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

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.NoSuchElementException;
import java.util.Objects;
import java.util.PriorityQueue;
import java.util.Set;
import java.util.TreeSet;


public class Main {
    public static void main(String[] args) {
        Out out = new Out();
        solve(out);
        out.outln();
        out.flush();
        out.close();
    }

    public static void solve(Out out) {
        ModuloArithmetic ma = new ModuloArithmetic(Const.MOD7);
        int n = In.ni();
        int m = In.ni();
        int s = In.ni() - 1;
        int t = In.ni() - 1;
        Graph g = new Graph(n);
        for (int i = 0; i < m; i++) {
            int u = In.ni() - 1;
            int v = In.ni() - 1;
            int d = In.ni();
            g.addEdge(u, v, d);
        }
        long[] sd = new Dijkstra(g, s).d;
        long st = sd[t];
        long[] td = new Dijkstra(g, t).d;
        TreeSet<Integer> onPath = new TreeSet<>((u, v) -> {
            if (sd[u] == sd[v]) {
                return u - v;
            } else {
                return Long.compare(sd[u], sd[v]);
            }
        });
        for (int i = 0; i < n; i++) {
            if (sd[i] + td[i] == st) {
                onPath.add(i);
            }
        }
        Digraph stg = new Digraph(n);
        Digraph tsg = new Digraph(n);
        for (int u : onPath) {
            for (Edge e : g.getEdges(u)) {
                int v = e.to;
                long c = e.cost;
                if (onPath.contains(v) && sd[u] < sd[v] && sd[u] + c == sd[v]) {
                    stg.addEdge(u, v, c);
                    tsg.addEdge(v, u, c);
                }
            }
        }
        long[] dpstg = new long[n];
        dpstg[s] = 1;
        Iterator<Integer> it = onPath.iterator();
        while (it.hasNext()) {
            int u = it.next();
            for (Edge e : stg.getEdges(u)) {
                int v = e.to;
                dpstg[v] = ma.add(dpstg[v], dpstg[u]);
            }
        }
        long[] dptsg = new long[n];
        dptsg[t] = 1;
        Iterator<Integer> rvit = onPath.descendingIterator();
        HashSet<Long> eSet = new HashSet<>();
        HashSet<Integer> vSet = new HashSet<>();
        while (rvit.hasNext()) {
            int u = rvit.next();
            if (td[u] == sd[u]) {
                vSet.add(u);
            }
            for (Edge e : tsg.getEdges(u)) {
                int v = e.to;
                dptsg[v] = ma.add(dptsg[v], dptsg[u]);
                if (td[u] << 1 < st && td[v] << 1 > st) {
                    eSet.add((long) u * n + v);
                }
            }
        }
        long sub = 0;
        for (long e : eSet) {
            int u = (int) (e / n);
            int v = (int) (e % n);
            sub += ma.prod(dptsg[u], dpstg[v], dpstg[v], dptsg[u]);
        }
        for (int v : vSet) {
            sub += ma.mul(dpstg[v], dptsg[v]);
        }
        sub = ma.mod(sub);
        long ans = ma.sub(ma.mul(dpstg[t], dptsg[s]), sub);
        out.append(ans);
    }
}

class Dijkstra {
    public static final long INF = Const.LINF;
    public int[] prev;
    public long[] d;

    public Dijkstra(AbstractGraph g, int s) {
        if (g.getE() * 10 >= (long) g.getV() * g.getV()) {
            dijkstra2(g, s);
        } else {
            dijkstra(g, s);
        }
    }

    /**
     * O((V+E)logV) dijkstra
     */
    private void dijkstra(AbstractGraph g, int s) {
        int n = g.getV();
        this.prev = new int[n];
        this.d = new long[n];
        PriorityQueue<long[]> pq = new PriorityQueue<>((u, v) -> Long.compare(u[0], v[0]));
        Arrays.fill(d, Const.LINF);
        d[s] = 0;
        prev[s] = -1;
        pq.add(new long[]{0, s});
        while (pq.size() > 0) {
            long[] dv = pq.poll();
            int u = (int) dv[1];
            if (dv[0] != d[u]) {
                continue;
            }
            for (Edge e : g.getEdges(u)) {
                int v = e.to;
                long c = e.cost;
                if (d[u] + c < d[v]) {
                    d[v] = d[u] + c;
                    prev[v] = u;
                    pq.add(new long[]{d[v], v});
                }
            }
        }
    }

    /**
     * O(V^2) dijkstra
     */
    private void dijkstra2(AbstractGraph g, int s) {
        int n = g.getV();
        this.d = new long[n];
        this.prev = new int[n];
        Arrays.fill(d, INF);
        d[s] = 0;
        prev[s] = -1;
        HashSet<Integer> set = new HashSet<>();
        for (int i = 0; i < n; i++) {
            set.add(i);
        }
        for (int i = 0; i < n; i++) {
            long minD = INF;
            int minV = -1;
            for (int v : set) {
                if (d[v] < minD) {
                    minD = d[v];
                    minV = v;
                }
            }
            if (minV < 0) {
                break;
            }
            int from = minV;
            for (Edge e : g.getEdges(from)) {
                int to = e.to;
                long cost = e.cost;
                if (d[from] + cost < d[to]) {
                    d[to] = d[from] + cost;
                    prev[to] = from;
                }
            }
            set.remove(from);
        }
    }

    public int[] path(int from, int to) {
        ArrayList<Integer> rev = new ArrayList<>();
        for (int now = to; now != from; now = this.prev[now]) {
            rev.add(now);
        }
        rev.add(from);
        int[] path = new int[rev.size()];
        for (int i = 0; i < path.length; i++) {
            path[i] = rev.get(path.length - i - 1);
        }
        return path;
    }
}


class Out {
    private StringBuilder sb = new StringBuilder();
    private PrintWriter w = new PrintWriter(System.out);

    public Out() {
    }

    public void append(int v) {
        sb.append(v);
    }

    public void append(long v) {
        sb.append(v);
    }

    public void append(char c) {
        sb.append(c);
    }

    public void append(double v) {
        sb.append(v);
    }

    public void append(char[] c) {
        sb.append(c);
    }

    public void append(String s) {
        sb.append(s);
    }

    public void append(Object e) {
        sb.append(e);
    }

    public void append(int[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void append(long[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void append(char[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void append(double[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void append(char[][] c, String sep) {
        for (int i = 0; i < c.length; i++) {
            append(c[i]);
            if (i < c.length - 1) {
                append(sep);
            }
        }
    }

    public void append(String[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void append(Object[] a, String sep) {
        for (int i = 0; i < a.length; i++) {
            append(a[i]);
            if (i < a.length - 1) {
                append(sep);
            }
        }
    }

    public void out() {
        w.print(sb);
        clearBuilder();
    }

    public void outln() {
        w.println(sb);
        clearBuilder();
    }

    public void print(int v) {
        w.print(v);
    }

    public void print(long v) {
        w.print(v);
    }

    public void print(char c) {
        w.print(c);
    }

    public void print(double v) {
        w.print(v);
    }

    public void print(String s) {
        w.print(s);
    }

    public void print(Object e) {
        w.print(e);
    }

    public void print(String sep, int... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void print(String sep, long... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void print(String sep, char... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void print(String sep, double... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void print(String sep, String... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void print(String sep, Object... a) {
        StringBuilder s = new StringBuilder();
        for (int i = 0; i < a.length; i++) {
            s.append(a[i]);
            if (i < a.length - 1) {
                s.append(sep);
            }
        }
        print(s);
    }

    public void println() {
        w.println();
    }

    public void println(int v) {
        w.println(v);
    }

    public void println(long v) {
        w.println(v);
    }

    public void println(double v) {
        w.println(v);
    }

    public void println(char c) {
        w.println(c);
    }

    public void println(String s) {
        w.println(s);
    }

    public void println(Object e) {
        w.println(e);
    }

    public void println(String sep, int... a) {
        print(sep, a);
        println();
    }

    public void println(String sep, long... a) {
        print(sep, a);
        println();
    }

    public void println(String sep, double... a) {
        print(sep, a);
        println();
    }

    public void println(String sep, char... a) {
        print(sep, a);
        println();
    }

    public void println(String sep, String... a) {
        print(sep, a);
        println();
    }

    // public void println(String sep, Object... a) {
    //     print(sep, a);
    //     println();
    // }

    public void flush() {
        w.flush();
    }

    public void close() {
        w.close();
    }

    public void clearBuilder() {
        sb = new StringBuilder();
    }

    public void reverse() {
        sb.reverse();
    }

    public void delete(int start, int end) {
        sb.delete(start, end);
    }

    public void delete(int length) {
        sb.delete(sb.length() - length, sb.length());
    }

    public int length() {
        return sb.length();
    }
}

class IntDeque implements Cloneable {
    private int head = -1;
    private int tail = 0;
    private int pSize;
    private int nSize;
    private int p[];
    private int n[];

    private static final int DEFAULT_SIZE = 64;

    public IntDeque(int initialSize) {
        this(Math.max(1, initialSize - initialSize / 2), Math.max(1, initialSize / 2));
        if (initialSize <= 0) {
            System.err.println("Error in IntDeque::new. initial size must be positive.");
        }
    }

    private IntDeque(int pSize, int nSize) {
        this.pSize = pSize;
        this.nSize = nSize;
        this.p = new int[this.pSize];
        this.n = new int[this.nSize];
    }

    public IntDeque() {
        this(DEFAULT_SIZE);
    }

    public static IntDeque asDeque(int initialSize) {
        return new IntDeque(initialSize);
    }

    public static IntDeque asDeque() {
        return new IntDeque(DEFAULT_SIZE);
    }

    public static IntDeque asQueue(int initialSize) {
        return new IntDeque(Math.max(1, initialSize), 1);
    }

    public static IntDeque asQueue() {
        return new IntDeque(DEFAULT_SIZE, 1);
    }

    public static IntDeque asStack(int initialSize) {
        return new IntDeque(1, Math.max(1, initialSize));
    }

    public static IntDeque asStack() {
        return new IntDeque(1, DEFAULT_SIZE);
    }

    public void addLast(int v) {
        putByRawIndex(tail++, v);
    }

    public void addFirst(int v) {
        putByRawIndex(head--, v);
    }

    public int pollFirst() {
        if (size() == 0) {
            System.err.println("Error in IntDeque::pollFirst. cannot poll element from \"EMPTY\" IntDeque.");
        }
        return getByRawIndex(++head);
    }

    public int pollLast() {
        if (size() == 0) {
            System.err.println("Error in IntDeque::pollLast. cannot poll element from \"EMPTY\" IntDeque.");
        }
        return getByRawIndex(--tail);
    }

    public int get(int index) {
        index += head + 1;
        if (index >= tail) {
            System.err.println("Error in IntDeque::get(" + index + "). the size of IntDeque is " + size() + ".");
        }
        return getByRawIndex(index);
    }

    public int getFirst() {
        return get(0);
    }

    public int getLast() {
        return get(size() - 1);
    }

    public int size() {
        return tail - head - 1;
    }

    public void clear() {
        head = -1;
        tail = 0;
    }

    public int[] toArray() {
        int[] a = new int[size()];
        for (int i = 0; i < size(); i++) {
            a[i] = get(i);
        }
        return a;
    }

    private void resize(boolean positive) {
        if (positive) {
            int[] pCopy = new int[pSize << 1];
            System.arraycopy(p, 0, pCopy, 0, pSize);
            p = pCopy;
            pSize <<= 1;
        } else {
            int[] nCopy = new int[nSize << 1];
            System.arraycopy(n, 0, nCopy, 0, nSize);
            n = nCopy;
            nSize <<= 1;
        }
    }

    private int getByRawIndex(int index) {
        if (index >= 0) {
            return p[index];
        } else {
            return n[-(index + 1)];
        }
    }

    private void putByRawIndex(int index, int v) {
        if (index < 0) {
            if (index <= -(nSize + 1)) {
                resize(false);
            }
            n[-(index + 1)] = v;
        } else {
            if (index >= pSize) {
                resize(true);
            }
            p[index] = v;
        }
    }

    public boolean isEmpty() {
        return size() == 0;
    }

    public boolean nonEmpty() {
        return size() != 0;
    }

    @Override
    public String toString() {
        StringBuilder sb = new StringBuilder();
        sb.append('[');
        for (int i = 0; i < size(); i++) {
            sb.append(get(i));
            if (i < size() - 1) {
                sb.append(", ");
            }
        }
        sb.append(']');
        return sb.toString();
    }
}

class Const {
    public static final long MOD7 = 1_000_000_007;
    public static final long MOD9 = 1_000_000_009;
    public static final long MOD99 = 998_244_353;

    public static final long LINF = 1_000_000_000_000_000_000l;
    public static final int IINF = 1_000_000_000;
    public static final double DINF = 1e150;

    public static final double DELTA = 0.000_000_000_1;
    public static final double LDELTA = 0.000_001;

    public static final String YES = "YES";
    public static final String NO = "NO";
    public static final String Yes = "Yes";
    public static final String No = "No";
    public static final String POSSIBLE = "POSSIBLE";
    public static final String IMPOSSIBLE = "IMPOSSIBLE";
    public static final String Possible = "Possible";
    public static final String Impossible = "Impossible";

    public static final int[] dx8 = {1, 0, -1, 0, 1, -1, -1, 1};
    public static final int[] dy8 = {0, 1, 0, -1, 1, 1, -1, -1};
    public static final int[] dx = {1, 0, -1, 0};
    public static final int[] dy = {0, 1, 0, -1};

    private Const(){}
}

class Edge {
    public int from;
    public int to;
    public long cost;

    public Edge(int from, int to, long cost) {
        this.from = from;
        this.to = to;
        this.cost = cost;
    }

    public Edge(int from, int to) {
        this(from, to, 1);
    }

    protected Edge reverse() {
        return new Edge(to, from, cost);
    }
}


class Graph extends AbstractGraph {
    public Graph(int n) {
        super(n);
    }

    public void addEdge(int from, int to, long cost) {
        Edge e = new Edge(from, to, cost);
        super.addEdge(e);
        super.addEdge(e.reverse());
        this.m++;
    }

    public void addEdge(int from, int to) {
        this.addEdge(from, to, 1);
    }

    @Override
    public void addEdge(Edge e) {
        super.addEdge(e);
        super.addEdge(e.reverse());
        this.m++;
    }

    @Override
    protected void generateComponentsIfAbsent() {
        if (Objects.isNull(getNullableComponents())) {
            HashMap<Integer, ArrayList<Integer>> components = new HashMap<>();
            for (int i = 0; i < n; i++) {
                if (!components.containsKey(getRoot(i))) {
                    ArrayList<Integer> component = new ArrayList<>();
                    component.add(i);
                    components.put(getRoot(i), component);
                } else {
                    components.get(getRoot(i)).add(i);
                }
            }
            setComponents(components);
        }
    }

    @Override
    public boolean isBipartite() {
        int[] color = new int[getV()];
        ArrayDeque<Integer> dq = new ArrayDeque<>();
        for (int root : getRoots()) {
            dq.add(root);
            color[root] = 1;
        }
        while (dq.size() > 0) {
            int now = dq.pollFirst();
            for (Edge e : getEdges(now)) {
                int next = e.to;
                if (color[next] == 0) {
                    color[next] = -color[now];
                    dq.add(next);
                } else if (color[now] == color[next]) {
                    return false;
                }
            }
        }
        return true;
    }
}


class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        } else {
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }

    private int readByte() {
        if (hasNextByte()) {
            return buffer[ptr++];
        } else {
            return -1;
        }
    }

    private static boolean isPrintableChar(int c) {
        return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr])) {
            ptr++;
        }
        return hasNextByte();
    }

    public String next() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext()) {
            throw new NoSuchElementException();
        }
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while (true) {
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            } else if (b == -1 || !isPrintableChar(b)) {
                return minus ? -n : n;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) {
            throw new NumberFormatException();
        }
        return (int) nl;
    }

    public double nextDouble() {
        return Double.parseDouble(next());
    }
}


class Digraph extends AbstractGraph {
    private Graph g;

    public Digraph(int n) {
        super(n);
    }

    public void addEdge(int from, int to, long cost) {
        Edge e = new Edge(from, to, cost);
        super.addEdge(e);
        this.m++;
    }

    public void addEdge(int from, int to) {
        this.addEdge(from, to, 1);
    }

    @Override
    public void addEdge(Edge e) {
        super.addEdge(e);
        this.m++;
    }

    @Override
    protected void generateComponentsIfAbsent() {
        if (Objects.isNull(getNullableComponents())) {
            setComponents(getG().getComponents());
        }
    }

    @Override
    public boolean isBipartite() {
        return getG().isBipartite();
    }

    private void generateNonDirectGraphIfAbsent() {
        if (Objects.isNull(this.g)) {
            this.g = new Graph(n);
            for (int i = 0; i < n; i++) {
                for (Edge e : getEdges(i)) {
                    this.g.addEdge(e);
                }
            }
        }
    }
    
    private Graph getG() {
        generateNonDirectGraphIfAbsent();
        return this.g;
    }

    public boolean isDAG() {
        return isConnected() && Objects.nonNull(TopologicalSort.topologicalSort(this));
    }
}

class ModuloArithmetic {
    public final long MOD;
    private long[] fact;
    private long[] pow;

    /**
     * support modulo p arithmetic
     * 
     * @param p p s.t. p is a prime number.
     */
    public ModuloArithmetic(long p) {
        this.MOD = p;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a mod p.
     * 
     * @param a
     * @return x s.t. 0 <= x < MOD /\ x = a mod p.
     */
    public long mod(long a) {
        a %= MOD;
        return a < 0 ? a + MOD : a;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 + 1_2 + ... + a_k mod p.
     * 
     * @param a
     * @return x s.t. 0 <= x < MOD /\ x = a_1 + 1_2 + ... + a_k mod p.
     */
    public long sum(long... a) {
        long ret = 0;
        for (long c : a) ret += c;
        return mod(ret);
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a + b mod p.
     * 
     * @param a
     * @param b
     * @return x s.t. 0 <= x < MOD /\ x = a + b mod p.
     */
    public long add(long a, long b) {
        long s = a + b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a - b mod p.
     * 
     * @param a
     * @param b
     * @return x s.t. 0 <= x < MOD /\ x = a - b mod p.
     */
    public long sub(long a, long b) {
        long s = a - b;
        return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 * 1_2 * ... * a_k mod p.
     * 
     * @param a
     * @return x s.t. 0 <= x < MOD /\ x = a_1 * 1_2 * ... * a_k mod p.
     */
    public long prod(long... a) {
        long ret = 1;
        for (long c : a) {
            ret = (ret * c) % MOD;
        }
        return ret < 0 ? ret + MOD : ret;
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a * b mod p.
     * 
     * @param a
     * @param b
     * @return x s.t. 0 <= x < MOD /\ x = a * b mod p.
     */
    public long mul(long a, long b) {
        return mod(a * b);
    }

    /**
     * Calculate x s.t. 0 <= x < MOD /\ b * x = a mod p.
     * 
     * @param a
     * @param b
     * @return x s.t. 0 <= x < MOD /\ b * x = a mod p.
     */
    public long div(long a, long b) {
        return mul(a, inv(b));
    }

    /**
     * Calculate the value b s.t. a*b mod MOD = 1.
     * 
     * @param a
     * @return inverse of a
     */
    public long inv(long a) {
        long b = MOD;
        long u = 1, v = 0;
        while (b >= 1) {
            long t = a / b;
            a -= t * b;
            long tmp1 = a; a = b; b = tmp1;
            u -= t * v;
            long tmp2 = u; u = v; v = tmp2;
        }
        return mod(u);
    }

    /**
     * Calculate the permutation nPr.
     * 
     * @param n
     * @param r
     * @return nPr
     */
    public long perm(long n, long r) {
        if (n < r) {
            return 0;
        }
        if (fact != null && n <= fact.length) {
            return div(fact[(int) n], fact[(int) (n - r)]);
        }
        long ret = 1;
        for (long i = n; i > n - r; i--) {
            ret = mul(ret, i);
        }
        return ret;
    }

    /**
     * Calculate the combination nCr.
     * 
     * @param n
     * @param r
     * @return nCr
     */
    public long comb(long n, long r) {
        if (n < r) {
            return 0;
        }
        if (fact != null && n <= fact.length) {
            return div(fact[(int) n], mul(fact[(int) (n - r)], fact[(int) r]));
        }
        r = Math.min(r, n - r);
        long res = 1;
        for (long d = 1; d <= r; d++) {
            res = div(mul(res, n--), d);
        }
        return res;
    }
    
    /**
     * call this method after building factorial array.
     * 
     * @param n
     * @return n! mod p
     */
    public long fact(int n) {
        return fact[n];
    }

    /**
     * call this nethod after building power array.
     * 
     * @param n
     * @return (base)^n
     */
    public long pow(int n) {
        return pow[n];
    }

    /**
     * Calculate a^b (mod {@code MOD}) in O(log b) time.
     * 
     * @param a base
     * @param b index
     * @return a^b (mod {@code MOD})
     */
    public long pow(long a, long b) {
        if (b == 0) {
            return 1;
        }
        long half = pow(a, b >> 1);
        if ((a & 1) == 0) {
            return mul(half, half);
        } else {
            return prod(half, half, a);
        }
    }

    /**
     * build factorial array from 0 to n (closed)
     * 
     * @param n
     */
    public void buildFactorial(int n) {
        this.fact = new long[n + 1];
        for (int i = 0; i <= n; i++) {
            if (i <= 1) {
                fact[i] = 1;
            } else {
                fact[i] = mul(fact[i - 1], i);
            }
        }
    }

    /**
     * build power array based on a from 0 to n (closed)
     * 
     * @param a base of power
     * @param n
     */
    public void buildPower(long a, int n) {
        if (a == 2) {
            buildPower(n);
        } else {
            this.pow = new long[n + 1];
            for (int i = 0; i <= n; i++) {
                if (i == 0) {
                    pow[i] = 1;
                } else {
                    pow[i] = mul(pow[i - 1], a);
                }
            }
        }
    }

    /**
     * build power array based on 2 from 0 to n (closed)
     * 
     * @param n
     */
    public void buildPower(int n) {
        this.pow = new long[n + 1];
        for (int i = 0; i <= n; i++) {
            if (i == 0) {
                pow[i] = 1;
            } else {
                pow[i] = pow[i - 1] << 1;
                if (pow[i] >= MOD) {
                    pow[i] -= MOD;
                }
            }
        }
    }
}


abstract class AbstractGraph {
    protected final int n;
    protected int m = 0;
    private ArrayList<Edge>[] adj;
    private int[] in;
    private HashMap<Integer, ArrayList<Integer>> components;
    private UnionFindTree uft;
    
    @SuppressWarnings("unchecked")
    protected AbstractGraph(int n) {
        this.n = n;
        this.in = new int[n];
        this.adj = new ArrayList[this.n];
        for (int i = 0; i < this.n; i++) {
            this.adj[i] = new ArrayList<>();
        }
        this.uft = new UnionFindTree(this.n);
    }

    protected void addEdge(Edge e) {
        this.adj[e.from].add(e);
        this.in[e.to]++;
        this.uft.unite(e.from, e.to);
    }

    public int in(int i) {
        return this.in[i];
    }

    public int out(int i) {
        return this.adj[i].size();
    }

    public Edge getEdge(int from, int index) {
        return this.adj[from].get(index);
    }

    public ArrayList<Edge> getEdges(int i) {
        return this.adj[i];
    }

    public int getV() {
        return this.n;
    }

    public int getE() {
        return this.m;
    }

    public int getRoot(int i) {
        return uft.root(i);
    }

    public HashMap<Integer, ArrayList<Integer>> getComponents() {
        generateComponentsIfAbsent();
        return this.components;
    }

    protected void setComponents(HashMap<Integer, ArrayList<Integer>> components) {
        this.components = components;
    }

    protected HashMap<Integer, ArrayList<Integer>> getNullableComponents() {
        return this.components;
    }

    public ArrayList<Integer> getComponent(int i) {
        return getComponents().get(getRoot(i));
    }

    public Set<Integer> getRoots() {
        return getComponents().keySet();
    }

    public Collection<ArrayList<Integer>> getComponentSet() {
        return getComponents().values();
    }

    public boolean isConnected() {
        return getComponents().size() == 1;
    }

    public boolean isConnected(int u, int v) {
        return getRoot(u) == getRoot(v);
    }

    public boolean isTree() {
        return m == n - 1 && isConnected();
    }

    abstract public boolean isBipartite();

    abstract protected void generateComponentsIfAbsent();
}


class UnionFindTree {
    private int[] parent;
    private int[] rank;
    private int[] size;
    private int groups;

    public UnionFindTree(int n) {
        this.parent = new int[n];
        this.rank = new int[n];
        this.size = new int[n];
        this.groups = n;
        Arrays.fill(this.size, 1);
        for (int i = 0; i < n; i++) {
            parent[i] = i;
        }
    }

    public int root(int x) {
        if (parent[x] == x) {
            return x;
        }
        return parent[x] = root(parent[x]);
    }

    public boolean isSame(int x, int y) {
        return root(x) == root(y);
    }

    public void unite(int x, int y) {
        int xr = root(x);
        int yr = root(y);
        if (xr == yr) {
            return;
        }
        if (rank[xr] < rank[yr]) {
            parent[xr] = yr;
            size[yr] += size[xr];
        } else {
            parent[yr] = xr;
            size[xr] += size[yr];
            if (rank[xr] == rank[yr]) {
                rank[xr]++;
            }
        }
        groups--;
    }

    public int size(int x) {
        return size[root(x)];
    }

    public boolean isRoot(int x) {
        return parent[x] == x;
    }

    public int groups() {
        return groups;
    }
}

class TopologicalSort {
    public static int[] topologicalSort(Digraph g) {
        int n = g.getV();
        int[] ret = new int[n];
        int[] in = new int[n];
        for (int i = 0; i < n; i++) {
            in[i] = g.in(i);
        }
        IntDeque zeros = new IntDeque();
        for (int i = 0; i < n; i++) {
            if (g.in(i) == 0) {
                zeros.addLast(i);
            }
        }
        int index = 0;
        while (zeros.size() > 0) {
            int v = zeros.pollFirst();
            for (Edge e : g.getEdges(v)) {
                if (--in[e.to] == 0) {
                    zeros.addLast(e.to);
                }
            }
            ret[index++] = v;
        }
        if (index < n) {
            return null;
        }
        return ret;
    }

    // for example (n=9)
    // [1, 8] -> [2, 5, 6] -> [3, 7] -> [4] -> [9]
    // => length:5
    public static int shortest(Digraph g) {
        int n = g.getV();
        int ret = 0;
        int[] in = new int[n];
        for (int i = 0; i < n; i++) {
            in[i] = g.in(i);
        }
        IntDeque zeros = new IntDeque();
        for (int i = 0; i < n; i++) {
            if (in[i] == 0) {
                zeros.addLast(i);
            }
        }
        int num = zeros.size();
        while (zeros.size() > 0) {
            int size = zeros.size();
            while (size > 0) {
                int v = zeros.pollFirst();
                size--;
                for (Edge e : g.getEdges(v)) {
                    if (--in[e.to] == 0) {
                        zeros.addLast(e.to);
                        num++;
                    }
                }
            }
            ret++;
        }
        if (num < n) {
            return -1;
        }
        return ret;
    }
}


class In {
    public static final FastScanner fsc = new FastScanner();

    public static int ni() {
        return fsc.nextInt();
    }

    public static int[] ni(int n) {
        int[] a = new int[n];
        Arrays.setAll(a, i -> fsc.nextInt());
        return a;
    }

    public static int[][] ni(int n, int m) {
        int[][] a = new int[n][m];
        Arrays.setAll(a, i -> ni(m));
        return a;
    }

    public static long nl() {
        return fsc.nextLong();
    }

    public static long[] nl(int n) {
        long[] a = new long[n];
        Arrays.setAll(a, i -> fsc.nextLong());
        return a;
    }

    public static long[][] nl(int n, int m) {
        long[][] a = new long[n][m];
        Arrays.setAll(a, i -> nl(m));
        return a;
    }

    public static char[] nc() {
        return fsc.next().toCharArray();
    }

    public static char[][] nc(int n, int m) {
        char[][] c = new char[n][m];
        Arrays.setAll(c, i -> nc());
        return c;
    }

    public static double nd() {
        return fsc.nextDouble();
    }

    public static double[] nd(int n) {
        double[] a = new double[n];
        Arrays.setAll(a, i -> nd());
        return a;
    }

    public static double[][] nd(int n, int m) {
        double[][] a = new double[n][m];
        Arrays.setAll(a, i -> nd(m));
        return a;
    }

    public static String ns() {
        return fsc.next();
    }

    public static String[] ns(int n) {
        String[] s = new String[n];
        Arrays.setAll(s, i -> ns());
        return s;
    }

    public static boolean[][] inputGrid(int h, int w, char truechar) {
        boolean[][] grid = new boolean[h][w];
        for (int i = 0; i < h; i++) {
            char[] s = fsc.next().toCharArray();
            for (int j = 0; j < w; j++) {
                grid[i][j] = s[j] == truechar;
            }
        }
        return grid;
    }
}