Problem:
Score: 300 points

Problem Statement

We have a canvas divided into a grid with H rows and W columns. The square at the i-th row from the top and the j-th column from the left is represented as (i, j).

Initially, all the squares are white. square1001 wants to draw a picture with black paint. His specific objective is to make Square (i, j) black when s_{i, j}= #, and to make Square (i, j) white when s_{i, j}= ..

However, since he is not a good painter, he can only choose two squares that are horizontally or vertically adjacent and paint those squares black, for some number of times (possibly zero). He may choose squares that are already painted black, in which case the color of those squares remain black.

Determine if square1001 can achieve his objective.

Constraints

H is an integer between 1 and 50 (inclusive).

W is an integer between 1 and 50 (inclusive).

For every (i, j) (1 \leq i \leq H, 1 \leq j \leq W), s_{i, j} is # or ..

Input

Input is given from Standard Input in the following format:

H W
s_{1, 1} s_{1, 2} s_{1, 3} ... s_{1, W}
s_{2, 1} s_{2, 2} s_{2, 3} ... s_{2, W}
: :
s_{H, 1} s_{H, 2} s_{H, 3} ... s_{H, W}

Output

If square1001 can achieve his objective, print Yes; if he cannot, print No.

Sample Input 1

3 3
.#.
###
.#.

Sample Output 1

Yes

One possible way to achieve the objective is shown in the figure below. Here, the squares being painted are marked by stars.

Sample Input 2

5 5
#.#.#
.#.#.
#.#.#
.#.#.
#.#.#

Sample Output 2

No

square1001 cannot achieve his objective here.

Sample Input 3

11 11
...#####...
.##.....##.
#..##.##..#
#..##.##..#
#.........#
#...###...#
.#########.
.#.#.#.#.#.
##.#.#.#.##
..##.#.##..
.##..#..##.

Sample Output 3

Yes

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Threading.Tasks;
using System.Globalization;
using System.Collections;
namespace CompetitiveProgramming
{
    public class CPair<T, U> : IComparable where T : IComparable<T>
    {
        public readonly T Item1;
        public readonly U Item2;

        public int CompareTo(object obj)
        {
            CPair<T, U> castedObj = (CPair<T, U>)obj;
            return this.Item1.CompareTo(castedObj.Item1);
        }

        public CPair(T t, U u)
        {
            Item1 = t;
            Item2 = u;
        }
    }
    class Pair : IComparable
    {
        public long x;
        public long a;
        public long b;
        public long c;
        public Pair(long X, long Y, long Z, long D)
        {
            x = X; a = Y; b = Z;
            c = D;
        }
        public Pair(long a, long x)
        {
            this.a = a; this.x = x;
        }
        public Pair(long a, long b, long c)
        {
            this.a = a; this.b = b; this.c = c;
        }

        public int CompareTo(object obj)
        {
            Pair kore = (Pair)obj;
            return x.CompareTo(kore.x);
        }
        public static bool operator <=(Pair p1, Pair p2)
        {
            if (p1.x != p2.x) { return p1.x <= p2.x; }
            else { return p1.a <= p2.a; }
        }
        public static bool operator >=(Pair p1, Pair p2)
        {
            if (p1.x != p2.x) { return p1.x >= p2.x; }
            else { return p1.a >= p2.a; }
        }
    }

    class Zettiti : IComparable
    {
        public long num;
        public Zettiti(long num)
        {
            this.num = num;
        }
        public int CompareTo(object obj)
        {
            Zettiti kore = (Zettiti)obj;
            long knum = Math.Abs(kore.num);
            long mynum = Math.Abs(num);
            return mynum.CompareTo(knum);
        }

    }
    class Program
    {
        static void Main(string[] args)
        {
            //    Utils.BinarySearch.Search
            StreamScanner scan = new StreamScanner(System.Console.OpenStandardInput());
            int h = scan.Integer();
            int w = scan.Integer();
            char[,] s = scan.Char(h, w);
            int[] dx = new int[] { -1, 1, 0, 0 };
            int[] dy = new int[] { 0, 0, -1, 1 };
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    bool clear = false;
                    if (s[i, j] == '.')
                        continue;
                    for (int k = 0; k < 4; k++)
                    {
                        int nx = i + dx[k];
                        int ny = j + dy[k];
                        if (0 > nx || h <= nx || 0 > ny || w <= ny)
                            continue;
                        if(s[nx,ny]=='#')
                        {
                            clear = true;
                            break;
                        }
                    }
                    if(!clear)
                    {
                        Console.WriteLine("No");
                        return;
                    }
                }
            }
            Console.WriteLine("Yes");
        }

        static bool isOK(long mid, long[] a, long[] f, long k)
        {
            long count = 0;
            for (long i = 0; i < a.Length; i++)
            {
                long togo = mid / f[i]; ;
                count += Math.Max(0, a[i] - togo);
            }
            return (count <= k);
        }

        static void Output<T>(List<T> list, string kugiri = "\n")
        {
            for (int i = 0; i < list.Count; i++)
            {
                Console.Write(list[i]);
                if (i != list.Count - 1)
                {
                    Console.Write(kugiri);
                }
            }
        }
        static void Output<T>(T[] list, string kugiri = "\n")
        {
            for (int i = 0; i < list.Length; i++)
            {
                Console.Write(list[i]);
                if (i != list.Length - 1)
                {
                    Console.Write(kugiri);
                }
            }
        }
        static void Output<T>(T[,] list, string kugiri1 = " ",string kugiri2 = "\n")
        {
            int len1 = list.GetLength(0);
            int len2 = list.GetLength(1);
            for (int i = 0; i < len1; i++)
            {
                for (int j = 0; j < len2; j++)
                {
                    Console.Write(list[i,j]);
                    if (j != len2 - 1)
                    {
                        Console.Write(kugiri1);
                    }
                }
                if (i != len1 - 1)
                {
                    Console.Write(kugiri2);
                }
               
            }
        }



        static int binary_search<T>(T key, T[] moto, Func<int, T, T[], bool> isok)
        {
            int ng = -1;
            int ok = moto.Length;
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;
                if (isok(mid, key, moto)) { ok = mid; }
                else { ng = mid; }
            }
            return ok;
        }
        static int binary_search(int key, int[] moto)
        {
            int ng = -1;
            int ok = moto.Length;
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;
                if (isOK(mid, key, moto)) { ok = mid; }
                else { ng = mid; }
            }
            return ok;
        }
        static int[] Togo;
        static bool CheckBipartite(int now, int prev, List<int>[] g, int c = 0)//0,-1が想定
        {
            if (Togo == null) Togo = Enumerable.Repeat(-1, g.Length).ToArray();
            foreach (var next in g[now])
            {
                if (next == prev) continue;
                var nextc = Togo[next];
                if (nextc >= 0 && nextc == c) return false;
                if (nextc >= 0) continue;
                Togo[next] = (c + 1) % 2;
                var res = CheckBipartite(next, now, g, (c + 1) % 2);
                if (res == false) return false;
            }
            return true;
        }
        static bool isOK(int index, int key, int[] moto)
        {
            return moto[index] > key;
        }
        public static int bitCount(int k)
        {
            uint i = (uint)k;
            i = i - ((i >> 1) & 0x55555555);
            i = (i & 0x33333333) + ((i >> 2) & 0x33333333);
            i = (i + (i >> 4)) & 0x0f0f0f0f;
            i = i + (i >> 8);
            i = i + (i >> 16);
            return (int)i & 0x3f;
        }
        static int check(long moto, long tai)
        {
            if (tai % moto != 0) { return 0; }
            while (tai > 1)
            {
                if (tai % moto == 0) { tai /= moto; }
                else if (tai > moto) { tai %= moto; }
                else { tai -= moto; }
            }
            if (tai == 1) { return 1; }
            else { return 0; }
        }

        static bool NextPermutation(int[] array)
        {
            var size = array.Length;
            var ok = false;

            //array[i]<array[i+1]を満たす最大のiを求める
            int i = size - 2;
            for (; 0 <= i; i--)
            {
                if (array[i] < array[i + 1])
                {
                    ok = true;
                    break;
                }
            }

            //全ての要素が降順の場合、NextPermutationは存在しない
            if (ok == false) return false;

            //array[i]<array[j]を満たす最大のjを求める
            int j = size - 1;
            for (; ; j--)
            {
                if (array[i] < array[j]) break;
            }

            //iとjの要素をswapする
            var tmp = array[i];
            array[i] = array[j];
            array[j] = tmp;

            //i以降の要素を反転させる
            Array.Reverse(array, i + 1, size - (i + 1));

            return true;
        }
        static List<int[]> GetAllPermutation(int N)
        {
            var res = new List<int[]>();
            var array = Enumerable.Range(0, N).ToArray();
            do
            {
                var copy = new int[N];
                array.CopyTo(copy, 0);
                res.Add(copy);
            } while (NextPermutation(array));

            return res;
        }
        static List<int[]> GetAllPermutation(int[] array)
        {
            var size = array.Length;
            var res = new List<int[]>();
            do
            {
                var copy = new int[size];
                array.CopyTo(copy, 0);
                res.Add(copy);
            } while (NextPermutation(array));

            return res;
        }

        static long Syakutori(int[] a, long k)//しゃくとり法モデル:連続部分列に含まれるすべての要素の値の和がK以上
        {
            long sum = 0;
            int right = 0;
            long ans = 0;
            for (int left = 0; left < a.Length; left++)
            {
                while (right < a.Length && sum < k)
                {
                    sum += a[right]; right++;

                }
                if (k <= sum) { ans += a.Length - right + 1; }
                if (left == right) { right++; }
                else { sum -= a[left]; }
            }
            return ans;
        }
        static int[] zalgo(String s)
        {
            int n = s.Length;
            int[] ret = new int[n];
            ret[0] = n;
            int i = 1, j = 0;
            while (i < n)
            {
                while (i + j < n && s[j] == s[i + j])
                    ++j;
                ret[i] = j;
                if (j == 0)
                {
                    ++i;
                    continue;
                }
                int k = 1;
                while (i + k < n && k + ret[k] < j)
                {
                    ret[i + k] = ret[k];
                    ++k;
                }
                i += k;
                j -= k;
            }
            return ret;
        }
        static int gcd(int a, int b)
        {
            if (a % b == 0) { return b; }
            return gcd(b, a % b);
        }
        static long gcd(long a, long b)
        {
            if (a % b == 0) { return b; }
            return gcd(b, a % b);
        }
        public static List<long> FactorialMemo = new List<long>() { 1 };
        static long Fact(int x)
        {
            for (int i = FactorialMemo.Count; i <= x; i++) FactorialMemo.Add(FactorialMemo.Last() * i);
            return FactorialMemo[x];
        }
        static int[] yousogotoplus(int[] a, int[] b)
        {
            int[] c = new int[a.Length];
            for (int i = 0; i < a.Length; i++)
            {
                c[i] = a[i] + b[i];
            }
            return c;
        }
        static int[] yousogotoplus(int[] a, int[,] b, int l)
        {
            int[] c = new int[a.Length];
            for (int i = 0; i < a.Length; i++)
            {
                c[i] = a[i] + b[l, i];
            }
            return c;
        }

        static int[] yousogotoplus(int[,] a, int l)
        {
            int[] c = new int[a.GetLength(1)];
            for (int i = 0; i < a.GetLength(1); i++)
            {
                c[i] = a[l, i];
            }
            return c;
        }

        public static long LowerBound(long[] a, long key)
        {
            long l = 0;
            long r = a.Length;
            long mid;
            while (l < r)
            {
                mid = ((r - l) / 2) + l;
                if (a[mid] <= key)
                    l = mid + 1;
                else
                    r = mid;
            }
            return l;
        }
        public static long LIS(long[] a)
        {
            var dp = Enumerable.Range(0, a.Length).Select(x => long.MaxValue).ToArray();
            for (int i = 0; i < a.Length; i++)
            {
                dp[LowerBound(dp, a[i])] = a[i];
            }
            return LowerBound(dp, long.MaxValue - 1);
        }

        static long DivideWith(long m, long n, List<long> primes)
        {
            while (m % n == 0)
            {
                primes.Add(n);
                m = m / n;
            }
            return m;
        }
        static List<CPair<long, long>> SoinsuBunkai(long num)
        {
            List<CPair<long, long>> primes = new List<CPair<long, long>>();
            List<long> yakusu = new List<long>();
            List<long> yakusu2 = new List<long>();
            for (int i = 2; i * i <= num; i++)
            {

                if (num % i != 0) { continue; }
                yakusu.Add(i);
                if (i * i != num) { yakusu2.Add(num / i); }
            }
            for (int i = 0; i < yakusu.Count; i++)
            {
                if (num % yakusu[i] != 0) { continue; }
                int j = 0;
                for (j = 0; num % yakusu[i] == 0; j++)
                {
                    num /= yakusu[i];
                }
                primes.Add(new CPair<long, long>(yakusu[i], j));
            }
            for (int i = yakusu2.Count - 1; i > 0; i--)
            {
                if (num % yakusu2[i] != 0) { continue; }
                int j = 0;
                for (j = 0; num % yakusu2[i] == 0; j++)
                {
                    num /= yakusu2[i];
                }
                primes.Add(new CPair<long, long>(yakusu2[i], j));
            }
            return primes;
        }
        static IEnumerable<int> Primes(int maxnum)
        {
            // var sieve = new BitArray(maxnum + 1, true);
            bool[] sieve = Enumerable.Repeat(true, maxnum + 1).ToArray();
            int squareroot = (int)Math.Sqrt(maxnum);
            for (int i = 2; i <= squareroot; i++)
            {
                if (sieve[i] == false)
                    continue;
                for (int n = i * 2; n <= maxnum; n += i)
                    sieve[n] = false;
            }
            for (int i = 2; i <= maxnum; i++)
                if (sieve[i] == true)
                    yield return i;
        }
        /*   static long[] SoinsuBunkai(long num)
           {
               if (num == 1)
                   return new long[] { 1 };
               List<long> primes = new List<long>();
               long n = DivideWith(num, 2, primes);
               n = DivideWith(n, 3, primes);
               n = DivideWith(n, 5, primes);
               long i = 0;
               while (n > 1)
               {
                   int[] ps = { 7, 11, 13, 17, 19, 23, 29, 31 };
                   foreach (var p in ps)
                   {
                       // 30m+2, 30m+3, 30m+4, 30m+5, 30m+6、30m+8... は割る必要はない。
                       n = DivideWith(n, i + p, primes);
                       if (n == 1)
                           break;
                   }
                   i += 30;
               }
               return primes.ToArray();
           }*/
        public static int h;
        public static int w;
        public static int BFS(int height, int width, char[,] map)
        {
            int[] dx = new int[] { -1, 0, 0, 1 };
            int[] dy = new int[] { 0, 1, -1, 0 };
            bool[,] visited = new bool[h, w];
            int[,] len = new int[h, w];
            var queue = new Queue<Tuple<int, int>>();
            queue.Enqueue(new Tuple<int, int>(height, width));
            int max = 0;
            visited[height, width] = true;

            while (queue.Count > 0)
            {
                var now = queue.Dequeue();
                for (int i = 0; i < 4; i++)
                {
                    int ny = now.Item1 + dy[i];
                    int nx = now.Item2 + dx[i];
                    if (ny >= h || ny < 0 || nx >= w || nx < 0)
                        continue;
                    if (visited[ny, nx] || map[ny, nx] == '#')
                        continue;
                    visited[ny, nx] = true;
                    len[ny, nx] = len[now.Item1, now.Item2] + 1;
                    max = Math.Max(max, len[ny, nx]);
                    queue.Enqueue(new Tuple<int, int>(ny, nx));
                }
            }
            return max;
        }
        static public bool Bit(long x, int dig) => ((1L << dig) & x) != 0;
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = Array.ConvertAll(array, x => initialvalue);
        }
        static int[] Manacher(string S)
        {
            int i = 0, j = 0;
            int[] R = new int[S.Length];
            while (i < S.Length)
            {
                while (i - j >= 0 && i + j < S.Length && S[i - j] == S[i + j]) ++j;
                R[i] = j;
                int k = 1;
                while (i - k >= 0 && k + R[i - k] < j) { R[i + k] = R[i - k]; ++k; }
                i += k; j -= k;
            }
            return R;
        }


        /*    static Pair[] a;
    static int SaikiZentansaku(List<int>now,int count,int last,int n,int m)
    {
        List<int> next = new List<int>();
        for (int i = 0; i < now.Count; i++)
        {
            next.Add(now[i]);
        }
        next.Add(last);
        var kore = next.ToArray();
        int ans = 0;
        if(count==n)
        {
            for (int i = 0; i < a.Length; i++)
            {
                int A = a[i].a;
                int b = a[i].b;
                int c = a[i].c;
                int d = a[i].x;
                if (kore[b] - kore[A] == c) { ans += d; }
            }
            return ans;
        }
        for (int i = last; i <= m; i++)
        {
            ans = Math.Max(ans, SaikiZentansaku(next, count + 1, i, n, m));
        }
        return ans;
    }*/
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        private readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private long len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        public char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        public char[] Char(long n)
        {
            char[] kore = new char[n];
            for (long i = 0; i < n; i++)
            {
                kore[i] = Char();
            }
            return kore;
        }
        public char[,] Char(long n, long m)
        {
            char[,] kore = new char[n, m];
            for (long i = 0; i < n; i++)
            {
                for (int j = 0; j < m; j++)
                {
                    kore[i, j] = Char();
                }
            }
            return kore;
        }
        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public 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 * 10 + b - '0';
            }
        }
        public long[] Long(long n,bool minus1=false)
        {
            long[] kore = new long[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Long();
                if (minus1) kore[i]--;
            }
            return kore;
        }
        public Mlong[] MLong(long n)
        {
            Mlong[] kore = new Mlong[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Long();
            }
            return kore;
        }
        public long[,] Long(long h, long w, bool minus1 = false)
        {
            long[,] kore = new long[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = Long();
                    if (minus1) kore[i,j]--;
                }
            }
            return kore;
        }
        public int Integer() { return (int)Long(); }
        public Mint[] MInteger(long n)
        {
            Mint[] kore = new Mint[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Integer();
            }
            return kore;
        }
        public int[] Integer(long n, bool minus1 = false)
        {
            int[] kore = new int[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = (int)Long();
                if (minus1) kore[i]--;
            }
            return kore;
        }
        public int[,] Integer(long h, long w, bool minus1 = false)
        {
            int[,] kore = new int[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = (int)Long();
                    if (minus1) kore[i, j]--;
                }

            }
            return kore;
        }

        public int[] StringToInt(int n)
        {
            char[] s = Scan().ToCharArray();
            int[] kore = new int[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = s[i] - '0';
            }
            return kore;
        }
        public int[,] StringToInt(int h, int w)
        {

            int[,] kore = new int[h, w];
            for (int i = 0; i < h; i++)
            {
                char[] s = Scan().ToCharArray();
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = s[j] - '0';
                }

            }
            return kore;
        }

        public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }

    public class UnionFind
    {
        public int[] data;

        public UnionFind(int size)
        {
            data = new int[size];
            for (int i = 0; i < size; i++) data[i] = -1;
        }

        public bool Unite(int x, int y)
        {
            x = Root(x);
            y = Root(y);

            if (x != y)
            {
                if (data[y] < data[x])
                {
                    var tmp = y;
                    y = x;
                    x = tmp;
                }
                data[x] += data[y];
                data[y] = x;
            }
            return x != y;
        }

        public bool IsSameGroup(int x, int y)
        {
            return Root(x) == Root(y);
        }

        public int Root(int x)
        {
            return data[x] < 0 ? x : data[x] = Root(data[x]);
        }
    }
    public class Mlong : IComparable
    {
        private static long mod = 1000000007;
        public long num;
        public Mlong(long num) { if (num < 0) { num += mod; } this.num = num % mod; }
        public override int GetHashCode()
        {
            return num.GetHashCode();
        }
        public override bool Equals(object obj)
        {
            return num.Equals(obj);
        }
        public static Mlong operator +(Mlong x, Mlong y)
        {
            return new Mlong((x.num + y.num) % mod);
        }
        public static Mlong operator -(Mlong x, Mlong y)
        {
            return new Mlong((x.num - y.num + mod) % mod);
        }
        public static Mlong operator *(Mlong x, Mlong y)
        {
            return new Mlong((long)(((1L * x.num) * y.num) % mod));
        }
        public Mlong hutujyo(Mlong x)
        {
            return new Mlong(num / x.num);
        }
        static long GetInverse(long a)
        {
            long div, p = mod, x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return x2 + mod; div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p;
                if (a == 1) return x1 + mod; div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a;
            }
        }
        public Mlong modjyo(Mlong x)
        {
            return num * GetInverse(x.num) % mod;
        }
        public static implicit operator Mlong(long x)
        {
            return new Mlong(x);
        }
        public static Mlong operator %(Mlong x, Mlong y)
        {
            return new Mlong(x.num % y.num);
        }
        public static explicit operator long(Mlong x)
        {
            return x.num;
        }
        public static bool operator ==(Mlong x, Mlong y)
        {
            return x.num == y.num;
        }
        public static bool operator !=(Mlong x, Mlong y)
        {
            return x.num != y.num;
        }
        public static bool operator <(Mlong x, Mlong y)
        {
            return x.num < y.num;
        }
        public static bool operator >(Mlong x, Mlong y)
        {
            return x.num > y.num;
        }
        public static bool operator <=(Mlong x, Mlong y)
        {
            return x.num <= y.num;
        }
        public static bool operator >=(Mlong x, Mlong y)
        {
            return x.num >= y.num;
        }
        public Mlong Pow(Mlong n)
        {
            Mlong pow = this;
            Mlong res = 1;
            while (n > 0)
            {
                if (n % 2 == 1) { res *= pow; }
                pow *= pow;
                n = n.hutujyo(2);
            }
            /*   if (n == 1) { return this % mod; }
               if (n % 2 == 1) { return (this * Pow(n - 1)) % mod; }
               Mint buf = Pow(n.hutujyo(2));
               return (buf * buf) % mod;*/
            return res;
        }
        public static Mlong Pow(Mlong a, Mlong n)
        {
            return a.Pow(n);
        }
        public static List<Mlong> FactorialMemo = new List<Mlong>() { 1 };
        static Mlong Factorial(int x)
        {
            for (int i = FactorialMemo.Count; i <= x; i++) FactorialMemo.Add(FactorialMemo.Last() * i);
            return FactorialMemo[x];
        }
        public static Mlong Comb(Mlong n, Mlong a)
        {
            Mlong buf1 = 1;
            Mlong buf2 = 1;
            for (long i = 0; i < a.num; i++)
            {
                buf1 *= n - i;
                buf2 *= (i + 1);
            }
            return buf1.modjyo(buf2);

        }

        public static Mlong Comb(int n, int r)
        {
            return Factorial(n).modjyo(Factorial(r)).modjyo(Factorial(n - r));
        }
        public static void WriteLine(Mlong x)
        {
            Console.WriteLine(x.num);

        }

        public int CompareTo(object obj)
        {
            Mlong kore = (Mlong)obj;
            return num.CompareTo(kore.num);
        }
    }
    public class Mint : IComparable
    {
        public static int mod = 1000000007;
        public int num;
        public Mint(int Num) { num = Num; }
        public override int GetHashCode()
        {
            return num.GetHashCode();
        }
        public override bool Equals(object obj)
        {
            return num.Equals(obj);
        }
        public static Mint operator +(Mint x, Mint y)
        {
            return new Mint((x.num + y.num) % mod);
        }
        public static Mint operator -(Mint x, Mint y)
        {
            return new Mint((x.num - y.num + mod) % mod);
        }
        public static Mint operator *(Mint x, Mint y)
        {
            return new Mint((int)(((1L * x.num) * y.num) % mod));
        }
        public Mint hutujyo(Mint x)
        {
            return new Mint(num / x.num);
        }

        static int GetInverse(long a)
        {
            long div, p = mod, x1 = 1, y1 = 0, x2 = 0, y2 = 1;
            while (true)
            {
                if (p == 1) return (int)(x2 + mod); div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p;
                if (a == 1) return (int)(x1 + mod); div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a;
            }
        }
        public Mint modjyo(Mint x)
        {
            return num * GetInverse(x.num) % mod;
        }
        /*  public Mint modjyo(Mint x)
          {
              return num * x.Pow(mod - 2);
          }*/
        public static implicit operator Mint(int x)
        {
            return new Mint(x);
        }
        public static Mint[] Toarray(int[] x)
        {
            Mint[] buf = new Mint[x.Length];
            for (int i = 0; i < x.Length; i++)
            {
                buf[i] = x[i];
            }
            return buf;
        }
        public static explicit operator int(Mint x)
        {
            return x.num;
        }
        public static Mint operator %(Mint x, Mint y)
        {
            return new Mint(x.num % y.num);
        }
        public static bool operator ==(Mint x, Mint y)
        {
            return x.num == y.num;
        }
        public static bool operator !=(Mint x, Mint y)
        {
            return x.num != y.num;
        }
        public static bool operator <(Mint x, Mint y)
        {
            return x.num < y.num;
        }
        public static bool operator >(Mint x, Mint y)
        {
            return x.num > y.num;
        }
        public static bool operator <=(Mint x, Mint y)
        {
            return x.num <= y.num;
        }
        public static bool operator >=(Mint x, Mint y)
        {
            return x.num >= y.num;
        }
        public Mint Pow(Mint n)
        {
            Mint pow = this;
            Mint res = 1;
            while (n > 0)
            {
                if (n % 2 == 1) { res *= pow; }
                pow *= pow;
                n.hutujyo(2);
            }
            /*   if (n == 1) { return this % mod; }
               if (n % 2 == 1) { return (this * Pow(n - 1)) % mod; }
               Mint buf = Pow(n.hutujyo(2));
               return (buf * buf) % mod;*/
            return res;
        }
        public static Mint Pow(Mint a, Mint n)
        {
            return a.Pow(n);
        }
        public static List<Mint> FactorialMemo = new List<Mint>() { 1 };
        static Mint Factorial(int x)
        {
            for (int i = FactorialMemo.Count; i <= x; i++) FactorialMemo.Add(FactorialMemo.Last() * i);
            return FactorialMemo[x];
        }


        public static Mint Comb(int n, int r)
        {
            return Factorial(n).modjyo(Factorial(r)).modjyo(Factorial(n - r));
        }
        public static void WriteLine(Mint x)
        {
            Console.WriteLine(x.num);

        }
        public int CompareTo(object obj)
        {
            Mint kore = (Mint)obj;
            return num.CompareTo(kore.num);
        }
    }
    public class PriorityQueue<T> : IEnumerable<T>
    {
        private readonly List<T> _data = new List<T>();
        private readonly IComparer<T> _comparer;
        private readonly bool _isDescending;

        public PriorityQueue(IComparer<T> comparer, bool isDescending = true)
        {
            _comparer = comparer;
            _isDescending = isDescending;
        }

        public PriorityQueue(Comparison<T> comparison, bool isDescending = true)
            : this(Comparer<T>.Create(comparison), isDescending)
        {
        }

        public PriorityQueue(bool isDescending = true)
            : this(Comparer<T>.Default, isDescending)
        {
        }

        public void Enqueue(T item)
        {
            _data.Add(item);
            var childIndex = _data.Count - 1;
            while (childIndex > 0)
            {
                var parentIndex = (childIndex - 1) / 2;
                if (Compare(_data[childIndex], _data[parentIndex]) >= 0)
                    break;
                Swap(childIndex, parentIndex);
                childIndex = parentIndex;
            }
        }

        public T Dequeue()
        {
            var lastIndex = _data.Count - 1;
            var firstItem = _data[0];
            _data[0] = _data[lastIndex];
            _data.RemoveAt(lastIndex--);
            var parentIndex = 0;
            while (true)
            {
                var childIndex = parentIndex * 2 + 1;
                if (childIndex > lastIndex)
                    break;
                var rightChild = childIndex + 1;
                if (rightChild <= lastIndex && Compare(_data[rightChild], _data[childIndex]) < 0)
                    childIndex = rightChild;
                if (Compare(_data[parentIndex], _data[childIndex]) <= 0)
                    break;
                Swap(parentIndex, childIndex);
                parentIndex = childIndex;
            }
            return firstItem;
        }

        public T Peek()
        {
            return _data[0];
        }

        private void Swap(int a, int b)
        {
            var tmp = _data[a];
            _data[a] = _data[b];
            _data[b] = tmp;
        }

        private int Compare(T a, T b)
        {
            return _isDescending ? _comparer.Compare(b, a) : _comparer.Compare(a, b);
        }

        public int Count => _data.Count;

        public IEnumerator<T> GetEnumerator()
        {
            return _data.GetEnumerator();
        }

        IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
    }
    public class Tree
    {
        public Node[] Nodes;
        public List<int[]> hen;
        public class Node
        {
            public int? parent = null;
            public List<int[]> children = new List<int[]>();
            public int parentcolor = -1;
        }
        public Tree(int kosu)
        {
            Nodes = new Node[kosu];
            for (int i = 0; i < Nodes.Length; i++)
            {
                Nodes[i] = new Node();
            }
            hen = new List<int[]>();
        }
        public void Connect(int oya, int ko, int id)
        {
            hen.Add(new int[] { oya, ko, 0 });
            Nodes[oya].children.Add(new int[] { ko, id });
            Nodes[ko].parent = oya;
        }
    }
    /// <summary>
    /// Self-Balancing Binary Search Tree
    /// (using Randamized BST)
    /// </summary>
    public class SB_BinarySearchTree<T> where T : IComparable
    {

        public class Node
        {
            public T Value;
            public Node LChild;
            public Node RChild;
            public int Count;     //size of the sub tree

            public Node(T v)
            {
                Value = v;
                Count = 1;
            }
        }

        static Random _rnd = new Random();

        public static int Count(Node t)
        {
            return t == null ? 0 : t.Count;
        }

        static Node Update(Node t)
        {
            t.Count = Count(t.LChild) + Count(t.RChild) + 1;
            return t;
        }

        public static Node Merge(Node l, Node r)
        {
            if (l == null || r == null) return l == null ? r : l;

            if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
            {
                l.RChild = Merge(l.RChild, r);
                return Update(l);
            }
            else
            {
                r.LChild = Merge(l, r.LChild);
                return Update(r);
            }
        }

        /// <summary>
        /// split as [0, k), [k, n)
        /// </summary>
        public static Tuple<Node, Node> Split(Node t, int k)
        {
            if (t == null) return new Tuple<Node, Node>(null, null);
            if (k <= Count(t.LChild))
            {
                var s = Split(t.LChild, k);
                t.LChild = s.Item2;
                return new Tuple<Node, Node>(s.Item1, Update(t));
            }
            else
            {
                var s = Split(t.RChild, k - Count(t.LChild) - 1);
                t.RChild = s.Item1;
                return new Tuple<Node, Node>(Update(t), s.Item2);
            }
        }

        public static Node Remove(Node t, T v)
        {
            if (Find(t, v) == null) return t;
            return RemoveAt(t, LowerBound(t, v));
        }

        public static Node RemoveAt(Node t, int k)
        {
            var s = Split(t, k);
            var s2 = Split(s.Item2, 1);
            return Merge(s.Item1, s2.Item2);
        }

        public static bool Contains(Node t, T v)
        {
            return Find(t, v) != null;
        }

        public static Node Find(Node t, T v)
        {
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp > 0) t = t.LChild;
                else if (cmp < 0) t = t.RChild;
                else break;
            }
            return t;
        }

        public static Node FindByIndex(Node t, int idx)
        {
            if (t == null) return null;

            var currentIdx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                if (currentIdx == idx) return t;
                if (currentIdx > idx)
                {
                    t = t.LChild;
                    currentIdx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else
                {
                    t = t.RChild;
                    currentIdx += (Count(t == null ? null : t.LChild) + 1);
                }
            }

            return null;
        }

        public static int UpperBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var ret = Int32.MaxValue;
            var idx = Count(t) - Count(t.RChild) - 1;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);

                if (cmp > 0)
                {
                    ret = Math.Min(ret, idx);
                    t = t.LChild;
                    idx -= (Count(t == null ? null : t.RChild) + 1);
                }
                else if (cmp <= 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                }
            }
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static int LowerBound(Node t, T v)
        {
            var torg = t;
            if (t == null) return -1;

            var idx = Count(t) - Count(t.RChild) - 1;
            var ret = Int32.MaxValue;
            while (t != null)
            {
                var cmp = t.Value.CompareTo(v);
                if (cmp >= 0)
                {
                    if (cmp == 0) ret = Math.Min(ret, idx);
                    t = t.LChild;
                    if (t == null) ret = Math.Min(ret, idx);
                    idx -= t == null ? 0 : (Count(t.RChild) + 1);
                }
                else if (cmp < 0)
                {
                    t = t.RChild;
                    idx += (Count(t == null ? null : t.LChild) + 1);
                    if (t == null) return idx;
                }
            }
            // return ret;
            return ret == Int32.MaxValue ? Count(torg) : ret;
        }

        public static Node Insert(Node t, T v)
        {

            var ub = LowerBound(t, v);
            return InsertByIdx(t, ub, v);
        }

        static Node InsertByIdx(Node t, int k, T v)
        {
            var s = Split(t, k);
            return Merge(Merge(s.Item1, new Node(v)), s.Item2);
        }

        public static IEnumerable<T> Enumerate(Node t)
        {
            var ret = new List<T>();
            Enumerate(t, ret);
            return ret;
        }

        static void Enumerate(Node t, List<T> ret)
        {
            if (t == null) return;
            Enumerate(t.LChild, ret);
            ret.Add(t.Value);
            Enumerate(t.RChild, ret);
        }
    }

    /// <summary>
    /// C-like set
    /// </summary>
    public class Set<T> where T : IComparable
    {
        protected SB_BinarySearchTree<T>.Node _root;
        public T this[int idx] { get { return ElementAt(idx); } }

        public int Count()
        {
            return SB_BinarySearchTree<T>.Count(_root);
        }
        public bool Any()
        {
            return _root != null;
        }
        public virtual void Insert(T v)
        {
            if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
            else
            {
                if (SB_BinarySearchTree<T>.Find(_root, v) != null) return;
                _root = SB_BinarySearchTree<T>.Insert(_root, v);
            }
        }

        public void Clear()
        {
            _root = null;
        }

        public void Remove(T v)
        {

            _root = SB_BinarySearchTree<T>.Remove(_root, v);
        }

        public bool Contains(T v)
        {
            return SB_BinarySearchTree<T>.Contains(_root, v);
        }

        public T ElementAt(int k)
        {
            var node = SB_BinarySearchTree<T>.FindByIndex(_root, k);
            if (node == null) throw new IndexOutOfRangeException();
            return node.Value;
        }

        public int Count(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v) - SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int LowerBound(T v)
        {
            return SB_BinarySearchTree<T>.LowerBound(_root, v);
        }

        public int UpperBound(T v)
        {
            return SB_BinarySearchTree<T>.UpperBound(_root, v);
        }
        public T max()
        {
            SB_BinarySearchTree<T>.Node ima = _root;
            while (ima.RChild != null)
            {
                ima = ima.RChild;
            }
            return ima.Value;
        }
        public T min()
        {
            SB_BinarySearchTree<T>.Node ima = _root;
            while (ima.LChild != null)
            {
                ima = ima.LChild;
            }
            return ima.Value;
        }
        public Tuple<int, int> EqualRange(T v)
        {
            if (!Contains(v)) return new Tuple<int, int>(-1, -1);
            return new Tuple<int, int>(SB_BinarySearchTree<T>.LowerBound(_root, v), SB_BinarySearchTree<T>.UpperBound(_root, v) - 1);
        }

        public List<T> ToList()
        {
            return new List<T>(SB_BinarySearchTree<T>.Enumerate(_root));
        }
    }

    /// <summary>
    /// C-like multiset
    /// </summary>
    public class MultiSet<T> : Set<T> where T : IComparable
    {
        public override void Insert(T v)
        {
            if (_root == null) _root = new SB_BinarySearchTree<T>.Node(v);
            else _root = SB_BinarySearchTree<T>.Insert(_root, v);
        }
    }
    public class RollingHashSlow
    {
        public const long B = (long)1e5 + 7;
        public const long MOD = (long)1e9 + 7;
        public string S { get; set; }
        public int N { get; set; }
        public long[] Power { get; set; }
        public long[] Hash { get; set; }
        public RollingHashSlow(string s)
        {
            this.S = s;
            this.N = s.Length;
            // B^nを計算しておく
            this.Power = new long[this.N + 1];
            this.Power[0] = 1;
            for (int i = 0; i < N; i++)
            {
                this.Power[i + 1] = (this.Power[i] * B) % MOD;
            }
            // ハッシュを前計算する
            this.Hash = new long[this.N + 1];
            for (int i = 0; i < N; i++)
            {
                this.Hash[i + 1] = (this.Hash[i] * B + S[i]) % MOD;
            }
        }
        // S[l, r) のハッシュ値を求める
        public long Get(int l, int r)
        {
            var hash = this.Hash[r] - (this.Hash[l] * this.Power[r - l] % MOD);
            if (hash < 0) hash += MOD;
            return hash;
        }
    }
    class RollingHashFast
    {
        public const ulong B = (ulong)1e9 + 7;
        public string S { get; set; }
        public int N { get; set; }
        public ulong[] Power { get; set; }
        public ulong[] Hash { get; set; }
        public RollingHashFast(string s)
        {
            this.S = s;
            this.N = s.Length;
            // B^nを計算しておく
            this.Power = new ulong[this.N + 1];
            this.Power[0] = 1;
            for (int i = 0; i < N; i++)
            {
                this.Power[i + 1] = this.Power[i] * B;
            }
            // ハッシュを前計算する
            this.Hash = new ulong[this.N + 1];
            for (int i = 0; i < N; i++)
            {
                this.Hash[i + 1] = this.Hash[i] * B + S[i];
            }
        }
        // [l, r) のハッシュ値を求める
        public ulong Get(int l, int r)
        {
            var hash = this.Hash[r] - (this.Hash[l] * this.Power[r - l]);
            return hash;
        }
    }
    class Segtree<T>
    {
        int n;
        T[] tree;
        Func<T, T, T> f;
        T exnum;
        public Segtree(int m, Func<T, T, T> f, T ex)
        {
            this.f = f;
            this.exnum = ex;
            n = 1;
            while (n < m) n <<= 1;

            tree = new T[(n << 1) - 1];
            for (int i = 0; i < tree.Length; i++) tree[i] = ex;
        }
        public Segtree(int m, T ini, Func<T, T, T> f, T ex) : this(m, f, ex)
        {
            for (int i = 0; i < m; ++i) tree[i + n - 1] = ini;
            all_update();
        }
        public void assign_without_update(int j, T x)
        {
            tree[j + n - 1] = x;
        }
        public void update(int j, T x)
        {
            int i = j + n - 1;
            tree[i] = x;
            while (i > 0)
            {
                i = (i - 1) >> 1;
                tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
            }
        }
        public void all_update()
        {
            for (int i = n - 2; i >= 0; i--)
                tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
        }
        public T look(int i) { return tree[i + n - 1]; }

        // [s, t)
        public T run(int s, int t) { return query(s, t, 0, 0, n); }
        T query(int s, int t, int k, int l, int r)
        {
            if (r <= s || t <= l) return exnum;
            if (s <= l && r <= t) return tree[k];

            return f(query(s, t, (k << 1) + 1, l, (l + r) >> 1), query(s, t, (k + 1) << 1, (l + r) >> 1, r));
        }

    }
    public class WarshallFloyd
    {
        public int V { get; set; }
        public long[,] DP { get; set; }
        public long INF = long.MaxValue;
        public bool[] Negative;
        public WarshallFloyd(int v)
        {
            this.V = v;
            this.DP = new long[V, V];
            for (int i = 0; i < V; i++)
            {
                for (int j = 0; j < V; j++)
                {
                    this.DP[i, j] = INF;
                }
                this.DP[i, i] = 0;
            }
        }
        public void SetCost(int a, int b, int c = 1)
        {
            this.DP[a, b] = c;
        }
        public long[,] CalcMinCost()
        {
            this.Negative = new bool[V];
            for (int k = 0; k < V; k++)
            {
                for (int i = 0; i < V; i++)
                {
                    for (int j = 0; j < V; j++)
                    {
                        if (DP[i, k] == INF || DP[k, j] == INF) continue;
                        DP[i, j] = Math.Min(DP[i, j], DP[i, k] + DP[k, j]);
                    }
                }
            }
            for (int k = 0; k < V; k++)
            {
                for (int i = 0; i < V; i++)
                {
                    for (int j = 0; j < V; j++)
                    {
                        if (DP[i, k] == INF || DP[k, j] == INF) continue;
                        if (DP[i, j] > DP[i, k] + DP[k, j])
                        {
                            this.Negative[i] = true;
                            this.Negative[j] = true;
                            this.Negative[k] = true;
                        }
                    }
                }
            }
            return DP;
        }
        public long GetMinCost(int a, int b)
        {
            return DP[a, b];
        }
        public bool IsINF(int a, int b)
        {
            return DP[a, b] >= INF;
        }
        public bool IsNegative(params int[] vs)
        {
            if (vs.Length == 0) vs = Enumerable.Range(0, V).ToArray();
            foreach (var v in vs)
            {
                if (Negative[v]) return true;
            }
            return false;
        }
    }
    /// <summary>
    /// Get min cost between two points
    /// </summary>
    public class Dijkstra
    {
        private int maxIndex = -1;
        private const int INF = Int32.MaxValue;

        private int[,] _edgeArray;

        /// <summary>
        /// Dijkstra, get min cost between two points
        /// should not contain negatie cost path
        /// </summary>
        /// <param name="size">max index of vertices</param>
        public Dijkstra(int size)
        {
            maxIndex = size + 1;
            _edgeArray = new int[maxIndex, maxIndex];

            for (int i = 0; i < _edgeArray.GetLength(0); i++)
            {
                for (int j = 0; j < _edgeArray.GetLength(1); j++)
                {
                    _edgeArray[i, j] = INF;
                    if (i == j) _edgeArray[i, j] = 0;
                }
            }
        }

        // Add a path(no direction) with its cost
        public void AddPath(int s, int t, int cost)
        {
            _edgeArray[s, t] = Math.Min(_edgeArray[s, t], cost);
            _edgeArray[t, s] = _edgeArray[s, t];
        }

        //Get the min cost between s and t
        //return Int32.MaxValue if no path
        public int GetMinCost(int s, int t)
        {
            int[] cost = new int[maxIndex];
            for (int i = 0; i < cost.Length; i++) cost[i] = INF;
            cost[s] = 0;

            var priorityQueue = new PriorityQueue<CPair<int, int>>();
            priorityQueue.Enqueue(new CPair<int, int>(0, s));

            while (priorityQueue.Count() > 0)
            {
                var costDestinationPair = priorityQueue.Dequeue();
                if (cost[costDestinationPair.Item2] < costDestinationPair.Item1) continue;

                for (int i = 0; i < maxIndex; i++)
                {
                    int newCostToi = _edgeArray[costDestinationPair.Item2, i] == INF ? INF :
                        costDestinationPair.Item1 + _edgeArray[costDestinationPair.Item2, i];
                    if (newCostToi < cost[i])
                    {
                        cost[i] = newCostToi;
                        priorityQueue.Enqueue(new CPair<int, int>(newCostToi, i));
                    }
                }
            }

            return cost[t];
        }
    }
    class Deque<T> : IEnumerable<T>
    {
        public T this[int i]
        {
            get { return this.Buffer[(this.FirstIndex + i) % this.Capacity]; }
            set
            {
                if (i < 0) throw new ArgumentOutOfRangeException();
                this.Buffer[(this.FirstIndex + i) % this.Capacity] = value;
            }
        }
        private T[] Buffer;
        private int Capacity;
        private int FirstIndex;
        private int LastIndex
        {
            get { return (this.FirstIndex + this.Length) % this.Capacity; }
        }
        public int Length;
        public Deque(int capacity = 16)
        {
            this.Capacity = capacity;
            this.Buffer = new T[this.Capacity];
            this.FirstIndex = 0;
        }
        public void PushBack(T data)
        {
            if (this.Length == this.Capacity) this.Resize();
            this.Buffer[this.LastIndex] = data;
            this.Length++;
        }
        public void PushFront(T data)
        {
            if (this.Length == this.Capacity) this.Resize();
            var index = this.FirstIndex - 1;
            if (index < 0) index = this.Capacity - 1;
            this.Buffer[index] = data;
            this.Length++;
            this.FirstIndex = index;
        }
        public T PopBack()
        {
            if (this.Length == 0) throw new InvalidOperationException("データが空です。");
            var data = this[this.Length - 1];
            this.Length--;
            return data;
        }
        public T PopFront()
        {
            if (this.Length == 0) throw new InvalidOperationException("データが空です。");
            var data = this[0];
            this.FirstIndex++;
            this.FirstIndex %= this.Capacity;
            this.Length--;
            return data;
        }
        private void Resize()
        {
            var newArray = new T[this.Capacity * 2];
            for (int i = 0; i < this.Length; i++)
            {
                newArray[i] = this[i];
            }
            this.FirstIndex = 0;
            this.Capacity *= 2;
            this.Buffer = newArray;
        }
        public IEnumerator<T> GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++)
            {
                yield return this[i];
            }
        }
        IEnumerator IEnumerable.GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++)
            {
                yield return this[i];
            }
        }
    }
    public struct Edge
    {
        public int from;

        public int to;
        public long dist;

        public Edge(int t, long c)
        {
            from = -1;
            to = t;
            dist = c;
        }
        public Edge(int f, int t, long c)
        {
            from = f;
            to = t;
            dist = c;
        }
    }
    public class PotentializedUnionFind
    {
        int _count;
        int[] par, sizes, rank;
        long[] weight;
        public PotentializedUnionFind(int N)
        {
            _count = N;
            par = new int[N];
            sizes = new int[N];
            rank = new int[N];
            weight = new long[N];
            for (int i = 0; i < N; ++i)
            {
                par[i] = i;
                sizes[i] = 1;
                rank[i] = 0;
            }
        }
        int Root(int X)
        {
            if (par[X] == X)
            {
                return X;
            }
            else
            {
                int r = Root(par[X]);
                weight[X] += weight[par[X]];
                return (par[X] = r);
            }
        }
        public bool Merge(int X, int Y, long W)
        {
            W += GetWeight(X);
            W -= GetWeight(Y);
            X = Root(X);
            Y = Root(Y);
            if (X == Y) return false;
            --_count;
            if (rank[X] < rank[Y])
            {
                par[X] = Y;
                sizes[Y] += sizes[X];
                weight[X] = -W;
            }
            else
            {
                par[Y] = X;
                sizes[X] += sizes[Y];
                if (rank[X] == rank[Y]) ++rank[X];
                weight[Y] = W;
            }
            return true;
        }
        public bool IsSame(int X, int Y) => Root(X) == Root(Y);
        public int Count => _count;
        public bool IsParent(int X) => X == Root(X);
        public int GetParent(int X) => Root(X);
        public int Size(int X) => sizes[Root(X)];
        public long GetWeight(int X) { Root(X); return weight[X]; }
        public long GetDiff(int X, int Y) => GetWeight(Y) - GetWeight(X);
    }
    public static class BellmanFord
    {
        public class Vertex
        {
            public long Distance { get; set; }

            public Vertex()
            {
                Distance = long.MaxValue;
            }
        }

        public class Edge
        {
            public int From { get; private set; }
            public int To { get; private set; }
            public long Weight { get; private set; }

            public Edge(int from, int to, long weight)
            {
                From = from;
                To = to;
                Weight = weight;
            }
        }

        private static void GetReachable(int origin, ref HashSet<int> reached, ref Dictionary<int, int[]> paths)
        {
            if (reached.Contains(origin))
                return;

            reached.Add(origin);

            if (paths.ContainsKey(origin) == false)
                return;

            foreach (var p in paths[origin])
                GetReachable(p, ref reached, ref paths);
        }

        /// <summary>
        /// null: 負の無限大
        /// long.MaxValue: たどり着けない
        /// その他: 距離
        /// </summary>
        public static long? RunBellmanFord(int vertexCount, List<Edge> rawEdges, int source, int dest)
        {
            var forwards = rawEdges.GroupBy(x => x.From).ToDictionary(x => x.Key, x => x.Select(xs => xs.To).ToArray());
            var reverses = rawEdges.GroupBy(x => x.To).ToDictionary(x => x.Key, x => x.Select(xs => xs.From).ToArray());

            var fromSource = new HashSet<int>();
            var toDest = new HashSet<int>();

            GetReachable(source, ref fromSource, ref forwards);
            GetReachable(dest, ref toDest, ref reverses);

            var edges = rawEdges
                .Where(e => fromSource.Contains(e.From))
                .Where(e => fromSource.Contains(e.To))
                .Where(e => toDest.Contains(e.From))
                .Where(e => toDest.Contains(e.To))
                .ToArray();

            // initialize distances
            var vertices = new List<Vertex>();
            for (int i = 0; i < vertexCount; i++)
                vertices.Add(new Vertex());
            vertices[source].Distance = 0L;

            // update distances
            for (int i = 0; i < vertices.Count; i++)
            {
                foreach (var e in edges)
                {
                    var from = vertices[e.From];
                    var to = vertices[e.To];

                    if (from.Distance == long.MaxValue)
                        continue;

                    var newDistance = from.Distance + e.Weight;
                    if (to.Distance > newDistance)
                    {
                        to.Distance = newDistance;
                    }
                }
            }

            // check negative cycle
            foreach (var e in edges)
            {
                var from = vertices[e.From];
                var to = vertices[e.To];

                if (from.Distance + e.Weight < to.Distance)
                    return null;
            }

            return vertices[dest].Distance;
        }
    }
    public class SortedMultiSet<T> : IEnumerable<T>
    {
        private SortedDictionary<T, int> _dict;

        public SortedMultiSet()
        {
            _dict = new SortedDictionary<T, int>();
        }

        public SortedMultiSet(IEnumerable<T> items) : this()
        {
            Add(items);
        }

        public bool Contains(T item)
        {
            return _dict.ContainsKey(item);
        }

        public void Add(T item)
        {
            if (_dict.ContainsKey(item))
                _dict[item]++;
            else
                _dict[item] = 1;
        }

        public void Add(IEnumerable<T> items)
        {
            foreach (var item in items)
                Add(item);
        }

        public void Remove(T item)
        {
            if (!_dict.ContainsKey(item))
                throw new ArgumentException();
            if (--_dict[item] == 0)
                _dict.Remove(item);
        }

        // Return the last value in the multiset
        public T Peek()
        {
            if (!_dict.Any())
                throw new NullReferenceException();
            return _dict.Last().Key;
        }

        // Return the last value in the multiset and remove it.
        public T Pop()
        {
            T item = Peek();
            Remove(item);
            return item;
        }

        public IEnumerator<T> GetEnumerator()
        {
            foreach (var kvp in _dict)
                for (int i = 0; i < kvp.Value; i++)
                    yield return kvp.Key;
        }

        IEnumerator IEnumerable.GetEnumerator()
        {
            return this.GetEnumerator();
        }
    }
    class Treap<T> : IEnumerable<T> where T : IComparable
    {
        public int Length
        {
            get { return this.Root == null ? 0 : this.NodeCount(this.Root); }
        }
        public T this[int i]
        {
            get
            {
                if (this.Root.Count < i - 1) throw new IndexOutOfRangeException();
                return this.Root.Get(i);
            }
        }
        public Node Root;
        private IComparer<T> Cmp = Comparer<T>.Default;
        public Treap(IComparer<T> cmp = null)
        {
            if (cmp != null) this.Cmp = cmp;
        }
        public Node Merge(Node l, Node r)
        {
            if (l == null || r == null) return l == null ? r : l;

            if (l.Priority > r.Priority)
            {
                l.R = this.Merge(l.R, r);
                return this.Update(l);
            }
            else
            {
                r.L = this.Merge(l, r.L);
                return this.Update(r);
            }
        }
        // 左にk個、右に残り(n-k)個に分割する
        public (Node, Node) Split(int k) { return this.Split(this.Root, k); }
        public (Node, Node) Split(Node t, int k)
        {
            if (t == null) return (null, null);
            if (k <= this.NodeCount(t.L))
            {
                var s = this.Split(t.L, k);
                t.L = s.Item2;
                return (s.Item1, this.Update(t));
            }
            else
            {
                var s = this.Split(t.R, k - this.NodeCount(t.L) - 1);
                t.R = s.Item1;
                return (this.Update(t), s.Item2);
            }
        }
        public int NodeCount(Node node)
        {
            return node == null ? 0 : node.Count;
        }
        public int LowerBouond(T value) { return this.LowerBouond(this.Root, value); }
        public int LowerBouond(Node node, T value)
        {
            if (node == null) return 0;
            if (this.Cmp.Compare(value, node.Value) <= 0)
                return this.LowerBouond(node.L, value);
            else
                return this.NodeCount(node.L) + 1 + this.LowerBouond(node.R, value);
        }
        public int UpperBound(T value) { return this.UpperBound(this.Root, value); }
        public int UpperBound(Node node, T value)
        {
            if (node == null) return 0;
            if (this.Cmp.Compare(value, node.Value) >= 0)
                return this.NodeCount(node.L) + 1 + this.UpperBound(node.R, value);
            else
                return this.UpperBound(node.L, value);
        }
        public int ValueCount(T value)
        {
            return this.UpperBound(value) - this.LowerBouond(value);
        }
        public void Insert(T value)
        {
            var sub = this.Split(this.Root, this.LowerBouond(value));
            var newNode = new Node(value);
            var merged = this.Merge(sub.Item1, newNode);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public void Erase(T value)
        {
            if (this.ValueCount(value) == 0) return;
            var sub = this.Split(this.Root, this.LowerBouond(value));
            this.Root = this.Merge(sub.Item1, this.Split(sub.Item2, 1).Item2);
        }
        public void InsertAt(Node node, int k)
        {
            var sub = this.Split(this.Root, k);
            var merged = this.Merge(sub.Item1, node);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public void InsertAt(T value, int k)
        {
            var node = new Node(value);
            var sub = this.Split(this.Root, k);
            var merged = this.Merge(sub.Item1, node);
            this.Root = this.Merge(merged, sub.Item2);
        }
        public Node EraseAt(int k)
        {
            var sub = this.Split(this.Root, k);
            var sub2 = this.Split(sub.Item2, 1);
            var res = sub2.Item1;
            this.Root = this.Merge(sub.Item1, sub2.Item2);
            return res;
        }
        public Node Update(Node node)
        {
            node.Count = this.NodeCount(node.L) + this.NodeCount(node.R) + 1;
            return node;
        }
        public class Node
        {
            private static Random Rand = new Random();
            public T Value;
            public Node L = null;
            public Node R = null;
            public int Count;
            public double Priority;
            public Node(T value, double priority)
            {
                this.Value = value;
                this.Priority = priority;
                this.Count = 1;
            }
            public Node(T value) : this(value, Rand.NextDouble()) { }
            public T Get(int i) { return this.GetNode(i).Value; }
            public Node GetNode(int i)
            {
                var lcount = 0;
                if (this.L != null) lcount = this.L.Count;
                if (lcount == i) return this;
                else if (lcount < i) return this.R.GetNode(i - lcount - 1);
                else return this.L.GetNode(i);
            }
        }
        public IEnumerator<T> GetEnumerator()
        {
            for (int i = 0; i < this.Length; i++) yield return this.Root.Get(i);
        }
        IEnumerator IEnumerable.GetEnumerator() { return this.GetEnumerator(); }
    }


    public static class ObjectExtension
    {
        // ディープコピーの複製を作る拡張メソッド
        public static T DeepClone<T>(this T src)
        {
            using (var memoryStream = new System.IO.MemoryStream())
            {
                var binaryFormatter
                  = new System.Runtime.Serialization
                        .Formatters.Binary.BinaryFormatter();
                binaryFormatter.Serialize(memoryStream, src); // シリアライズ
                memoryStream.Seek(0, System.IO.SeekOrigin.Begin);
                return (T)binaryFormatter.Deserialize(memoryStream); // デシリアライズ
            }
        }
    }
}