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-csc 3.5.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Text;
using System.Text.RegularExpressions;
using System.Diagnostics;
using static util;
using P = pair<int, int>;
using Number = System.Int64;

class Program {
    static void Main(string[] args) {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var solver = new Solver(sw);
        // var t = new Thread(solver.solve, 1 << 26); // 64 MB
        // t.Start();
        // t.Join();
        solver.solve();
        sw.Flush();
    }
}

class Solver {
    StreamWriter sw;
    Scan sc;
    void Prt(string a) => sw.WriteLine(a);
    void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    void Prt(params object[] a) => Prt(string.Join(" ", a));
    public Solver(StreamWriter sw) {
        this.sw = sw;
        this.sc = new Scan();
    }

    public void solve() {
        int n, m;
        sc.Multi(out n, out m);
        int s, t;
        sc.Multi(out s, out t);
        --s;
        --t;
        var edge = new List<pair<long, int>>[n];
        for (int i = 0; i < n; i++)
        {
            edge[i] = new List<pair<long, int>>();
        }
        for (int i = 0; i < m; i++)
        {
            int u, v, d;
            sc.Multi(out u, out v, out d);
            --u;
            --v;
            edge[u].Add(new pair<long, int>(d, v));
            edge[v].Add(new pair<long, int>(d, u));
        }
        var ds = ShortestPath.Dijkstra(edge, s, M, LM);
        var dt = ShortestPath.Dijkstra(edge, t, M, LM);
        long ans = ds.v2[t] * dt.v2[s] % M;
        long dist = ds.v1[t];
        for (int i = 0; i < n; i++)
        {
            if (ds.v1[i] == dt.v1[i] && ds.v1[i] + dt.v1[i] == dist) {
                long c = ds.v2[i] * dt.v2[i] % M;
                ans = (ans - c * c % M + M) % M;
            }
            foreach (var item in edge[i])
            {
                if (ds.v1[i] + item.v1 + dt.v1[item.v2] == dist && ds.v1[i] * 2 < dist && dt.v1[item.v2] * 2 < dist) {
                    long c = ds.v2[i] * dt.v2[item.v2] % M;
                    ans = (ans - c * c % M + M) % M;
                }
            }
        }
        Prt(ans);
    }
}

class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
}
static class util {
    public static readonly int M = 1000000007;
    // public static readonly int M = 998244353;
    public static readonly long LM = 1L << 60;
    public static readonly double eps = 1e-11;
    public static void DBG(string a) => Console.Error.WriteLine(a);
    public static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    public static void DBG(params object[] a) => DBG(string.Join(" ", a));
    public static void Assert(params bool[] conds) {
        if (conds.Any(x => !x)) throw new Exception();
    }
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static int CompareList<T>(IList<T> a, IList<T> b) where T : IComparable<T> {
        for (int i = 0; i < a.Count && i < b.Count; i++)
            if (a[i].CompareTo(b[i]) != 0) return a[i].CompareTo(b[i]);
        return a.Count.CompareTo(b.Count);
    }
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    public static readonly int[] dd = { 0, 1, 0, -1 };
    // static readonly string dstring = "RDLU";
    public static IEnumerable<P> adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1]));
    public static IEnumerable<P> adjacents(int i, int j, int h, int w)
        => adjacents(i, j).Where(p => inside(p.v1, p.v2, h, w));
    public static IEnumerable<P> adjacents(this P p) => adjacents(p.v1, p.v2);
    public static IEnumerable<P> adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static IEnumerable<int> all_subset(this int p) {
        for (int i = 0; ; i = i - p & p) {
            yield return i;
            if (i == p) break;
        }
    }
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.SelectMany(x => x));
    public static T[] inv<T>(this Dictionary<T, int> dic) {
        var res = new T[dic.Count];
        foreach (var item in dic) res[item.Value] = item.Key;
        return res;
    }
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => sr.ReadLine().Trim();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]);
    }
    public void Multi<T, U, V>(out T a, out U b, out V c) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]);
    }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]);
    }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]);
    }
}


static class ShortestPath {
    public static void WarshallFloyd(Number[][] g) {
        int n = g.Length;
        for (int i = 0; i < n; ++i)
            for (int j = 0; j < n; ++j)
                for (int k = 0; k < n; ++k)
                    g[j][k] = Math.Min(g[j][k], g[j][i] + g[i][k]);
    }

    class Heap {
        int n;
        Number[] values;
        int[] keys, indices;

        void Update(int i, Number val) {
            if (indices[i] == -1) {
                keys[n] = i;
                indices[i] = n;
                ++n;
            }
            values[i] = val;
            int p = indices[i];
            while (p > 0) {
                int par = p - 1 >> 1;
                if (values[keys[par]] > val) {
                    keys[p] = keys[par];
                    indices[keys[p]] = p;
                    p = par;
                }
                else break;
            }
            keys[p] = i;
            indices[i] = p;
        }
        int Pop() {
            --n;
            int ret = keys[0];
            indices[ret] = -1;
            int i = keys[n];
            keys[n] = -1;
            Number val = values[i];
            if (n == 0) return ret;
            int p = 0;
            while ((p << 1 | 1) < n) {
                int ch = p << 1 | 1;
                if (ch < n - 1 && values[keys[ch]] > values[keys[ch + 1]]) ++ch;
                if (val > values[keys[ch]]) {
                    keys[p] = keys[ch];
                    indices[keys[p]] = p;
                    p = ch;
                }
                else break;
            }
            keys[p] = i;
            indices[i] = p;
            return ret;
        }

        void init(int m, int s, Number inf) {
            values = new Number[m];
            keys = new int[m];
            indices = new int[m];
            for (int i = 0; i < m; i++) {
                values[i] = inf;
                keys[i] = -1;
                indices[i] = -1;
            }
            values[s] = 0;
            keys[0] = s;
            indices[s] = 0;
        }

        public Number[] Run(List<pair<Number, int>>[] edges, int s, Number inf) {
            init(edges.Length, s, inf);

            n = 1;
            while (n > 0) {
                int p = Pop();
                foreach (var e in edges[p])
                    if (values[e.v2] > values[p] + e.v1)
                        Update(e.v2, values[p] + e.v1);
            }
            return values;
        }
        public pair<Number[], long[]> Run(List<pair<Number, int>>[] edges, int s, long Mod, Number inf) {
            init(edges.Length, s, inf);
            var cnts = new long[edges.Length];
            cnts[s] = 1;

            n = 1;
            while (n > 0) {
                int p = Pop();
                foreach (var e in edges[p])
                    if (values[e.v2] > values[p] + e.v1) {
                        cnts[e.v2] = cnts[p];
                        Update(e.v2, values[p] + e.v1);
                    }
                    else if (values[e.v2] == values[p] + e.v1)
                        cnts[e.v2] = (cnts[e.v2] + cnts[p]) % Mod;
            }
            return new pair<Number[], long[]>(values, cnts);
        }
    }

    public static Number[] Dijkstra(List<pair<Number, int>>[] edges, int s, Number inf)
        => new Heap().Run(edges, s, inf);

    public static pair<Number[], long[]> Dijkstra(List<pair<Number, int>>[] edges, int s, long Mod, Number inf)
        => new Heap().Run(edges, s, Mod, inf);
}