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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

// package arc.arc093;

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.*;

public class Main {
    private static final long MOD = 1000000007;

    public static void main(String[] args) {
        InputReader in = new InputReader(System.in);
        PrintWriter out = new PrintWriter(System.out);

        int n = in.nextInt();
        int m = in.nextInt();
        long X = in.nextLong();

        int[][] edges = new int[m][3];
        for (int i = 0; i < m ; i++) {
            edges[i][0] = in.nextInt()-1;
            edges[i][1] = in.nextInt()-1;
            edges[i][2] = in.nextInt();
        }
        Arrays.sort(edges, Comparator.comparingInt(u -> u[2]));


        UnionFind uf = new UnionFind(n);
        int[][] unionEdges = new int[n-1][3];
        boolean[] used = new boolean[m];
        long baseCost = 0;

        int ni = 0;
        for (int i = 0; i < m ; i++) {
            int u = edges[i][0];
            int v = edges[i][1];
            if (uf.issame(u, v)) {
                continue;
            }
            used[i] = true;
            baseCost += edges[i][2];
            uf.unite(u, v);
            unionEdges[ni++] = edges[i].clone();
        }

        if (baseCost > X) {
            // muri
            out.println(0);
            out.flush();
            return;
        }

        tree = buildWeightedGraph(n, unionEdges);
        parent = new int[n];
        plength = new int[n];
        depth = new int[n];
        dfs(0, -1);

        long[] pow2mod = new long[m+10];
        pow2mod[0] = 1;
        for (int i = 1 ; i < m+10 ; i++) {
            pow2mod[i] = pow2mod[i-1] * 2 % MOD;
        }

        long ans = 0;
        for (int i = 0; i < m ; i++) {
            int u = edges[i][0];
            int v = edges[i][1];
            long w = edges[i][2];

            long elseEdgePtn = pow2mod[m-i-1];
            if (i == 0) {
                elseEdgePtn = (elseEdgePtn + MOD - 1) % MOD;
            }

            if (used[i]) {
                if (baseCost == X) {
                    ans += elseEdgePtn;
                }
            } else {
                long ecost = largestCost(u, v);
                long toCost = baseCost + w - ecost;
                if (toCost == X) {
                    ans += elseEdgePtn * ((baseCost == X) ? 1L : 2L);
                }
            }
        }

        out.println(ans % MOD);
        out.flush();
    }

    static int[][][] tree;
    static int[] parent;
    static int[] plength;
    static int[] depth;

    static int largestCost(int u, int v) {
        if (depth[u] == depth[v]) {
            if (u == v) {
                return 0;
            } else {
                int p = largestCost(parent[u], parent[v]);
                return Math.max(plength[u], Math.max(plength[v], p));
            }
        } else if (depth[u] < depth[v]) {
            int p = largestCost(u, parent[v]);
            return Math.max(plength[v], p);
        } else {
            int p = largestCost(parent[u], v);
            return Math.max(plength[u], p);
        }
    }


    static void dfs(int now, int par) {
        for (int[] e : tree[now]) {
            int to = e[0];
            int w = e[1];
            if (to == par) {
                continue;
            }
            dfs(to, now);

            parent[to] = now;
            plength[to] = w;
            depth[to] = depth[now] + 1;
        }
    }


    static int[][][] buildWeightedGraph(int n, int[][] edges) {
        int m = edges.length;
        int[][][] graph = new int[n][][];
        int[] deg = new int[n];
        for (int i = 0 ; i < m ; i++) {
            deg[edges[i][0]]++;
            deg[edges[i][1]]++;
        }
        for (int i = 0 ; i < n ; i++) {
            graph[i] = new int[deg[i]][2];
        }
        for (int i = 0 ; i < m ; i++) {
            int a = edges[i][0];
            int b = edges[i][1];
            int w = edges[i][2];
            graph[a][--deg[a]][0] = b;
            graph[b][--deg[b]][0] = a;
            graph[a][deg[a]][1] = w;
            graph[b][deg[b]][1] = w;
        }
        return graph;
    }

    static class UnionFind {
        int[] rank;
        int[] parent;
        int[] cnt;

        public UnionFind(int n) {
            rank = new int[n];
            parent = new int[n];
            cnt = new int[n];
            for (int i = 0; i < n; i++) {
                parent[i] = i;
                cnt[i] = 1;
            }
        }

        public int find(int a) {
            if (parent[a] == a) {
                return a;
            }
            parent[a] = find(parent[a]);
            return parent[a];
        }

        public void unite(int a, int b) {
            a = find(a);
            b = find(b);
            if (a == b) {
                return;
            }
            if (rank[a] < rank[b]) {
                parent[a] = b;
                cnt[b] += cnt[a];
                cnt[a] = cnt[b];
            } else {
                parent[b] = a;
                cnt[a] += cnt[b];
                cnt[b] = cnt[a];
                if (rank[a] == rank[b]) {
                    rank[a]++;
                }
            }
        }

        public int groupCount(int a) {
            return cnt[find(a)];
        }

        private boolean issame(int a, int b) {
            return find(a) == find(b);
        }
    }


    public static void debug(Object... o) {
        System.err.println(Arrays.deepToString(o));
    }

    public static class InputReader {
        private static final int BUFFER_LENGTH = 1 << 12;
        private InputStream stream;
        private byte[] buf = new byte[BUFFER_LENGTH];
        private int curChar;
        private int numChars;

        public InputReader(InputStream stream) {
            this.stream = stream;
        }

        private int next() {
            if (numChars == -1) {
                throw new InputMismatchException();
            }
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0)
                    return -1;
            }
            return buf[curChar++];
        }

        public char nextChar() {
            return (char) skipWhileSpace();
        }

        public String nextToken() {
            int c = skipWhileSpace();
            StringBuilder res = new StringBuilder();
            do {
                res.append((char) c);
                c = next();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        public int nextInt() {
            return (int) nextLong();
        }

        public long nextLong() {
            int c = skipWhileSpace();
            long sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = next();
            }
            long res = 0;
            do {
                if (c < '0' || c > '9') {
                    throw new InputMismatchException();
                }
                res *= 10;
                res += c - '0';
                c = next();
            } while (!isSpaceChar(c));
            return res * sgn;
        }

        public double nextDouble() {
            return Double.valueOf(nextToken());
        }

        int skipWhileSpace() {
            int c = next();
            while (isSpaceChar(c)) {
                c = next();
            }
            return c;
        }

        boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }
    }
}