using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using Library;

namespace Program
{
    public static class ProblemA
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static int A = 5;
        static int N = 5;
        static int M = 5;
        static int TAYOU = 4;
        static public long Func1((int x, int y)[] ab, List<(int from, int to, int keiyu)> pathList)
        {
            var startTime = DateTime.Now;

            Func<int, int, long> calcDist = (f, t) =>
            {
                var fromX = 0;
                var fromY = 0;
                var toX = 0;
                var toY = 0;
                var fromCoef = 1L;
                var toCoef = 1L;
                fromX = ab[f - 1].x;
                fromY = ab[f - 1].y;
                if (f <= N)
                {
                    fromCoef = A;
                }
                toX = ab[t - 1].x;
                toY = ab[t - 1].y;
                if (t <= N)
                {
                    toCoef = A;
                }
                return fromCoef * toCoef * ((fromX - toX) * (fromX - toX) + (fromY - toY) * (fromY - toY));
            };

            var score = 0L;
            {
                var done = new bool[N + 1];
                var from = 1;
                done[from] = true;
                for (var i = 1; i < N; ++i)
                {
                    var kouho = new LIB_PriorityQueue();
                    for (var j = 1; j <= N; ++j)
                    {
                        if (done[j]) continue;
                        kouho.Push(-calcDist(from, j), j);
                        if (kouho.Count > TAYOU) kouho.Pop();
                    }
                    var kouhoList = new List<int>();
                    while (kouho.Count > 0) kouhoList.Add(kouho.Pop().Value);
                    var minNum = kouhoList[(int)(xorshift % kouhoList.Count)];
                    score += calcDist(from, minNum);
                    pathList.Add((from, minNum, -1));
                    from = minNum;
                    done[from] = true;
                }
                score += calcDist(from, 1);
                pathList.Add((from, 1, -1));
            }

            var hasEdge = ab.Select(_ =>
            {
                var ret = new int[2];
                ret[0] = ret[1] = -1;
                return ret;
            }).ToArray();
            for (var i = 0; i < pathList.Count; ++i)
            {
                if (hasEdge[pathList[i].from - 1][0] == -1) hasEdge[pathList[i].from - 1][0] = i;
                else hasEdge[pathList[i].from - 1][1] = i;
                if (hasEdge[pathList[i].to - 1][0] == -1) hasEdge[pathList[i].to - 1][0] = i;
                else hasEdge[pathList[i].to - 1][1] = i;
            }

            while ((DateTime.Now - startTime).TotalMilliseconds < 170)
            {
                var p1 = (int)(xorshift % pathList.Count);
                var p2 = (int)(xorshift % (pathList.Count - 1));
                if (p2 >= p1) ++p2;
                if (pathList[p1].from == pathList[p2].from || pathList[p1].from == pathList[p2].to ||
                    pathList[p1].to == pathList[p2].from || pathList[p1].to == pathList[p2].to)
                {
                    continue;
                }
                var goal = 0;
                {
                    var from = pathList[p1].from;
                    goal = pathList[p1].to;
                    while (true)
                    {
                        if (goal == pathList[p2].from || goal == pathList[p2].to) break;
                        var to = 0;
                        var pos1 = pathList[hasEdge[goal - 1][0]];
                        var pos2 = pathList[hasEdge[goal - 1][1]];
                        if (pos1.from == from || pos1.to == from)
                        {
                            if (pos2.from == goal) to = pos2.to;
                            else to = pos2.from;
                        }
                        else
                        {
                            if (pos1.from == goal) to = pos1.to;
                            else to = pos1.from;
                        }
                        from = goal;
                        goal = to;
                    }
                }
                var newScore = score;
                newScore -= calcDist(pathList[p1].from, pathList[p1].to);
                newScore -= calcDist(pathList[p2].from, pathList[p2].to);
                if (goal == pathList[p2].from)
                {
                    newScore += calcDist(pathList[p1].from, pathList[p2].from);
                    newScore += calcDist(pathList[p2].to, pathList[p1].to);
                    if (newScore < score)
                    {
                        score = newScore;
                        if (hasEdge[pathList[p1].to - 1][0] == p1)
                        {
                            hasEdge[pathList[p1].to - 1][0] = p2;
                        }
                        else
                        {
                            hasEdge[pathList[p1].to - 1][1] = p2;
                        }
                        if (hasEdge[pathList[p2].from - 1][0] == p2)
                        {
                            hasEdge[pathList[p2].from - 1][0] = p1;
                        }
                        else
                        {
                            hasEdge[pathList[p2].from - 1][1] = p1;
                        }
                        var newp1 = (pathList[p1].from, pathList[p2].from, -1);
                        var newp2 = (pathList[p2].to, pathList[p1].to, -1);
                        pathList[p1] = newp1;
                        pathList[p2] = newp2;
                    }
                }
                else
                {
                    newScore += calcDist(pathList[p1].from, pathList[p2].to);
                    newScore += calcDist(pathList[p2].from, pathList[p1].to);
                    if (newScore < score)
                    {
                        score = newScore;
                        if (hasEdge[pathList[p1].to - 1][0] == p1)
                        {
                            hasEdge[pathList[p1].to - 1][0] = p2;
                        }
                        else
                        {
                            hasEdge[pathList[p1].to - 1][1] = p2;
                        }
                        if (hasEdge[pathList[p2].to - 1][0] == p2)
                        {
                            hasEdge[pathList[p2].to - 1][0] = p1;
                        }
                        else
                        {
                            hasEdge[pathList[p2].to - 1][1] = p1;
                        }
                        var newp1 = (pathList[p1].from, pathList[p2].to, -1);
                        var newp2 = (pathList[p2].from, pathList[p1].to, -1);
                        pathList[p1] = newp1;
                        pathList[p2] = newp2;
                    }
                }
            }

            var addedNum = 0;

            {
                var maxKaizen = 0L;
                var maxNum = 0;
                for (var i = 0; i < pathList.Count; ++i)
                {
                    var thisKaizen = calcDist(pathList[i].from, pathList[i].to) - calcDist(pathList[i].from, pathList[i].to) / 25;
                    ab[N] = ab[pathList[i].from - 1];
                    ab[N + 1] = ab[pathList[i].to - 1];
                    for (var j = 0; j < pathList.Count; ++j)
                    {
                        if (i == j) continue;
                        var item = pathList[j];

                        var kijunDist = calcDist(item.from, item.to);
                        var newDist1 = calcDist(item.from, N + 1) + calcDist(N + 1, item.to);
                        var newDist2 = calcDist(item.from, N + 2) + calcDist(N + 2, item.to);

                        thisKaizen += Max(0, kijunDist - Min(newDist1, newDist2));
                    }
                    if (maxKaizen.Chmax(thisKaizen)) maxNum = i;
                }
                ab[N] = ab[pathList[maxNum].from - 1];
                ab[N + 1] = ab[pathList[maxNum].to - 1];
                score -= calcDist(pathList[maxNum].from, pathList[maxNum].to);
                score += calcDist(N + 1, N + 2);
                var newForFrom = pathList.Count;
                pathList.Add((pathList[maxNum].from, N + 1, -1));
                var newForTo = pathList.Count;
                pathList.Add((pathList[maxNum].to, N + 2, -1));
                if (hasEdge[pathList[maxNum].from - 1][0] == maxNum) hasEdge[pathList[maxNum].from - 1][0] = newForFrom;
                else hasEdge[pathList[maxNum].from - 1][1] = newForFrom;
                if (hasEdge[pathList[maxNum].to - 1][0] == maxNum) hasEdge[pathList[maxNum].to - 1][0] = newForTo;
                else hasEdge[pathList[maxNum].to - 1][1] = newForTo;
                hasEdge[N + 0][0] = newForFrom;
                hasEdge[N + 1][0] = newForTo;
                hasEdge[N + 0][1] = maxNum;
                hasEdge[N + 1][1] = maxNum;
                pathList[maxNum] = (N + 1, N + 2, -1);

                addedNum += 2;
            }

            {
                var kouho = new List<(long kaizen, int x, int y)>();
                for (var i = 0; i <= 1000; i += 20)
                {
                    for (var j = 0; j <= 1000; j += 20)
                    {
                        ab[N + addedNum] = (i, j);
                        var sum = 0L;
                        foreach (var item in pathList)
                        {
                            var kijunDist = calcDist(item.from, item.to);
                            var newDist = calcDist(item.from, N + addedNum + 1) + calcDist(N + addedNum + 1, item.to);
                            sum += Max(0, kijunDist - newDist);
                        }
                        if (sum > 0) kouho.Add((sum, i, j));
                    }
                }
                {
                    var sum = 0L;
                    foreach (var item in pathList)
                    {
                        var kijunDist = calcDist(item.from, item.to);
                        var newDist = calcDist(item.from, N + 1) + calcDist(N + 1, item.to);
                        sum += Max(0, kijunDist - newDist);
                    }
                    if (sum > 0) kouho.Add((sum, -1, -1));
                }
                {
                    var sum = 0L;
                    foreach (var item in pathList)
                    {
                        var kijunDist = calcDist(item.from, item.to);
                        var newDist = calcDist(item.from, N + 2) + calcDist(N + 2, item.to);
                        sum += Max(0, kijunDist - newDist);
                    }
                    if (sum > 0) kouho.Add((sum, -2, -2));
                }
                kouho = kouho.OrderByDescending(e => e.kaizen).ToList();
                foreach (var item2 in kouho)
                {
                    var staNum = addedNum + N + 1;
                    if (item2.x < 0)
                    {
                        staNum = -item2.x + N;
                    }
                    else
                    {
                        staNum = addedNum + N + 1;
                        ab[staNum - 1] = (item2.x, item2.y);
                    }
                    var sum = 0L;
                    foreach (var item in pathList)
                    {
                        if (item.keiyu != -1) continue;
                        var kijunDist = calcDist(item.from, item.to);
                        var newDist = calcDist(item.from, staNum) + calcDist(staNum, item.to);
                        sum += Max(0, kijunDist - newDist);
                    }
                    if (sum >= item2.kaizen * 9 / 10 && sum > 0)
                    {
                        for (var i = 0; i < pathList.Count; ++i)
                        {
                            var item = pathList[i];
                            if (item.keiyu != -1) continue;
                            var kijunDist = calcDist(item.from, item.to);
                            var newDist = calcDist(item.from, staNum) + calcDist(staNum, item.to);
                            var thisKaizen = kijunDist - newDist;
                            if (thisKaizen > 0)
                            {
                                score -= thisKaizen;
                                pathList[i] = (item.from, item.to, staNum);
                            }
                        }
                        if (item2.x >= 0) ++addedNum;
                    }
                    if (addedNum == M) break;
                }
            }
            return score;
        }
        static public long Func2((int x, int y)[] ab, List<(int from, int to, int keiyu)> pathList)
        {
            var startTime = DateTime.Now;

            Func<int, int, long> calcDist = (f, t) =>
            {
                var fromX = 0;
                var fromY = 0;
                var toX = 0;
                var toY = 0;
                var fromCoef = 1L;
                var toCoef = 1L;
                fromX = ab[f - 1].x;
                fromY = ab[f - 1].y;
                if (f <= N)
                {
                    fromCoef = A;
                }
                toX = ab[t - 1].x;
                toY = ab[t - 1].y;
                if (t <= N)
                {
                    toCoef = A;
                }
                return fromCoef * toCoef * ((fromX - toX) * (fromX - toX) + (fromY - toY) * (fromY - toY));
            };

            var score = 0L;
            {
                var done = new bool[N + 1];
                var from = 1;
                done[from] = true;
                for (var i = 1; i < N; ++i)
                {
                    var kouho = new LIB_PriorityQueue();
                    for (var j = 1; j <= N; ++j)
                    {
                        if (done[j]) continue;
                        kouho.Push(-calcDist(from, j), j);
                        if (kouho.Count > TAYOU) kouho.Pop();
                    }
                    var kouhoList = new List<int>();
                    while (kouho.Count > 0) kouhoList.Add(kouho.Pop().Value);
                    var minNum = kouhoList[(int)(xorshift % kouhoList.Count)];
                    score += calcDist(from, minNum);
                    pathList.Add((from, minNum, -1));
                    from = minNum;
                    done[from] = true;
                }
                score += calcDist(from, 1);
                pathList.Add((from, 1, -1));
            }

            var hasEdge = ab.Select(_ =>
            {
                var ret = new int[2];
                ret[0] = ret[1] = -1;
                return ret;
            }).ToArray();
            for (var i = 0; i < pathList.Count; ++i)
            {
                if (hasEdge[pathList[i].from - 1][0] == -1) hasEdge[pathList[i].from - 1][0] = i;
                else hasEdge[pathList[i].from - 1][1] = i;
                if (hasEdge[pathList[i].to - 1][0] == -1) hasEdge[pathList[i].to - 1][0] = i;
                else hasEdge[pathList[i].to - 1][1] = i;
            }

            while ((DateTime.Now - startTime).TotalMilliseconds < 170)
            {
                var p1 = (int)(xorshift % pathList.Count);
                var p2 = (int)(xorshift % (pathList.Count - 1));
                if (p2 >= p1) ++p2;
                if (pathList[p1].from == pathList[p2].from || pathList[p1].from == pathList[p2].to ||
                    pathList[p1].to == pathList[p2].from || pathList[p1].to == pathList[p2].to)
                {
                    continue;
                }
                var goal = 0;
                {
                    var from = pathList[p1].from;
                    goal = pathList[p1].to;
                    while (true)
                    {
                        if (goal == pathList[p2].from || goal == pathList[p2].to) break;
                        var to = 0;
                        var pos1 = pathList[hasEdge[goal - 1][0]];
                        var pos2 = pathList[hasEdge[goal - 1][1]];
                        if (pos1.from == from || pos1.to == from)
                        {
                            if (pos2.from == goal) to = pos2.to;
                            else to = pos2.from;
                        }
                        else
                        {
                            if (pos1.from == goal) to = pos1.to;
                            else to = pos1.from;
                        }
                        from = goal;
                        goal = to;
                    }
                }
                var newScore = score;
                newScore -= calcDist(pathList[p1].from, pathList[p1].to);
                newScore -= calcDist(pathList[p2].from, pathList[p2].to);
                if (goal == pathList[p2].from)
                {
                    newScore += calcDist(pathList[p1].from, pathList[p2].from);
                    newScore += calcDist(pathList[p2].to, pathList[p1].to);
                    if (newScore < score)
                    {
                        score = newScore;
                        if (hasEdge[pathList[p1].to - 1][0] == p1)
                        {
                            hasEdge[pathList[p1].to - 1][0] = p2;
                        }
                        else
                        {
                            hasEdge[pathList[p1].to - 1][1] = p2;
                        }
                        if (hasEdge[pathList[p2].from - 1][0] == p2)
                        {
                            hasEdge[pathList[p2].from - 1][0] = p1;
                        }
                        else
                        {
                            hasEdge[pathList[p2].from - 1][1] = p1;
                        }
                        var newp1 = (pathList[p1].from, pathList[p2].from, -1);
                        var newp2 = (pathList[p2].to, pathList[p1].to, -1);
                        pathList[p1] = newp1;
                        pathList[p2] = newp2;
                    }
                }
                else
                {
                    newScore += calcDist(pathList[p1].from, pathList[p2].to);
                    newScore += calcDist(pathList[p2].from, pathList[p1].to);
                    if (newScore < score)
                    {
                        score = newScore;
                        if (hasEdge[pathList[p1].to - 1][0] == p1)
                        {
                            hasEdge[pathList[p1].to - 1][0] = p2;
                        }
                        else
                        {
                            hasEdge[pathList[p1].to - 1][1] = p2;
                        }
                        if (hasEdge[pathList[p2].to - 1][0] == p2)
                        {
                            hasEdge[pathList[p2].to - 1][0] = p1;
                        }
                        else
                        {
                            hasEdge[pathList[p2].to - 1][1] = p1;
                        }
                        var newp1 = (pathList[p1].from, pathList[p2].to, -1);
                        var newp2 = (pathList[p2].from, pathList[p1].to, -1);
                        pathList[p1] = newp1;
                        pathList[p2] = newp2;
                    }
                }
            }

            var addedNum = 0;

            {
                var kouho = new List<(long kaizen, int x, int y)>();
                for (var i = 0; i <= 1000; i += 20)
                {
                    for (var j = 0; j <= 1000; j += 20)
                    {
                        ab[N + addedNum] = (i, j);
                        var sum = 0L;
                        foreach (var item in pathList)
                        {
                            var kijunDist = calcDist(item.from, item.to);
                            var newDist = calcDist(item.from, N + addedNum + 1) + calcDist(N + addedNum + 1, item.to);
                            sum += Max(0, kijunDist - newDist);
                        }
                        if (sum > 0) kouho.Add((sum, i, j));
                    }
                }
                kouho = kouho.OrderByDescending(e => e.kaizen).ToList();
                foreach (var item2 in kouho)
                {
                    var staNum = addedNum + N + 1;
                    ab[staNum - 1] = (item2.x, item2.y);
                    var sum = 0L;
                    foreach (var item in pathList)
                    {
                        if (item.keiyu != -1) continue;
                        var kijunDist = calcDist(item.from, item.to);
                        var newDist = calcDist(item.from, staNum) + calcDist(staNum, item.to);
                        sum += Max(0, kijunDist - newDist);
                    }
                    if (sum >= item2.kaizen * 9 / 10 && sum > 0)
                    {
                        for (var i = 0; i < pathList.Count; ++i)
                        {
                            var item = pathList[i];
                            if (item.keiyu != -1) continue;
                            var kijunDist = calcDist(item.from, item.to);
                            var newDist = calcDist(item.from, staNum) + calcDist(staNum, item.to);
                            var thisKaizen = kijunDist - newDist;
                            if (thisKaizen > 0)
                            {
                                score -= thisKaizen;
                                pathList[i] = (item.from, item.to, staNum);
                            }
                        }
                        ++addedNum;
                    }
                    if (addedNum == M) break;
                }
            }
            return score;
        }
        static public void Solve()
        {
            N = (int)NN;
            M = (int)NN;
            (int x, int y)[] ab = Repeat(0, N).Select(_ => ((int)NN, (int)NN)).Concat(Repeat(0, M).Select(_ => (0, 0))).ToArray();
            var pathList = new List<(int from, int to, int keiyu)>();
            var ab1 = ab.ToArray();
            var pathList1 = new List<(int from, int to, int keiyu)>();
            var ab2 = ab.ToArray();
            var pathList2 = new List<(int from, int to, int keiyu)>();
            var ab3 = ab.ToArray();
            var pathList3 = new List<(int from, int to, int keiyu)>();
            var score1 = Func1(ab1, pathList1);
            var score2 = Func2(ab2, pathList2);
            var score3 = Func2(ab3, pathList3);
            var max = Min(score1, Min(score2, score3));
            if (max == score1)
            {
                ab = ab1;
                pathList = pathList1;
            }
            else if (max == score2)
            {
                ab = ab2;
                pathList = pathList2;
            }
            else
            {
                ab = ab3;
                pathList = pathList3;
            }
            {
                var hasEdge = ab.Select(_ =>
                {
                    var ret = new int[2];
                    ret[0] = ret[1] = -1;
                    return ret;
                }).ToArray();
                for (var i = 0; i < pathList.Count; ++i)
                {
                    if (hasEdge[pathList[i].from - 1][0] == -1) hasEdge[pathList[i].from - 1][0] = i;
                    else hasEdge[pathList[i].from - 1][1] = i;
                    if (hasEdge[pathList[i].to - 1][0] == -1) hasEdge[pathList[i].to - 1][0] = i;
                    else hasEdge[pathList[i].to - 1][1] = i;
                }

                for (var i = 0; i < M; ++i)
                {
                    Console.WriteLine($"{ab[i + N].x} {ab[i + N].y}");
                }
                var ansList = new List<string>();
                var from = -1;
                var now = 1;
                ansList.Add($"1 1");
                for (var i = 0; i < pathList.Count; ++i)
                {
                    var to = 0;
                    var p1 = pathList[hasEdge[now - 1][0]];
                    var p2 = pathList[hasEdge[now - 1][1]];
                    var keiyu = -1;
                    if (p1.from == from || p1.to == from)
                    {
                        if (p2.from == now) to = p2.to;
                        else to = p2.from;
                        keiyu = p2.keiyu;
                    }
                    else
                    {
                        if (p1.from == now) to = p1.to;
                        else to = p1.from;
                        keiyu = p1.keiyu;
                    }
                    from = now;
                    now = to;
                    if (keiyu != -1) ansList.Add($"2 {keiyu - N}");
                    if (now > N) ansList.Add($"2 {now - N}");
                    else ansList.Add($"1 {now}");
                }
                Console.WriteLine(ansList.Count);
                foreach (var item in ansList)
                {
                    Console.WriteLine(item);
                }
            }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
        static bool Chmax<T>(this ref T lhs, T rhs) where T : struct, IComparable<T> { if (lhs.CompareTo(rhs) < 0) { lhs = rhs; return true; } return false; }
        static bool Chmin<T>(this ref T lhs, T rhs) where T : struct, IComparable<T> { if (lhs.CompareTo(rhs) > 0) { lhs = rhs; return true; } return false; }
        static void Fill<T>(this T[] array, T value) => array.AsSpan().Fill(value);
        static void Fill<T>(this T[,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0], array.Length).Fill(value);
        static void Fill<T>(this T[,,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0, 0], array.Length).Fill(value);
        static void Fill<T>(this T[,,,] array, T value) => MemoryMarshal.CreateSpan(ref array[0, 0, 0, 0], array.Length).Fill(value);
    }
}
namespace Library {
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[2048];
        int len, ptr;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 2048)) <= 0)
                {
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = (ret << 3) + (ret << 1) + b - '0';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
    class LIB_PriorityQueue
    {
        long[] heap;
        int[] dat;
        public (long Key, int Value) Peek => (heap[0], dat[0]);
        public long Count
        {
            get;
            private set;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue()
        {
            heap = new long[8];
            dat = new int[8];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(long key) => Push(key, 0);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(long key, int val)
        {
            if (Count == heap.Length) Expand();
            var i = (int)Count++;
            ref long heapref = ref heap[0];
            ref int datref = ref dat[0];
            Unsafe.Add<long>(ref heapref, i) = key;
            Unsafe.Add<int>(ref datref, i) = val;
            while (i > 0)
            {
                var ni = (i - 1) / 2;
                var heapni = Unsafe.Add<long>(ref heapref, ni);
                if (key >= heapni) break;
                Unsafe.Add<long>(ref heapref, i) = heapni;
                Unsafe.Add<int>(ref datref, i) = Unsafe.Add<int>(ref datref, ni);
                i = ni;
            }
            Unsafe.Add<long>(ref heapref, i) = key;
            Unsafe.Add<int>(ref datref, i) = val;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (long Key, int Value) Pop()
        {
            ref long heapref = ref heap[0];
            ref int datref = ref dat[0];
            var ret = (heapref, datref);
            var cnt = (int)(--Count);
            var key = Unsafe.Add<long>(ref heapref, cnt);
            var val = Unsafe.Add<int>(ref datref, cnt);
            if (cnt == 0) return ret;
            var i = 0; while ((i << 1) + 1 < cnt)
            {
                var i1 = (i << 1) + 1;
                var i2 = (i << 1) + 2;
                if (i2 < cnt && Unsafe.Add<long>(ref heapref, i1) > Unsafe.Add<long>(ref heapref, i2)) i1 = i2;
                var heapi1 = Unsafe.Add<long>(ref heapref, i1);
                if (key <= heapi1) break;
                Unsafe.Add<long>(ref heapref, i) = heapi1;
                Unsafe.Add<int>(ref datref, i) = Unsafe.Add<int>(ref datref, i1);
                i = i1;
            }
            Unsafe.Add<long>(ref heapref, i) = key;
            Unsafe.Add<int>(ref datref, i) = val;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Expand()
        {
            var len = heap.Length;
            var tmp = new long[len << 1];
            var tmp2 = new int[len << 1];
            Unsafe.CopyBlock(ref Unsafe.As<long, byte>(ref tmp[0]), ref Unsafe.As<long, byte>(ref heap[0]), (uint)(8 * len));
            Unsafe.CopyBlock(ref Unsafe.As<int, byte>(ref tmp2[0]), ref Unsafe.As<int, byte>(ref dat[0]), (uint)(4 * len));
            heap = tmp;
            dat = tmp2;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (long Key, int Value)[] NoSortList()
        {
            var ret = new (long Key, int Value)[Count];
            ref long heapref = ref heap[0];
            ref int datref = ref dat[0];
            for (var i = 0; i < Count; ++i)
            {
                ret[i] = (Unsafe.Add<long>(ref heapref, i), Unsafe.Add<int>(ref datref, i));
            }
            return ret;
        }
    }
    class LIB_PriorityQueue<T>
    {
        T[] heap;
        Comparison<T> comp;
        public T Peek => heap[0];
        public long Count
        {
            get;
            private set;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, Comparison<T> cmp, bool asc = true)
        {
            heap = new T[cap];
            comp = asc ? cmp : (x, y) => cmp(y, x);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(Comparison<T> cmp, bool asc = true)
        {
            heap = new T[8];
            comp = asc ? cmp : (x, y) => cmp(y, x);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, bool asc = true) : this(cap, Comparer<T>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(bool asc = true) : this(Comparer<T>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(T val)
        {
            if (Count == heap.Length) Expand();
            var i = Count++;
            heap[i] = val;
            while (i > 0)
            {
                var ni = (i - 1) / 2;
                if (comp(val, heap[ni]) >= 0) break;
                heap[i] = heap[ni];
                i = ni;
            }
            heap[i] = val;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Pop()
        {
            var ret = heap[0];
            var val = heap[--Count];
            if (Count == 0) return ret;
            var i = 0; while ((i << 1) + 1 < Count)
            {
                var i1 = (i << 1) + 1;
                var i2 = (i << 1) + 2;
                if (i2 < Count && comp(heap[i1], heap[i2]) > 0) i1 = i2;
                if (comp(val, heap[i1]) <= 0) break;
                heap[i] = heap[i1]; i = i1;
            }
            heap[i] = val;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Expand()
        {
            var tmp = new T[Count << 1];
            for (var i = 0; i < heap.Length; ++i) tmp[i] = heap[i];
            heap = tmp;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T[] NoSortList()
        {
            var ret = new List<T>();
            for (var i = 0; i < Count; ++i)
            {
                ret.Add(heap[i]);
            }
            return ret.ToArray();
        }
    }
    class LIB_PriorityQueue<TK, TV>
    {
        LIB_PriorityQueue<KeyValuePair<TK, TV>> q;
        public KeyValuePair<TK, TV> Peek => q.Peek;
        public long Count => q.Count;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, Comparison<TK> cmp, bool asc = true)
        {
            q = new LIB_PriorityQueue<KeyValuePair<TK, TV>>(cap, (x, y) => cmp(x.Key, y.Key), asc);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(Comparison<TK> cmp, bool asc = true)
        {
            q = new LIB_PriorityQueue<KeyValuePair<TK, TV>>((x, y) => cmp(x.Key, y.Key), asc);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(long cap, bool asc = true) : this(cap, Comparer<TK>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_PriorityQueue(bool asc = true) : this(Comparer<TK>.Default.Compare, asc) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(TK k, TV v) => q.Push(new KeyValuePair<TK, TV>(k, v));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public KeyValuePair<TK, TV> Pop() => q.Pop();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public KeyValuePair<TK, TV>[] NoSortList() => q.NoSortList();
    }
}