach (var e in G[u])
                {
                    var v = e.Key;
                    var nd = d + e.Value;
                    if (nd < dist[v])
                    {
                        dist[v] = nd;
                        pat[v] = pat[u];
                        pq.Enqueue(new KeyValuePair<int, Number>(v, nd));
                    }
                    else if (nd == dist[v]) pat[v] += pat[u];
                }
            }

        }
        const long MOD = (long)1e9 + 7;
        const long INF = 1L << 60;
        //int[] dx = { -1, 0, 1, 0 };
        //int[] dy = { 0, 1, 0, -1 };

        int ri { get { return sc.Integer(); } }
        long rl { get { return sc.Long(); } }
        double rd { get { return sc.Double(); } }
        string rs { get { return sc.Scan(); } }
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());

        static T[] Enumerate<T>(int n, Func<int, T> f) {
            var a = new T[n];
            for (int i = 0; i < n; ++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.Select(x => x.ToString()).ToArray());
        //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 { get { return 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;
        public bool IsEndOfStream { get { return isEof; } }

        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

#region ModInt
public struct ModInt {
    public const long Mod = (long)1e9 + 7;
    public long num;
    public ModInt(long n) { num = n; }
    public override string ToString() { return num.ToString(); }
    public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
    public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
    public static ModInt operator *(ModInt l, ModInt r) { return new ModInt((int)(l.num * r.num % Mod)); }
    public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }
    public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }
    public static ModInt Pow(long v, long k) {
        long ret = 1;
        for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
            if ((k & 1) == 1) ret = ret * v % Mod;
        return new ModInt(ret);
    }
    public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
}
#endregion


#region BinaryHeap
public class PriorityQueue<T> {
    readonly List<T> heap = new List<T>();
    readonly Comparison<T> compare;
    public PriorityQueue() { compare = Comparer<T>.Default.Compare; }

    public PriorityQueue(Comparison<T> comparison) { compare = comparison; }
    public PriorityQueue(IComparer<T> comparer) { compare = comparer.Compare; }

    public void Enqueue(T item) {
        var pos = heap.Count;
        heap.Add(item);
        while (pos > 0)
        {
            var par = (pos - 1) / 2;
            if (compare(heap[par], item) <= 0)
                break;
            heap[pos] = heap[par];
            pos = par;
        }
        heap[pos] = item;

    }

    public T Dequeue() {
        var ret = heap[0];
        var pos = 0;
        var x = heap[heap.Count - 1];

        while (pos * 2 + 1 < heap.Count - 1)
        {
            var lch = pos * 2 + 1;
            var rch = pos * 2 + 2;
            if (rch < heap.Count - 1 && compare(heap[rch], heap[lch]) < 0) lch = rch;
            if (compare(heap[lch], x) >= 0)
                break;
            heap[pos] = heap[lch];
            pos = lch;
        }
        heap[pos] = x;
        heap.RemoveAt(heap.Count - 1);
        return ret;

    }
    public T Peek() { return heap[0]; }
    public int Count { get { return heap.Count; } }
    public bool Any() { return heap.Count > 0; }

    public T[] Items {
        get {
            var ret = heap.ToArray();
            Array.Sort(ret, compare);
            return ret;
        }
    }
}
#endregion