Problem:
Score : 500 points

Problem Statement

There are N Snuke Cats numbered 1, 2, \ldots, N, where N is even.

Each Snuke Cat wears a red scarf, on which his favorite non-negative integer is written.

Recently, they learned the operation called xor (exclusive OR).

What is xor?

For n non-negative integers x_1, x_2, \ldots, x_n, their xor, x_1~\textrm{xor}~x_2~\textrm{xor}~\ldots~\textrm{xor}~x_n is defined as follows:

When x_1~\textrm{xor}~x_2~\textrm{xor}~\ldots~\textrm{xor}~x_n is written in base two, the digit in the 2^k's place (k \geq 0) is 1 if the number of integers among x_1, x_2, \ldots, x_n whose binary representations have 1 in the 2^k's place is odd, and 0 if that count is even.

For example, 3~\textrm{xor}~5 = 6.

They wanted to use this operation quickly, so each of them calculated the xor of the integers written on their scarfs except his scarf.

We know that the xor calculated by Snuke Cat i, that is, the xor of the integers written on the scarfs except the scarf of Snuke Cat i is a_i.
Using this information, restore the integer written on the scarf of each Snuke Cat.

Constraints

All values in input are integers.

2 \leq N \leq 200000

N is even.

0 \leq a_i \leq 10^9

There exists a combination of integers on the scarfs that is consistent with the given information.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print a line containing N integers separated with space.

The i-th of the integers from the left should represent the integer written on the scarf of Snuke Cat i.

If there are multiple possible solutions, you may print any of them.

Sample Input 1

4
20 11 9 24

Sample Output 1

26 5 7 22

5~\textrm{xor}~7~\textrm{xor}~22 = 20

26~\textrm{xor}~7~\textrm{xor}~22 = 11

26~\textrm{xor}~5~\textrm{xor}~22 = 9

26~\textrm{xor}~5~\textrm{xor}~7 = 24

Thus, this output is consistent with the given information.

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 Atcoder
{
    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 a;
        public long b;
        public long kosu;
        public Pair(long A,long B)
        {
            a = A;b = B;
        }

        public int CompareTo(object obj)
        {
            Pair kore = (Pair)obj;
            if (a != kore.a) { return a.CompareTo(kore.a); }
            return b.CompareTo(kore.b);
        }
     /*   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 Pair2 : IComparable
    {
        public long a;
        public long b;
        public long kosu;
        public long teki;
        public Mlong jikkou = 1;
        public Pair2(long A, long B)
        {
            a = A; b = B;
        }

        public int CompareTo(object obj)
        {
            Pair2 kore = (Pair2)obj;
            if (a != kore.a) { return a.CompareTo(kore.a); }
            return b.CompareTo(kore.b);
        }
        /*   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 Dps:IComparable
    {
        long[,] dp;
        long i;
        public Dps(long i,long n)
        {
            this.i = i;
            dp = new long[n, 100001];
        }
        public int CompareTo(object obj)
        {
            Dps dps = (Dps)obj;
            return i.CompareTo(dps.i);
        }
    }
    class Program
    {
        static List<int>[] togo;
       static  long[][,] dpps;
      static  bool[] visited ;
        static void Main(string[] args)
        {
            StreamScanner scan = new StreamScanner(System.Console.OpenStandardInput());
            long n = scan.Long();
            long[] a = scan.Long(n);

            long[,] ans = new long[n, 36];
            for (int i = 0; i < 36; i++)
            {
                int[] buf = new int[n];
                int sum = 0;
                for (long k = 0; k < n; k++)
                {
                    if (((a[k] >> i) & 1) == 1)
                    {
                        buf[k] = 1;
                        sum += 1;
                    }
                }
                sum %= 2;
                for (long k = 0; k < n; k++)
                {
                    int ho = sum - buf[k];
                    if (ho < 0) { ho += 2; }
                    ho %= 2;
                    ans[k, i] = ho;
                }
            }
            long[] final = new long[n];
            for (long i = 0; i < n; i++)
            {
                long kore = 1;
                long an = 0;
                for (long k = 0; k < 36; k++)
                {
                    an += kore * ans[i, k];
                    kore *= 2;
                }
                final[i] = an;
            }
            Console.WriteLine(String.Join(" ",final));

        }
        private static string Change35toZ(int intZ)
        {
            string Change35toZ = "0";
            if (intZ >= 0 && intZ < 26)
            {
                //if (intZ > 9) intZ += 7;
                intZ += 97;
                Change35toZ = Convert.ToChar(intZ).ToString();
            }
            return Change35toZ;
        }
        // １０進数（0～2821109907455）を３６進数に変換
        private static string Convert10to36(long lngZ)
        {
            string Convert10to36 = "";
            long l;
            int i;
            for (int i1 = 15; i1 >= 0; i1--)
            {
                l = (long)Math.Pow(26, i1);
                i = (int)(lngZ / l);
                l = i * l;
                lngZ -= l;
                Convert10to36 += Change35toZ(i);
            }
            return Convert10to36;
        }


        static void shuffle(Pair2 []ary)
        {
            System.Random rng = new System.Random();
            int n = ary.Length;
            while (n > 1)
            {
                n--;
                int k = rng.Next(n + 1);
                var tmp = new Pair2(ary[k].a,ary[k].b);
                ary[k] = new Pair2(ary[n].a, ary[n].b);
                ary[n] = new Pair2(tmp.a, tmp.b);
            }
        }
        static bool DFS(int now, int last)
        {
            visited[now] = true;
            for (int i = 0; i < togo[now].Count; i++)
            {
                if (togo[now][i] == last) { continue; }
                if (visited[togo[now][i]]) { return false; }
                if (!DFS(togo[now][i], now)) { return false; }
            }
            return true;
        }



        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 void Soinsubunkai(long num)
        {
            long m = num;
            for (long i = 2; i*i < num; i++)
            {
                long j = 0;
                for ( j = 0; j < num; j++)
                {
                    if(m%i==0)
                    {
                        m /= i;
                    }
                }

                //i^j | num,i^{j+1} not divide num
            }

            if(m!=1)
            {
                //素因数が一つ残っている(k*k>numな素因数は高々一つ)
            }
        }
       /* 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 List<int> Topological_Sort(List<List<Edge>> G)
        {
            var ret = new List<int>();
            var Nodes = new Queue<int>();
            int N = G.Count();
            var Link = new int[N];
            for (var i = 0; i < N; i++) { Link[i] = 0; }
            for (var i = 0; i < N; i++)
            {
                foreach (var nx in G[i]) { Link[nx.to]++; }
            }
            for (var i = 0; i < N; i++)
            {
                if (Link[i] == 0) { Nodes.Enqueue(i); }
            }

            while (Nodes.Any())
            {
                var nd = Nodes.Dequeue();
                ret.Add(nd);
                foreach (var nx in G[nd])
                {
                    Link[nx.to]--;
                    if (Link[nx.to] == 0)
                    {
                        Nodes.Enqueue(nx.to);
                    }
                }
            }
            return ret;
        }
        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 int 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 (int i = 0; i < n; i++)
            {
                kore[i] = Char();
            }
            return kore;
        }
        public char[,] Char(long n, long m)
        {
            char[,] kore = new char[n, m];
            for (int 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)
        {
            long[] kore = new long[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = Long();
            }
            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)
        {
            long[,] kore = new long[h, w];
            for (int i = 0; i < h; i++)
            {
                for (int j = 0; j < w; j++)
                {
                    kore[i, j] = Long();
                }
            }
            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)
        {
            int[] kore = new int[n];
            for (int i = 0; i < n; i++)
            {
                kore[i] = (int)Long();
            }
            return kore;
        }
        public int[,] Integer(long h, long w)
        {
            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();
                }

            }
            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) { this.num = num; }
        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();
        }
    }
}