Problem:
Score : 900 points

Problem Statement

We have an undirected weighted graph with N vertices and M edges.
The i-th edge in the graph connects Vertex U_i and Vertex V_i, and has a weight of W_i.
Additionally, you are given an integer X.

Find the number of ways to paint each edge in this graph either white or black such that the following condition is met, modulo 10^9 + 7:

The graph has a spanning tree that contains both an edge painted white and an edge painted black. Furthermore, among such spanning trees, the one with the smallest weight has a weight of X.

Here, the weight of a spanning tree is the sum of the weights of the edges contained in the spanning tree.

Constraints

1 \leq N \leq 1 000

1 \leq M \leq 2 000

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

1 \leq W_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)

The given graph is connected.

1 \leq X \leq 10^{12}

All input values are integers.

Input

Input is given from Standard Input in the following format:

N M
X
U_1 V_1 W_1
U_2 V_2 W_2
:
U_M V_M W_M

Output

Print the answer.

Sample Input 1

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

Sample Output 1

6

Sample Input 2

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

Sample Output 2

2

Sample Input 3

5 4
1
1 2 3
1 3 3
2 4 6
2 5 8

Sample Output 3

0

Sample Input 4

8 10
49
4 6 10
8 4 11
5 8 9
1 8 10
3 8 128773450
7 8 10
4 2 4
3 4 1
3 1 13
5 2 2

Sample Output 4

4

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

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using SB = System.Text.StringBuilder;
//using System.Numerics;
using static System.Math;
namespace Program {
    public class Solver {
        Random rnd = new Random(0);
        public void Solve() {
            var n = ri;
            var m = ri;
            var X = rl;
            var f = new int[m];
            var t = new int[m];
            var w = new int[m];
            var set = new DisjointSet(n);
            for (int i = 0; i < m; i++)
            {
                f[i] = ri - 1; t[i] = ri - 1; w[i] = ri;
            }
            var G = Enumerate(n, x => new List<KeyValuePair<int, long>>());

            var sum = 0L;
            foreach (var id in Enumerate(m, x => x).OrderBy(x => w[x]))
            {
                if (set.Unite(f[id], t[id]))
                {
                    sum += w[id];
                    G[f[id]].Add(new KeyValuePair<int, long>(t[id], w[id]));
                    G[t[id]].Add(new KeyValuePair<int, long>(f[id], w[id]));
                }
            }
            if (sum > X)
            {
                Console.WriteLine(0); return;
            }
            var dist = Enumerate(n, x => new long[n]);
            for (int i = 0; i < n; i++)
            {
                Action<int, int, long> dfs = null;
                dfs = (prev, cur, v) =>
                {
                    dist[i][cur] = v;
                    foreach (var e in G[cur]) if (e.Key != prev) dfs(cur, e.Key, Max(v, e.Value));
                };
                dfs(-1, i, 0);
            }
            var eq = 0;
            var over = 0;
            for (int i = 0; i < m; i++)
            {
                var d = w[i] - dist[f[i]][t[i]];
                if (d == X - sum) eq++;
                else if (d > X - sum) over++;
            }
            ModInt ans = 0;
            if (X == sum)
            {
                ans += (ModInt.Pow(2, n - 1) - 2) * ModInt.Pow(2, m - (n - 1));
                ans += 2 * (ModInt.Pow(2, eq - (n - 1)) - 1) * ModInt.Pow(2, over);
            }
            else
            {
                ans += 2 * ModInt.Pow(2, over);
            }
            Console.WriteLine(ans);
        }
        const long INF = 5L << 60;
        static int[] dx = { -1, 0, 1, 0 };
        static int[] dy = { 0, 1, 0, -1 };
        const string URDL = "URDL";
        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);
    }
    static public void Main() {
        Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
        var solver = new Program.Solver();
        try
        {
            solver.Solve();
            Console.Out.Flush();
        }
        catch { }
    }
}
#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 = (int)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(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 DisjointSet
public class DisjointSet {
    int[] par;
    byte[] rank;
    public DisjointSet(int n) {
        par = new int[n];
        for (int i = 0; i < n; i++)
            par[i] = -1;
        rank = new byte[n];
    }
    public int this[int id] {
        get {
            if ((par[id] < 0)) return id;
            return par[id] = this[par[id]];
        }
    }
    public bool Unite(int x, int y) {
        x = this[x]; y = this[y];
        if (x == y) return false;
        if (rank[x] < rank[y]) { var z = x; x = y; y = z; }
        par[x] += par[y];
        par[y] = x;
        if (rank[x] == rank[y])
            rank[x]++;
        return true;
    }
    public int Size(int x) { return -par[this[x]]; }
    public bool IsUnited(int x, int y) { return this[x] == this[y]; }

}
#endregion