Problem:
Score : 400 points

Problem Statement

We have an undirected graph G with N vertices numbered 1 to N and N edges as follows:

For each i=1,2,...,N-1, there is an edge between Vertex i and Vertex i+1.

There is an edge between Vertex X and Vertex Y.

For each k=1,2,...,N-1, solve the problem below:

Find the number of pairs of integers (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j in G is k.

Constraints

3 \leq N \leq 2 \times 10^3

1 \leq X,Y \leq N

X+1 < Y

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X Y

Output

For each k=1, 2, ..., N-1 in this order, print a line containing the answer to the problem.

Sample Input 1

5 2 4

Sample Output 1

5
4
1
0

The graph in this input is as follows:

There are five pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 1: (1,2)\,,(2,3)\,,(2,4)\,,(3,4)\,,(4,5).

There are four pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 2: (1,3)\,,(1,4)\,,(2,5)\,,(3,5).

There is one pair (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 3: (1,5).

There are no pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 4.

Sample Input 2

3 1 3

Sample Output 2

3
0

The graph in this input is as follows:

Sample Input 3

7 3 7

Sample Output 3

7
8
4
2
0
0

Sample Input 4

10 4 8

Sample Output 4

10
12
10
8
4
1
0
0
0

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Diagnostics.CodeAnalysis;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Text;
using static System.Math;
using static Solve.Methods;
using static Solve.Input;
using static Solve.Output;
using pii = Solve.Pair<int, int>;
using pll = Solve.Pair<long, long>;
using pli = Solve.Pair<long, int>;
using pil = Solve.Pair<int, long>;
using pss = Solve.Pair<string, string>;
using psi = Solve.Pair<string, int>;
using lint = System.Collections.Generic.List<int>;
using llong = System.Collections.Generic.List<long>;
using lstr = System.Collections.Generic.List<string>;
using llint = System.Collections.Generic.List<System.Collections.Generic.List<int>>;
using llstr = System.Collections.Generic.List<System.Collections.Generic.List<long>>;
using lllong = System.Collections.Generic.List<System.Collections.Generic.List<string>>;
using lii = System.Collections.Generic.List<Solve.Pair<int, int>>;
using lll = System.Collections.Generic.List<Solve.Pair<long, long>>;
using lli = System.Collections.Generic.List<Solve.Pair<long, int>>;
using lil = System.Collections.Generic.List<Solve.Pair<int, long>>;
using ll = System.Int64;

namespace Solve
{
    public class Solver
    {
        public void Main()
        {
            int N = ReadInt;
            int X = ReadInt-1;
            int Y = ReadInt-1;

            int[,] dists = new int[N,N];
            for (int i = 0; i < N; i++)
            {
                for (int j = 0; j < N; j++)
                {
                    dists[i, j] = INF;
                }

                dists[i, i] = 0;
            }

            var dijkstra = new Dijkstra(N);
            for (int i = 0; i < N-1; i++)
            {
                dijkstra.Add(i, i+1, 1,true);
            }

            dijkstra.Add(X, Y, 1);

            var ans = new int[N];
            var d = Range(N).Select(x => new List<Pair<int, int>>()).ToArray();
            
            for (int i = 0; i < N; i++)
            {
                var res = dijkstra.Run(i);
                for (int j = i + 1; j < N; ++j)
                {
                    ans[res[j].Cost]++;
                    d[res[j].Cost].Add(make_pair(i + 1, res[j].Number + 1));
                }
            }
            
            for (int k = 1; k < N; ++k)
            {
                Console.WriteLine(ans[k]);
                Console.Error.WriteLine(d[k].JoinSpace());
            }
        }



        // ReSharper disable UnusedMember.Local
        private const int MOD = (int) 1e9 + 7,
            INF = 1000000010;

        private const long LINF = 1000000000000000100;
    }

    // ライブラリ置き場ここから

    /// <summary>
    /// 優先度付きキュー を表します。
    /// </summary>
    /// <typeparam name="T">キューの型</typeparam>
    public class PriorityQueue<T> where T : IComparable<T>
    {
        public bool Any() => Count > 0;
        public int Count { get; private set; }
        private bool Descendance;
        private T[] data = new T[65536];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public PriorityQueue(bool descendance = false) { Descendance = descendance; }

        /// <summary>
        /// キューの最小 (<see cref="Descendance"/> が <see cref="true"/> の場合は最大) の要素を取得します。
        /// </summary>
        public T Top
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                ValidateNonEmpty();
                return data[1];
            }
        }

        /// <summary>
        /// キューの最小 (<see cref="Descendance"/> が <see cref="true"/> の場合は最大) の要素を削除して返します。
        /// </summary>
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Pop()
        {
            var top = Top;
            var elem = data[Count--];
            int index = 1;
            while (true)
            {
                if ((index << 1) >= Count)
                {
                    if (index << 1 > Count) break;
                    if (elem.CompareTo(data[index << 1]) > 0 ^ Descendance) data[index] = data[index <<= 1];
                    else break;
                }
                else
                {
                    var nextIndex = data[index << 1].CompareTo(data[(index << 1) + 1]) <= 0 ^ Descendance
                        ? (index << 1)
                        : (index << 1) + 1;
                    if (elem.CompareTo(data[nextIndex]) > 0 ^ Descendance) data[index] = data[index = nextIndex];
                    else break;
                }
            }

            data[index] = elem;
            return top;
        }

        /// <summary>
        /// キューに要素を追加します。
        /// </summary>
        /// <param name="value">追加する要素</param>
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Push(T value)
        {
            int index = ++Count;
            if (data.Length == Count) Extend(data.Length * 2);
            while ((index >> 1) != 0)
            {
                if (data[index >> 1].CompareTo(value) > 0 ^ Descendance) data[index] = data[index >>= 1];
                else break;
            }

            data[index] = value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private void Extend(int newSize)
        {
            T[] newDatas = new T[newSize];
            data.CopyTo(newDatas, 0);
            data = newDatas;
        }
        private void ValidateNonEmpty()
        {
            if (Count == 0) throw new Exception();
        }
    }
    /// <summary>
    /// グラフの2頂点間の最小コストを求めるアルゴリズムです。
    /// </summary>
    public class Dijkstra
    {
        public struct Node : IComparable<Node>
        {
            public Node(int num, int from, long cost)
            {
                Number = num;
                From = from;
                Cost = cost;
                To = new List<Pair<int, long>>();
            }

            public readonly int Number;
            public int From { get; internal set; }
            public long Cost { get; internal set; }
            public List<Pair<int, long>> To;

            public static bool operator >(Node a, Node b) { return a.Cost > b.Cost; }

            public static bool operator <(Node a, Node b) { return a.Cost < b.Cost; }

            public int CompareTo(Node other) { return Cost.CompareTo(other.Cost); }
        }

        public const long INF = 1000000000000000100;
        private readonly int _n;
        private PriorityQueue<Node> _que;
        public readonly Node[] Nodes;

        public Dijkstra(int N)
        {
            Nodes = new Node[N];
            for (int i = 0; i < N; i++)
            {
                Nodes[i] = new Node(i, i, INF);
            }

            _n = N;
        }

        /// <summary>
        /// グラフに辺を追加します。
        /// </summary>
        /// <param name="start">辺の始点</param>
        /// <param name="end">辺の終点</param>
        /// <param name="cost">辺のコスト</param>
        /// <param name="undirected">逆方向の辺も追加するか</param>
        public void Add(int start, int end, long cost, bool undirected = false)
        {
            Validate(start, nameof(start));
            Validate(end, nameof(end));
            Nodes[start].To.Add(make_pair(end, cost));
            if (undirected) Nodes[end].To.Add(make_pair(start, cost));
        }

        /// <summary>
        /// Dijkstra を実行します。
        /// </summary>
        /// <param name="start">始点</param>
        /// <param name="end">終点</param>
        /// <param name="firstCost">初期コスト</param>
        /// <remarks>計算量 O((E+V) log V)</remarks>
        public Node[] Run(int start, long firstCost = 0)
        {
            Validate(start, nameof(start));
            _que = new PriorityQueue<Node>();
            var node = Nodes.Select(x => new Node(x.Number, x.From, x.Cost) {To = x.To.ToList()}).ToArray();
            node[start].Cost = firstCost;

            foreach (var t in node[start].To)
            {
                node[t.first].Cost = t.second + node[start].Cost;
                node[t.first].From = start;
                _que.Push(node[t.first]);
            }

            while (_que.Any())
            {
                var next = _que.Pop();
                if (next.Cost > node[next.Number].Cost) continue;
                foreach (var t in next.To
                    .Where(t => node[t.first].Cost > next.Cost + t.second))
                {
                    node[t.first].Cost = next.Cost + t.second;
                    node[t.first].From = next.Number;
                    _que.Push(node[t.first]);
                }
            }

            return node;
        }


        public static Node[] Restore(IReadOnlyList<Node> nodes, int to)
        {
            var node = nodes[to];
            var list = new List<Node> {node};
            while (node.Number != node.From)
            {
                node = nodes[node.From];
                list.Add(node);
            }

            list.Reverse();
            return list.ToArray();
        }

        private void Validate(int val, string argument)
        {
            if (val < 0 || val >= _n)
                throw new ArgumentOutOfRangeException(argument, $"{argument} は 0 以上 頂点数 未満 でなければなりません。");
        }
    }
    // ライブラリ置き場ここまで

    #region Templete

#if !LOCAL
namespace Library { }
#endif
    public static class Methods
    {
        public static readonly int[] dx = {-1, 0, 0, 1};
        public static readonly int[] dy = {0, 1, -1, 0};


        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void Assert(bool b, string message = null)
        {
            if (!b) throw new Exception(message ?? "Assert failed.");
        }

        /*
        public static Comparison<T> greater<T>() 
            where T : IComparable<T> 
            => (a, b) => b.CompareTo(a);
        */

        public static string JoinSpace<T>(this IEnumerable<T> source) => source.Join(" ");
        public static string JoinEndline<T>(this IEnumerable<T> source) => source.Join("\n");
        public static string Join<T>(this IEnumerable<T> source, string s) => string.Join(s, source);
        public static string Join<T>(this IEnumerable<T> source, char c) => string.Join(c.ToString(), source);

        /// <summary>
        /// <see cref="Pair{T1, T2}"/> クラスのインスタンスを作成します。
        /// </summary>
        /// <typeparam name="T1">firstの型</typeparam>
        /// <typeparam name="T2">secondの型</typeparam>
        /// <param name="first">firstの値</param>
        /// <param name="second">secondの値</param>
        /// <returns>作成した<see cref="Pair{T1, T2}"/> クラスのインスタンス</returns>
        public static Pair<T1, T2> make_pair<T1, T2>(T1 first, T2 second)
            where T1 : IComparable<T1>
            where T2 : IComparable<T2>
            => new Pair<T1, T2>(first, second);

        /// <summary>aとbをスワップします。</summary>
        public static void Swap<T>(ref T a, ref T b) where T : struct
        {
            var tmp = b;
            b = a;
            a = tmp;
        }

        /// <summary>aとbの最大公約数を求めます。</summary>
        /// <returns>aとbの最大公約数</returns>
        public static long Gcd(long a, long b)
        {
            while (true)
            {
                if (a < b) Swap(ref a, ref b);
                if (a % b == 0) return b;
                var x = a;
                a = b;
                b = x % b;
            }
        }

        /// <summary>aとbの最小公倍数を求めます。</summary>
        /// <returns>aとbの最小公倍数</returns>
        public static long Lcm(long a, long b) => a / Gcd(a, b) * b;

        /// <summary>
        /// 指定した数値が素数であるかを判定します。
        /// </summary>
        /// <remarks>計算量 (sqrt(value)) </remarks>
        /// <param name="value">判定する数値</param>
        /// <returns>value が素数であるか</returns>
        public static bool IsPrime(long value)
        {
            if (value <= 1) return false;
            for (long i = 2; i * i <= value; i++)
                if (value % i == 0)
                    return false;
            return true;
        }

        /// <summary>
        /// <see cref="a"/> ^ <see cref="b"/> (mod <see cref="p"/>) を求める
        /// </summary>
        /// <returns><see cref="a"/> ^ <see cref="b"/> (mod <see cref="p"/>) の値</returns>
        public static long PowMod(long a, long b, long p)
        {
            long res = 1;
            while (b > 0)
            {
                if (b % 2 != 0) res = res * a % p;
                a = a * a % p;
                b >>= 1;
            }

            return res;
        }

        /// <summary>
        /// mod pにおけるaの逆元を求めます。
        /// </summary>
        /// <param name="a"></param>
        /// <param name="p">法</param>
        /// <returns></returns>
        public static long ModInv(long a, long p)
            => PowMod(a, p - 2, p);

        public static int DivCeil(int left, int right)
            => left / right + (left % right == 0 ? 0 : 1);

        public static long DivCeil(long left, long right)
            => left / right + (left % right == 0L ? 0L : 1L);

        /// <summary>
        /// src の順列を求めます。
        /// </summary>
        /// <typeparam name="T">型</typeparam>
        /// <param name="src">順列を求める配列</param>
        /// <returns>src の順列</returns>
        public static IEnumerable<T[]> Permutations<T>(IEnumerable<T> src)
        {
            var ret = new List<T[]>();
            Search(ret, new Stack<T>(), src.ToArray());
            return ret;
        }

        private static void Search<T>(ICollection<T[]> perms, Stack<T> stack, T[] a)
        {
            int N = a.Length;
            if (N == 0) perms.Add(stack.Reverse().ToArray());
            else
            {
                var b = new T[N - 1];
                Array.Copy(a, 1, b, 0, N - 1);
                for (int i = 0; i < a.Length; ++i)
                {
                    stack.Push(a[i]);
                    Search(perms, stack, b);
                    if (i < b.Length) b[i] = a[i];
                    stack.Pop();
                }
            }
        }

        /// <summary>
        /// 指定した条件を満たす最小の数値を返します。
        /// </summary>
        /// <param name="low">検索する数値の最小値</param>
        /// <param name="high">検索する数値の最大値</param>
        /// <param name="expression">条件</param>
        /// <returns>条件を満たす最小の数値</returns>
        public static long BinarySearch(long low, long high, Func<long, bool> expression)
        {
            while (low < high)
            {
                long middle = (high - low) / 2 + low;
                if (!expression(middle))
                    high = middle;
                else
                    low = middle + 1;
            }

            return high;
        }

        /// <summary>
        /// 指定した値以上の先頭のインデクスを返します。
        /// </summary>
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="start">開始インデクス [inclusive]</param>
        /// <param name="end">終了インデクス [exclusive]</param>
        /// <param name="value">検索する値</param>
        /// <param name="comparer">比較関数(インターフェイス)</param>
        /// <returns>指定した値以上の先頭のインデクス</returns>
        public static int LowerBound<T>(T[] arr, int start, int end, T value, IComparer<T> comparer)
        {
            int low = start;
            int high = end;
            while (low < high)
            {
                var mid = ((high - low) >> 1) + low;
                if (comparer.Compare(arr[mid], value) < 0)
                    low = mid + 1;
                else
                    high = mid;
            }

            return low;
        }

        /// <summary>
        /// 指定した値以上の先頭のインデクスを返します。
        /// </summary>
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="value">検索する値</param>
        /// <returns>指定した値以上の先頭のインデクス</returns>
        public static int LowerBound<T>(T[] arr, T value) where T : IComparable
        {
            return LowerBound(arr, 0, arr.Length, value, Comparer<T>.Default);
        }

        /// <summary>
        /// 指定した値より大きい先頭のインデクスを返します。
        /// </summary>
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="start">開始インデクス [inclusive]</param>
        /// <param name="end">終了インデクス [exclusive]</param>
        /// <param name="value">検索する値</param>
        /// <param name="comparer">比較関数(インターフェイス)</param>
        /// <returns>指定した値より大きい先頭のインデクス</returns>
        public static int UpperBound<T>(T[] arr, int start, int end, T value, IComparer<T> comparer)
        {
            int low = start;
            int high = end;
            while (low < high)
            {
                var mid = ((high - low) >> 1) + low;
                if (comparer.Compare(arr[mid], value) <= 0)
                    low = mid + 1;
                else
                    high = mid;
            }

            return low;
        }

        /// <summary>
        /// 指定した値より大きい先頭のインデクスを返します。
        /// </summary>Z
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="value">検索する値</param>
        /// <returns>指定した値より大きい先頭のインデクス</returns>
        public static int UpperBound<T>(T[] arr, T value)
        {
            return UpperBound(arr, 0, arr.Length, value, Comparer<T>.Default);
        }

        public static IEnumerable<TResult> SelectNotNull<TSource, TResult>(this IEnumerable<TSource> source,
            Func<TSource, TResult> func)
            => source.Where(val => val != null).Select(func);

        public static IEnumerable<TSource> WhereNotNull<TSource>(this IEnumerable<TSource> source)
            => source.Where(val => val != null);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static T[] ArrayOf<T>(params T[] arr) => arr;

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static List<T> ListOf<T>(params T[] arr) => new List<T>(arr);

        public static IEnumerable<TResult> Repeat<TResult>(TResult value)
        {
            while (true) yield return value;
            // ReSharper disable once IteratorNeverReturns
        }

        public static IEnumerable<TResult> Repeat<TResult>(TResult value, int count)
            => Enumerable.Repeat(value, count);

        [SuppressMessage("ReSharper", "PossibleMultipleEnumeration")]
        public static IEnumerable<TResult> Repeat<TResult>(this IEnumerable<TResult> source, int count)
        {
            if (source == null) throw new ArgumentNullException(nameof(source));
            if (count < 0) throw new ArgumentOutOfRangeException(nameof(count));
            for (int i = 0; i < count; i++)
                foreach (var v in source)
                    yield return v;
        }

        [SuppressMessage("ReSharper", "PossibleMultipleEnumeration")]
        public static IEnumerable<TResult> Repeat<TResult>(this IEnumerable<TResult> source)
        {
            if (source == null) throw new ArgumentNullException(nameof(source));
            while (true)
                foreach (var v in source)
                    yield return v;
        }

        /// <summary>
        /// 文字の配列を文字列に変換します。
        /// </summary>
        /// <param name="source">文字の配列</param>
        /// <returns>変換した文字列</returns>
        public static string AsString(this IEnumerable<char> source) => new string(source.ToArray());

        /// <summary>
        /// <see cref="source"/> の累積和を返します。
        /// </summary>
        /// <returns><see cref="source"/> の累積和</returns>
        public static IEnumerable<long> CumSum(this IEnumerable<long> source)
        {
            long sum = 0;
            foreach (var item in source)
                yield return sum += item;
        }

        /// <summary>
        /// <see cref="source"/> の累積和を返します。
        /// </summary>
        /// <returns><see cref="source"/> の累積和</returns>
        public static IEnumerable<int> CumSum(this IEnumerable<int> source)
        {
            int sum = 0;
            foreach (var item in source)
                yield return sum += item;
        }

        /// <summary>
        /// <see cref="value"/>が l以上 r未満の範囲に含まれているかを返します。
        /// </summary>
        /// <typeparam name="T">要素の型</typeparam>
        /// <param name="value">判定する値</param>
        /// <param name="l">下限の値 (含まれる)</param>
        /// <param name="r">上限の値 (含まれない)</param>
        /// <returns><see cref="value"/> が指定した範囲に含まれているか</returns>
        public static bool IsIn<T>(this T value, T l, T r)
            where T : IComparable<T>
        {
            if (l.CompareTo(r) > 0) throw new ArgumentException();
            return l.CompareTo(value) <= 0 && value.CompareTo(r) < 0;
        }

        /// <summary>
        /// <see cref="start"/> 以上 <see cref="end"/> 未満の値を <see cref="step"/> ずつ増やした結果を列挙します。
        /// </summary>
        /// <param name="start">値の下限 (含まれる)</param>
        /// <param name="end">値の上限 (含まれない)</param>
        /// <param name="step">1要素ごとに増やす値</param>
        /// <returns>範囲の結果</returns>
        public static IEnumerable<int> Range(int start, int end, int step = 1)
        {
            for (var i = start; i < end; i += step) yield return i;
        }

        /// <summary>
        /// 0 以上 <see cref="end"/> 未満の値を 1 ずつ増やした結果を列挙します。
        /// </summary>
        /// <param name="end">値の上限 (含まれない)</param>
        /// <returns>範囲の結果</returns>
        public static IEnumerable<int> Range(int end) => Range(0, end);

        /// <summary>
        /// <see cref="start"/> 以上 <see cref="end"/> 未満の値を <see cref="step"/> ずつ増やした結果を逆順に列挙します。
        /// </summary>
        /// <param name="start">値の下限 (含まれる)</param>
        /// <param name="end">値の上限 (含まれない)</param>
        /// <param name="step">1要素ごとに増やす値</param>
        /// <returns>範囲の結果</returns>
        public static IEnumerable<int> RangeReverse(int start, int end, int step = 1)
        {
            for (var i = end - 1; i >= start; i -= step) yield return i;
        }

        /// <summary>
        /// 0 以上 <see cref="end"/> 未満の値を 1 ずつ増やした結果を逆順に列挙します。
        /// </summary>
        /// <param name="end">値の上限 (含まれない)</param>
        /// <returns>範囲の結果</returns>
        public static IEnumerable<int> RangeReverse(int end) => RangeReverse(0, end);


        /// <summary>
        /// 指定した配列をコピーして昇順ソートします。（非破壊的）
        /// </summary>
        /// <typeparam name="T">ソートする配列の型</typeparam>
        /// <param name="arr">ソートする配列</param>
        /// <returns>ソートされた配列</returns>
        public static T[] Sort<T>(this T[] arr)
        {
            var array = new T[arr.Length];
            arr.CopyTo(array, 0);
            Array.Sort(array);
            return array;
        }

        /// <summary>
        /// 指定した配列をコピーして降順ソートします。（非破壊的）
        /// </summary>
        /// <typeparam name="T">ソートする配列の型</typeparam>
        /// <param name="arr">ソートする配列</param>
        /// <returns>ソートされた配列</returns>
        public static T[] SortDescending<T>(this T[] arr)
        {
            var array = new T[arr.Length];
            arr.CopyTo(array, 0);
            Array.Sort(array);
            Array.Reverse(array);
            return array;
        }

        public static double Log2(double x) => Log(x, 2);

        public static bool chmin<T>(ref T a, T b) where T : IComparable<T>
        {
            if (a.CompareTo(b) > 0)
            {
                a = b;
                return true;
            }

            return false;
        }

        public static bool chmax<T>(ref T a, T b) where T : IComparable<T>
        {
            if (a.CompareTo(b) < 0)
            {
                a = b;
                return true;
            }

            return false;
        }

        public static T Min<T>(params T[] col) => col.Min();
        public static T Max<T>(params T[] col) => col.Max();


        /// <summary>
        /// 要素数 (a, b) の、defaultValue で満たされたジャグ配列を作成します。
        /// </summary>
        /// <typeparam name="T">配列の型</typeparam>
        /// <param name="a">1次元の要素数</param>
        /// <param name="b">2次元の要素数</param>
        /// <param name="defaultValue">デフォルト値</param>
        /// <returns>指定した条件で初期化された配列</returns>
        public static T[][] JaggedArray2D<T>(int a, int b, T defaultValue = default(T))
        {
            var ret = new T[a][];
            for (int i = 0; i < a; i++)
            {
                ret[i] = Enumerable.Repeat(defaultValue, b).ToArray();
            }

            return ret;
        }

        /// <summary>
        /// 要素数 (a, b) の，defaultValue で満たされた二次元配列を作成します。
        /// </summary>
        /// <param name="a">1次元の要素数</param>
        /// <param name="b">2次元の要素数</param>
        /// <param name="defaultValue">デフォルト値</param>
        /// <typeparam name="T">配列の型</typeparam>
        public static T[,] Array2D<T>(int a, int b, T defaultValue = default(T))
        {
            var ret = new T[a, b];
            for (int i = 0; i < a; i++)
            for (int j = 0; j < b; j++)
                ret[i, j] = defaultValue;
            return ret;
        }

        /// <summary>
        /// ジャグ配列を2次元配列に変換します。配列の各要素の長さがすべて同じである必要があります。
        /// </summary>
        /// <param name="array">ジャグ配列</param>
        /// <typeparam name="T">二次元配列</typeparam>
        public static T[,] To2DArray<T>(this T[][] array)
        {
            if (!array.Any()) return new T[0, 0];

            int len = array[0].Length;
            if (array.Any(x => x.Length != len))
                throw new ArgumentException("array の各要素の長さが異なります。", nameof(array));

            var ret = new T[array.Length, len];
            for (int i = 0; i < array.Length; i++)
            {
                for (int j = 0; j < len; j++)
                {
                    ret[i, j] = array[i][j];
                }
            }

            return ret;
        }

//        public static vector<T> ToVector<T>(this IEnumerable<T> source) => new vector<T>(source);
    }

    public static class Input
    {
        private const char _separator = ' ';
        private static readonly Queue<string> _input = new Queue<string>();
        private static readonly StreamReader sr =
#if FILE
            new StreamReader("in.txt");
#else
            new StreamReader(Console.OpenStandardInput());
#endif

        public static string ReadLine => sr.ReadLine();
        public static string ReadStr => Read;

        public static string Read
        {
            get
            {
                if (_input.Count != 0) return _input.Dequeue();

                // ReSharper disable once PossibleNullReferenceException
                var tmp = sr.ReadLine().Split(_separator);
                foreach (var val in tmp)
                {
                    _input.Enqueue(val);
                }

                return _input.Dequeue();
            }
        }

        public static int ReadInt => int.Parse(Read);
        public static long ReadLong => long.Parse(Read);
        public static double ReadDouble => double.Parse(Read);
        public static string[] StrArray() => ReadLine.Split(' ');
        public static int[] IntArray() => ReadLine.Split(' ').Select(int.Parse).ToArray();
        public static long[] LongArray() => ReadLine.Split(' ').Select(long.Parse).ToArray();

        public static string[] StrArray(int n)
        {
            var ret = new string[n];
            for (long i = 0; i < n; ++i) ret[i] = Read;
            return ret;
        }

        public static int[] IntArray(int n, int offset = 0, bool sorted = false)
        {
            var ret = StrArray(n).Select(x => int.Parse(x) + offset).ToArray();
            if (sorted) Array.Sort(ret);
            return ret;
        }

        public static long[] LongArray(int n, long offset = 0, bool sorted = false)
        {
            var ret = StrArray(n).Select(x => long.Parse(x) + offset).ToArray();
            if (sorted) Array.Sort(ret);
            return ret;
        }

        public static string[,] Str2DArray(int n, int m)
        {
            var ret = new string[n, m];
            for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                ret[i, j] = ReadStr;

            return ret;
        }

        public static int[,] Int2DArray(int n, int m, int offset = 0)
        {
            var ret = new int[n, m];
            for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                ret[i, j] = ReadInt + offset;
            return ret;
        }

        public static long[,] Long2DArray(int n, int m, long offset = 0)
        {
            var ret = new long[n, m];
            for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++)
                ret[i, j] = ReadLong + offset;
            return ret;
        }

        public static Tuple<string[], string[]> StrArrays2(int n)
        {
            var ret1 = new string[n];
            var ret2 = new string[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadStr;
                ret2[i] = ReadStr;
            }

            return Tuple.Create(ret1, ret2);
        }

        public static Tuple<int[], int[]> IntArrays2(int n, int offset1 = 0, int offset2 = 0)
        {
            var ret1 = new int[n];
            var ret2 = new int[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadInt + offset1;
                ret2[i] = ReadInt + offset2;
            }

            return Tuple.Create(ret1, ret2);
        }

        public static Tuple<long[], long[]> LongArrays2(int n, long offset1 = 0, long offset2 = 0)
        {
            var ret1 = new long[n];
            var ret2 = new long[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadLong + offset1;
                ret2[i] = ReadLong + offset2;
            }

            return Tuple.Create(ret1, ret2);
        }

        public static Tuple<string[], string[], string[]> StrArrays3(int n)
        {
            var ret1 = new string[n];
            var ret2 = new string[n];
            var ret3 = new string[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadStr;
                ret2[i] = ReadStr;
            }

            return Tuple.Create(ret1, ret2, ret3);
        }

        public static Tuple<int[], int[], int[]> IntArrays3(int n, int offset1 = 0, int offset2 = 0, int offset3 = 0)
        {
            var ret1 = new int[n];
            var ret2 = new int[n];
            var ret3 = new int[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadInt + offset1;
                ret2[i] = ReadInt + offset2;
                ret3[i] = ReadInt + offset3;
            }

            return Tuple.Create(ret1, ret2, ret3);
        }

        public static Tuple<long[], long[], long[]> LongArrays3(int n, long offset1 = 0, long offset2 = 0,
            long offset3 = 0)
        {
            var ret1 = new long[n];
            var ret2 = new long[n];
            var ret3 = new long[n];
            for (int i = 0; i < n; i++)
            {
                ret1[i] = ReadLong + offset1;
                ret2[i] = ReadLong + offset2;
                ret3[i] = ReadLong + offset3;
            }

            return Tuple.Create(ret1, ret2, ret3);
        }

        private static bool TypeEquals<T, U>() => typeof(T) == typeof(U);
        private static T ChangeType<T, U>(U a) => (T) System.Convert.ChangeType(a, typeof(T));

        private static T Convert<T>(string s) => TypeEquals<T, int>() ? ChangeType<T, int>(int.Parse(s))
            : TypeEquals<T, long>() ? ChangeType<T, long>(long.Parse(s))
            : TypeEquals<T, double>() ? ChangeType<T, double>(double.Parse(s))
            : TypeEquals<T, char>() ? ChangeType<T, char>(s[0])
            : ChangeType<T, string>(s);

        public static bool In<T>(out T a)
        {
            try
            {
                a = Convert<T>(Read);
                return true;
            }
            catch
            {
                a = default(T);
                return false;
            }
        }

        public static bool In<T, U>(out T a, out U b)
        {
            try
            {
                var ar = StrArray(2);
                a = Convert<T>(ar[0]);
                b = Convert<U>(ar[1]);
                return true;
            }
            catch
            {
                a = default(T);
                b = default(U);
                return false;
            }
        }

        public static bool In<T, U, V>(out T a, out U b, out V c)
        {
            try
            {
                var ar = StrArray(3);
                a = Convert<T>(ar[0]);
                b = Convert<U>(ar[1]);
                c = Convert<V>(ar[2]);
                return true;
            }
            catch
            {
                a = default(T);
                b = default(U);
                c = default(V);
                return false;
            }
        }

        public static bool In<T, U, V, W>(out T a, out U b, out V c, out W d)
        {
            try
            {
                var ar = StrArray(4);
                a = Convert<T>(ar[0]);
                b = Convert<U>(ar[1]);
                c = Convert<V>(ar[2]);
                d = Convert<W>(ar[3]);
                return true;
            }
            catch
            {
                a = default(T);
                b = default(U);
                c = default(V);
                d = default(W);
                return false;
            }
        }

        public static bool In<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
        {
            try
            {
                var ar = StrArray(5);
                a = Convert<T>(ar[0]);
                b = Convert<U>(ar[1]);
                c = Convert<V>(ar[2]);
                d = Convert<W>(ar[3]);
                e = Convert<X>(ar[4]);
                return true;
            }
            catch
            {
                a = default(T);
                b = default(U);
                c = default(V);
                d = default(W);
                e = default(X);
                return false;
            }
        }
    }

    public static class Output
    {
        public static void print<T>(T t) => Console.WriteLine(t);
        public static void print(params object[] o) => Console.WriteLine(o.Join(" "));

        public static void PrintBool(bool val, string yes = "Yes", string no = "No")
            => Console.WriteLine(val ? yes : no);

        public static void PrintYn(bool val) => PrintBool(val);
        public static void PrintYN(bool val) => PrintBool(val, "YES", "NO");
        public static void PrintPossible(bool val) => PrintBool(val, "Possible", "Impossible");
        public static void PrintYay(bool val) => PrintBool(val, "Yay!", ":(");

        public static void PrintDebug(params object[] args)
            => Console.Error.WriteLine(string.Join(" ", args));

        /// <summary>
        /// setter で設定された値を標準出力に出力します。
        /// </summary>
        public static object cout { set { Console.WriteLine(value); } }

        /// <summary>
        /// Local環境のみ，setter で設定された値を標準出力に出力します。
        /// </summary>
        public static object dout
        {
            set
            {
#if LOCAL
                Console.WriteLine(value);
#endif
            }
        }

        /// <summary>
        /// setter で設定された値を標準エラー出力に出力します。
        /// </summary>
        public static object cerr { set { Console.Error.WriteLine(value); } }

        public const string endl = "\n";
    }

    public class Program
    {
        public static void Main(string[] args)
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false};
            Console.SetOut(sw);
            new Solver().Main();

            Console.Out.Flush();
        }
    }

    [DebuggerDisplay("({first}, {second})")]
    public class Pair<T1, T2> : IComparable<Pair<T1, T2>>, IEquatable<Pair<T1, T2>>
        where T1 : IComparable<T1>
        where T2 : IComparable<T2>
    {
        public Pair(T1 first, T2 second)
        {
            this.first = first;
            this.second = second;
        }

        public T1 first;
        public T2 second;

        public int CompareTo(Pair<T1, T2> other)
        {
            if (ReferenceEquals(this, other)) return 0;
            if (ReferenceEquals(null, other)) return 1;
            var firstComparison = first.CompareTo(other.first);
            return firstComparison != 0 ? firstComparison : second.CompareTo(other.second);
        }

        public override string ToString() => $"({first}, {second})";

        public bool Equals(Pair<T1, T2> other)
        {
            if (ReferenceEquals(null, other)) return false;
            if (ReferenceEquals(this, other)) return true;
            return EqualityComparer<T1>.Default.Equals(first, other.first) &&
                   EqualityComparer<T2>.Default.Equals(second, other.second);
        }

        public override bool Equals(object obj)
        {
            if (ReferenceEquals(null, obj)) return false;
            if (ReferenceEquals(this, obj)) return true;
            return obj.GetType() == GetType() && Equals((Pair<T1, T2>) obj);
        }

        public override int GetHashCode()
        {
            unchecked
            {
                return (EqualityComparer<T1>.Default.GetHashCode(first) * 397) ^
                       EqualityComparer<T2>.Default.GetHashCode(second);
            }
        }
    }

    #endregion
}