Problem:
Score : 700 points

Problem Statement

You are given an integer L. Construct a directed graph that satisfies the conditions below. The graph may contain multiple edges between the same pair of vertices. It can be proved that such a graph always exists.

The number of vertices, N, is at most 20. The vertices are given ID numbers from 1 to N.

The number of edges, M, is at most 60. Each edge has an integer length between 0 and 10^6 (inclusive).

Every edge is directed from the vertex with the smaller ID to the vertex with the larger ID. That is, 1,2,...,N is one possible topological order of the vertices.

There are exactly L different paths from Vertex 1 to Vertex N. The lengths of these paths are all different, and they are integers between 0 and L-1.

Here, the length of a path is the sum of the lengths of the edges contained in that path, and two paths are considered different when the sets of the edges contained in those paths are different.

Constraints

2 \leq L \leq 10^6

L is an integer.

Input

Input is given from Standard Input in the following format:

L

Output

In the first line, print N and M, the number of the vertices and edges in your graph.
In the i-th of the following M lines, print three integers u_i,v_i and w_i, representing the starting vertex, the ending vertex and the length of the i-th edge.
If there are multiple solutions, any of them will be accepted.

Sample Input 1

4

Sample Output 1

8 10
1 2 0
2 3 0
3 4 0
1 5 0
2 6 0
3 7 0
4 8 0
5 6 1
6 7 1
7 8 1

In the graph represented by the sample output, there are four paths from Vertex 1 to N=8:

1 → 2 → 3 → 4 → 8 with length 0

1 → 2 → 3 → 7 → 8 with length 1

1 → 2 → 6 → 7 → 8 with length 2

1 → 5 → 6 → 7 → 8 with length 3

There are other possible solutions.

Sample Input 2

5

Sample Output 2

5 7
1 2 0
2 3 1
3 4 0
4 5 0
2 4 0
1 3 3
3 5 1

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

import java.io.IOException;
import java.io.InputStream;
import java.util.*;
import java.util.function.IntFunction;
import java.util.function.Supplier;
import java.util.stream.IntStream;
import java.util.stream.Stream;


public class Main {
    public static void main(String[] args) {
        Scanner scanner = new Scanner();
        int l=scanner.nextInt();
        int count=0;
        while(1<<count<=l){
            count++;
        }
        final int n=count;
        List<Tuple<Pair,Integer>> result=new ArrayList<>();
        for(int i=0;i<n-1;i++){
            result.add(new Tuple<>(new Pair(i+1,i+2),0));
            result.add(new Tuple<>(new Pair(i+1,i+2),1<<i));
        }
        int max=(1<<(n-1))-1;
        put(format("max=%d",max));
        for(int i=n-2;i>=0;i--){
            if(max+(1<<i)<=l-1){
                result.add(new Tuple<>(new Pair(i+1,n+1),max+1));
                max+=(1<<i);
            }
        }
        put(format("%d %d",n,result.size()));
        for(Tuple<Pair,Integer> tuple:result){
            put(format("%d %d %d",tuple.getT().x,tuple.getT().y,tuple.getV()));
        }



    }


    public static int max(int a,int b){
        return Math.max(a,b);
    }
    public static long max(long a,long b){
        return Math.max(a,b);
    }
    public static int abs(int a){
        return Math.abs(a);
    }
    public static long abs(long a){
        return Math.abs(a);
    }
    public static void print(Object object){
        System.out.print(object);
    }
    public static void put(Object object) {
        System.out.println(object);
    }
    public static void put(){
        System.out.println();
    }
    public static String format(String string, Object... args) {
        return String.format(string, args);
    }
}

final class Scanner {
    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 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());
    }
}
final class Pair {
    final public int x, y;
    Pair(int x, int y) {
        this.x = x;
        this.y = y;
    }

    @Override
    public int hashCode() {
        return x+y;
    }

    @Override
    public boolean equals(Object obj) {
        boolean result=super.equals(obj);
        if(obj.getClass()!=this.getClass()){
            return false;
        }
        Pair pair=(Pair)obj;
        if(this.x==pair.x&&this.y==pair.y) return true;
        return false;
    }

    @Override
    public String toString() {
        return String.format("(%d,%d)", x, y);
    }
}
final class Tuple<T,V>{
    //immutabl1でないことに注意(T,Vがmutableの場合変更可能)
    final private T t;
    final private V v;
    Tuple(T t,V v){
        this.t=t;
        this.v=v;
    }

    public T getT() {
        return t;
    }

    public V getV() {
        return v;
    }

    @Override
    public int hashCode() {
        return (t.hashCode()+v.hashCode());
    }

    @Override
    @SuppressWarnings("unchecked")
    public boolean equals(Object obj) {
        if(obj.getClass()!=this.getClass()){
            return false;
        }
        Tuple<T,V> tuple=(Tuple<T, V>) obj;
        return tuple.t.equals(this.t)&&tuple.v.equals(this.v);
    }

    @Override
    public String toString() {
        return String.format("<Tuple>=<%s,%s>",t,v);
    }
}
final class LowerBoundComparator<T extends Comparable<? super T>>
        implements Comparator<T>
{
    public int compare(T x, T y)
    {
        return (x.compareTo(y) >= 0) ? 1 : -1;
    }
}

final class UpperBoundComparator<T extends Comparable<? super T>>
        implements Comparator<T>
{
    public int compare(T x, T y)
    {
        return (x.compareTo(y) > 0) ? 1 : -1;
    }
}
final class Util {
    static long gcd(long a,long b){
        if(a%b==0)return b;
        return gcd(b,a%b);
    }
    static long lcm(long a,long b){
        long gcd=gcd(a,b);
        long result=b/gcd;
        return a*result;
    }
    static long kaijoMod(int n,int mod){
        if(n<1) return -1;
        long result=1;
        for(int i=n;i>1;i--){
            result*=i;
            result%=mod;
        }
        return result;
    }
    static <T extends Comparable<T>> Map<T,Integer> count(List<T> list){
        //副作用
        Collections.sort(list);
        Map<T,Integer> result=new HashMap<>();
        int l=0,r=0;
        while(l<list.size()){
            while(r<list.size()-1&&list.get(r).equals(r+1)){
                r++;
            }
            result.put(list.get(r),r-l+1);
            r++;
            l=r;
        }
        return result;

    }
    static Map<Integer,Integer> count(int[] array){
        //副作用
        Arrays.sort(array);
        Map<Integer,Integer> result=new HashMap<>();
        int l=0,r=0;
        while(l<array.length){
            while(r<array.length-1&&array[r]==array[r+1]){
                r++;
            }
            result.put(array[l],r-l+1);
            r++;
            l=r;
        }
        return result;
    }
    static <T> String toStringBWS(Iterable<T> iterable){
        Iterator<T> ite=iterable.iterator();
        return toStringBWS(ite);
    }
    static <T> String toStringBWS(Iterator<T> ite){
        StringBuilder sb=new StringBuilder();
        sb.append(ite.next());
        while(ite.hasNext()){
            sb.append(" ");
            sb.append(ite.next());
        }
        return sb.toString();
    }
    static String toStringBWS(int[] array){
        StringBuilder sb=new StringBuilder();
        for(int i=0;i<array.length-1;i++){
            sb.append(array[i]);
            sb.append(" ");
        }
        sb.append(array[array.length-1]);
        return sb.toString();
    }
    static String toStringBWS(long[] array){
        StringBuilder sb=new StringBuilder();
        for(int i=0;i<array.length-1;i++){
            sb.append(array[i]);
            sb.append(" ");
        }
        sb.append(array[array.length-1]);
        return sb.toString();
    }
}