Problem:
Score : 300 points

Problem Statement

We have a square grid with H rows and W columns.
Snuke wants to write 0 or 1 in each of the squares.
Here, all of the following conditions have to be satisfied:

For every row, the smaller of the following is A: the number of 0s contained in the row, and the number of 1s contained in the row. (If these two numbers are equal, “the smaller” should be read as “either”.)

For every column, the smaller of the following is B: the number of 0s contained in the column, and the number of 1s contained in the column.

Determine if these conditions can be satisfied by writing 0 or 1 in each of the squares. If the answer is yes, show one way to fill the squares so that the conditions are satisfied.

Constraints

1 \leq H,W \leq 1000

0 \leq A

2 \times A \leq W

0 \leq B

2 \times B \leq H

All values in input are integers.

Input

Input is given from Standard Input in the following format:

H W A B

Output

If the conditions cannot be satisfied by writing 0 or 1 in each of the squares, print -1.

If the conditions can be satisfied, print one way to fill the squares so that the conditions are satisfied, in the following format:

s_{11}s_{12}\cdots s_{1W}
s_{21}s_{22}\cdots s_{2W}
\vdots
s_{H1}s_{H2}\cdots s_{HW}

Here s_{ij} is the digit written in the square at the i-th row from the top and the j-th column from the left in the grid.

If multiple solutions exist, printing any of them will be accepted.

Sample Input 1

3 3 1 1

Sample Output 1

100
010
001

Every row contains two 0s and one 1, so the first condition is satisfied.
Also, every column contains two 0s and one 1, so the second condition is satisfied.

Sample Input 2

1 5 2 0

Sample Output 2

01010

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
namespace Program
{
    public static class AGC038A
    {
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var H = NN;
            var W = NN;
            var A = NN;
            var B = NN;
            if (A == 0)
            {
                for (var i = 0; i < B; i++)
                {
                    for (var j = 0; j < W; j++)
                    {
                        Console.Write("1");
                    }
                    Console.WriteLine();
                }
                for (var i = B; i < H; i++)
                {
                    for (var j = 0; j < W; j++)
                    {
                        Console.Write("0");
                    }
                    Console.WriteLine();
                }
                return;
            }
            if (B == 0)
            {
                for (var i = 0; i < H; i++)
                {
                    for (var j = 0; j < A; j++)
                    {
                        Console.Write("1");
                    }
                    for (var j = A; j < W; j++)
                    {
                        Console.Write("0");
                    }
                    Console.WriteLine();
                }
                return;
            }
            if (H % B == 0 && W % A == 0)
            {
                var acnt = W / A;
                var bcnt = H / B;
                if (acnt != bcnt)
                {
                    Console.WriteLine("No");
                    return;
                }
                for (var i = 0; i < H; i++)
                {
                    for (var j = 0; j < W; j++)
                    {
                        if ((i / B) * A <= j && j < ((i / B) * A + A))
                        {
                            Console.Write("1");
                        }
                        else
                        {
                            Console.Write("0");
                        }
                    }
                    Console.WriteLine();
                }
            }
            else
            {
                Console.WriteLine("No");
            }
        }
        static public void Main(string[] args) { if (args.Length == 0) { var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw); } var t = new Thread(Solve, 134217728); t.Start(); t.Join(); Console.Out.Flush(); }
        static class Console_
        {
            static Queue<string> param = new Queue<string>();
            public static string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        }
        static long NN => long.Parse(Console_.NextString());
        static double ND => double.Parse(Console_.NextString());
        static string NS => Console_.NextString();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static void RevSort<T>(T[] l) where T : IComparable { Array.Sort(l, (x, y) => y.CompareTo(x)); }
        static void RevSort<T>(T[] l, Comparison<T> comp) where T : IComparable { Array.Sort(l, (x, y) => comp(y, x)); }
        static IEnumerable<long> Primes(long x) { if (x < 2) yield break; yield return 2; var halfx = x / 2; var table = new bool[halfx + 1]; var max = (long)(Math.Sqrt(x) / 2); for (long i = 1; i <= max; ++i) { if (table[i]) continue; var add = 2 * i + 1; yield return add; for (long j = 2 * i * (i + 1); j <= halfx; j += add) table[j] = true; } for (long i = max + 1; i <= halfx; ++i) if (!table[i] && 2 * i + 1 <= x) yield return 2 * i + 1; }
        static IEnumerable<long> Factors(long x) { if (x < 2) yield break; while (x % 2 == 0) { x /= 2; yield return 2; } var max = (long)Math.Sqrt(x); for (long i = 3; i <= max; i += 2) while (x % i == 0) { x /= i; yield return i; } if (x != 1) yield return x; }
        static IEnumerable<long> Divisor(long x) { if (x < 1) yield break; var max = (long)Math.Sqrt(x); for (long i = 1; i <= max; ++i) { if (x % i != 0) continue; yield return i; if (i != x / i) yield return x / i; } }
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
        static long GCD(long a, long b) { while (b > 0) { var tmp = b; b = a % b; a = tmp; } return a; }
        static long LCM(long a, long b) => a * b / GCD(a, b);
        static Mat<T> Pow<T>(Mat<T> x, long y) => Mat<T>.Pow(x, y);
        static Mod Pow(Mod x, long y) { Mod a = 1; while (y != 0) { if ((y & 1) == 1) a.Mul(x); x.Mul(x); y >>= 1; } return a; }
        static long Pow(long x, long y) { long a = 1; while (y != 0) { if ((y & 1) == 1) a *= x; x *= x; y >>= 1; } return a; }
        static List<long> _fact = new List<long>() { 1 };
        static void _B(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
        static long Comb(long n, long k) { _B(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k]; }
        static long Perm(long n, long k) { _B(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)]; }
        static Func<TR> Lambda<TR>(Func<Func<TR>, TR> f) { Func<TR> t = () => default(TR); return t = () => f(t); }
        static Func<T1, TR> Lambda<T1, TR>(Func<T1, Func<T1, TR>, TR> f) { Func<T1, TR> t = x1 => default(TR); return t = x1 => f(x1, t); }
        static Func<T1, T2, TR> Lambda<T1, T2, TR>(Func<T1, T2, Func<T1, T2, TR>, TR> f) { Func<T1, T2, TR> t = (x1, x2) => default(TR); return t = (x1, x2) => f(x1, x2, t); }
        static Func<T1, T2, T3, TR> Lambda<T1, T2, T3, TR>(Func<T1, T2, T3, Func<T1, T2, T3, TR>, TR> f) { Func<T1, T2, T3, TR> t = (x1, x2, x3) => default(TR); return t = (x1, x2, x3) => f(x1, x2, x3, t); }
        static Func<T1, T2, T3, T4, TR> Lambda<T1, T2, T3, T4, TR>(Func<T1, T2, T3, T4, Func<T1, T2, T3, T4, TR>, TR> f) { Func<T1, T2, T3, T4, TR> t = (x1, x2, x3, x4) => default(TR); return t = (x1, x2, x3, x4) => f(x1, x2, x3, x4, t); }
        static List<T> LCS<T>(T[] s, T[] t) where T : IEquatable<T> { int sl = s.Length, tl = t.Length; var dp = new int[sl + 1, tl + 1]; for (var i = 0; i < sl; i++) for (var j = 0; j < tl; j++) dp[i + 1, j + 1] = s[i].Equals(t[j]) ? dp[i, j] + 1 : Max(dp[i + 1, j], dp[i, j + 1]); { var r = new List<T>(); int i = sl, j = tl; while (i > 0 && j > 0) if (s[--i].Equals(t[--j])) r.Add(s[i]); else if (dp[i, j + 1] > dp[i + 1, j]) ++j; else ++i; r.Reverse(); return r; } }
        static long LIS<T>(T[] array, bool strict) { var l = new List<T>(); foreach (var e in array) { var i = l.BinarySearch(e); if (i < 0) i = ~i; else if (!strict) ++i; if (i == l.Count) l.Add(e); else l[i] = e; } return l.Count; }
        class PQ<T> where T : IComparable
        {
            List<T> h; Comparison<T> c; public T Peek => h[0]; public int Count => h.Count;
            public PQ(int cap, Comparison<T> c, bool asc = true) { h = new List<T>(cap); this.c = asc ? c : (x, y) => c(y, x); }
            public PQ(Comparison<T> c, bool asc = true) { h = new List<T>(); this.c = asc ? c : (x, y) => c(y, x); }
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            public void Push(T v) { var i = h.Count; h.Add(v); while (i > 0) { var ni = (i - 1) / 2; if (c(v, h[ni]) >= 0) break; h[i] = h[ni]; i = ni; } h[i] = v; }
            public T Pop() { var r = h[0]; var v = h[h.Count - 1]; h.RemoveAt(h.Count - 1); if (h.Count == 0) return r; var i = 0; while (i * 2 + 1 < h.Count) { var i1 = i * 2 + 1; var i2 = i * 2 + 2; if (i2 < h.Count && c(h[i1], h[i2]) > 0) i1 = i2; if (c(v, h[i1]) <= 0) break; h[i] = h[i1]; i = i1; } h[i] = v; return r; }
        }
        class PQ<TK, TV> where TK : IComparable
        {
            PQ<Tuple<TK, TV>> q; public Tuple<TK, TV> Peek => q.Peek; public int Count => q.Count;
            public PQ(int cap, Comparison<TK> c, bool asc = true) { q = new PQ<Tuple<TK, TV>>(cap, (x, y) => c(x.Item1, y.Item1), asc); }
            public PQ(Comparison<TK> c, bool asc = true) { q = new PQ<Tuple<TK, TV>>((x, y) => c(x.Item1, y.Item1), asc); }
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            public void Push(TK k, TV v) => q.Push(Tuple.Create(k, v));
            public Tuple<TK, TV> Pop() => q.Pop();
        }
        public class UF
        {
            long[] d;
            public UF(long s) { d = Repeat(-1L, s).ToArray(); }
            public bool Unite(long x, long y) { x = Root(x); y = Root(y); if (x != y) { if (d[y] < d[x]) { var t = y; y = x; x = t; } d[x] += d[y]; d[y] = x; } return x != y; }
            public bool IsSame(long x, long y) => Root(x) == Root(y);
            public long Root(long x) => d[x] < 0 ? x : d[x] = Root(d[x]);
            public long Count(long x) => -d[Root(x)];
        }
        struct Mod : IEquatable<Mod>, IEquatable<long>
        {
            static public long _mod = 1000000007; long v;
            public Mod(long x) { if (x < _mod && x >= 0) v = x; else if ((v = x % _mod) < 0) v += _mod; }
            static public implicit operator Mod(long x) => new Mod(x);
            static public implicit operator long(Mod x) => x.v;
            public void Add(Mod x) { if ((v += x.v) >= _mod) v -= _mod; }
            public void Sub(Mod x) { if ((v -= x.v) < 0) v += _mod; }
            public void Mul(Mod x) => v = (v * x.v) % _mod;
            public void Div(Mod x) => v = (v * Inverse(x.v)) % _mod;
            static public Mod operator +(Mod x, Mod y) { var t = x.v + y.v; return t >= _mod ? new Mod { v = t - _mod } : new Mod { v = t }; }
            static public Mod operator -(Mod x, Mod y) { var t = x.v - y.v; return t < 0 ? new Mod { v = t + _mod } : new Mod { v = t }; }
            static public Mod operator *(Mod x, Mod y) => x.v * y.v;
            static public Mod operator /(Mod x, Mod y) => x.v * Inverse(y.v);
            static public bool operator ==(Mod x, Mod y) => x.v == y.v;
            static public bool operator !=(Mod x, Mod y) => x.v != y.v;
            static public long Inverse(long x) { long b = _mod, r = 1, u = 0, t = 0; while (b > 0) { var q = x / b; t = u; u = r - q * u; r = t; t = b; b = x - q * b; x = t; } return r < 0 ? r + _mod : r; }
            public bool Equals(Mod x) => v == x.v;
            public bool Equals(long x) => v == x;
            public override bool Equals(object x) => x == null ? false : Equals((Mod)x);
            public override int GetHashCode() => v.GetHashCode();
            public override string ToString() => v.ToString();
            static List<Mod> _fact = new List<Mod>() { 1 };
            static void B(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
            static public Mod Comb(long n, long k) { B(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k]; }
            static public Mod Perm(long n, long k) { B(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)]; }
        }
        struct Mat<T>
        {
            T[,] m;
            public Mat(T[,] v) { m = (T[,])v.Clone(); }
            static public implicit operator Mat<T>(T[,] v) => new Mat<T>(v);
            public T this[int r, int c] { get { return m[r, c]; } set { m[r, c] = value; } }
            static public Mat<T> operator +(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)x; return tm; }
            static public Mat<T> operator +(Mat<T> a, Mat<T> b) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)b[r, c]; return tm; }
            static public Mat<T> operator -(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)x; return tm; }
            static public Mat<T> operator -(Mat<T> a, Mat<T> b) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)b[r, c]; return tm; }
            static public Mat<T> operator *(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] *= (dynamic)x; return tm; }
            static public Mat<T> operator *(Mat<T> a, Mat<T> b) { var nr = a.m.GetLength(0); var nc = b.m.GetLength(1); var tm = new T[nr, nc]; for (int i = 0; i < nr; ++i) for (int j = 0; j < nc; ++j) tm[i, j] = (dynamic)0; for (int r = 0; r < nr; ++r) for (int c = 0; c < nc; ++c) for (int i = 0; i < a.m.GetLength(1); ++i) tm[r, c] += a[r, i] * (dynamic)b[i, c]; return tm; }
            static public Mat<T> Pow(Mat<T> x, long y) { var n = x.m.GetLength(0); var t = (Mat<T>)new T[n, n]; for (int i = 0; i < n; ++i) for (int j = 0; j < n; ++j) t[i, j] = (dynamic)(i == j ? 1 : 0); while (y != 0) { if ((y & 1) == 1) t *= x; x *= x; y >>= 1; } return t; }
        }
        class Tree
        {
            long N; int l; List<long>[] p; int[] d; long[][] pr; long r; Tuple<long, long, int>[] e; Tuple<long, long>[] b; bool lca; bool euler; bool bfs;
            public Tree(List<long>[] p_, long r_) { N = p_.Length; p = p_; r = r_; lca = false; euler = false; }
            public Tuple<long, long>[] BFSRoot() { if (!bfs) { var nb = new List<Tuple<long, long>>(); var q = new Queue<long>(); var d = new bool[N]; nb.Add(Tuple.Create(r, -1L)); d[r] = true; q.Enqueue(r); while (q.Count > 0) { var w = q.Dequeue(); foreach (var i in p[w]) { if (d[i]) continue; d[i] = true; q.Enqueue(i); nb.Add(Tuple.Create(i, w)); } } b = nb.ToArray(); bfs = true; } return b; }
            public Tuple<long, long>[] BFSLeaf() => BFSRoot().Reverse().ToArray();
            public Tuple<long, long, int>[] Euler() { if (!euler) { var ne = new List<Tuple<long, long, int>>(); var s = new Stack<Tuple<long, long>>(); var d = new bool[N]; d[r] = true; s.Push(Tuple.Create(r, -1L)); while (s.Count > 0) { var w = s.Peek(); var ad = true; foreach (var i in p[w.Item1]) { if (d[i]) continue; d[i] = true; ad = false; s.Push(Tuple.Create(i, w.Item1)); } if (!ad || p[w.Item1].Count == 1) ne.Add(Tuple.Create(w.Item1, w.Item2, 1)); if (ad) { s.Pop(); ne.Add(Tuple.Create(w.Item1, w.Item2, -1)); } } e = ne.ToArray(); euler = true; } return e; }
            public long LCA(long u, long v) { if (!lca) { l = 0; while (N > (1 << l)) l++; d = new int[N]; pr = Repeat(0, l).Select(_ => new long[N]).ToArray(); d[r] = 0; pr[0][r] = -1; var q = new Stack<long>(); q.Push(r); while (q.Count > 0) { var w = q.Pop(); foreach (var i in p[w]) { if (i == pr[0][w]) continue; q.Push(i); d[i] = d[w] + 1; pr[0][i] = w; } } for (var k = 0; k + 1 < l; k++) for (var w = 0; w < N; w++) if (pr[k][w] < 0) pr[k + 1][w] = -1; else pr[k + 1][w] = pr[k][pr[k][w]]; lca = true; } if (d[u] > d[v]) { var t = u; u = v; v = t; } for (var k = 0; k < l; k++) if ((((d[v] - d[u]) >> k) & 1) != 0) v = pr[k][v]; if (u == v) return u; for (var k = l - 1; k >= 0; k--) if (pr[k][u] != pr[k][v]) { u = pr[k][u]; v = pr[k][v]; } return pr[0][u]; }
        }
        class Graph
        {
            int n; List<Tuple<int, int, long>> pathList; Dictionary<int, long>[] vtxPath; long INF = (long.MaxValue >> 1) - 1;
            public Graph(long _n)
            {
                n = (int)_n;
                pathList = new List<Tuple<int, int, long>>();
                vtxPath = Repeat(0, n).Select(_ => new Dictionary<int, long>()).ToArray();
            }
            public void AddPath(long a, long b, long c)
            {
                pathList.Add(Tuple.Create((int)a, (int)b, c));
                vtxPath[a][(int)b] = vtxPath[a].ContainsKey((int)b) ? Min(vtxPath[a][(int)b], c) : c;
            }
            public long[,] WarshallFloyd()
            {
                var ret = new long[n, n];
                for (var i = 0; i < n; i++)
                    for (var j = 0; j < n; j++)
                        ret[i, j] = vtxPath[i].ContainsKey(j) ? vtxPath[i][j] : INF;
                for (var k = 0; k < n; k++)
                    for (var i = 0; i < n; i++)
                        for (var j = 0; j < n; j++)
                            ret[i, j] = Min(ret[i, j], ret[i, k] + ret[k, j]);
                return ret;
            }
            public Tuple<long[], int?[], bool[]> BellmanFord(long s)
            {
                var dist = Repeat(INF, n).ToArray();
                var pred = new int?[n];
                var neg = new bool[n];
                dist[s] = 0;
                for (var i = 1; i < n; i++)
                    foreach (var path in pathList)
                        if (dist[path.Item2] > (dist[path.Item1] == INF ? INF : dist[path.Item1] + path.Item3))
                        {
                            dist[path.Item2] = dist[path.Item1] + path.Item3;
                            pred[path.Item2] = path.Item1;
                        }
                for (var i = 0; i < n; i++)
                    foreach (var path in pathList)
                        if (dist[path.Item2] > (dist[path.Item1] == INF ? INF : dist[path.Item1] + path.Item3) || neg[path.Item1])
                        {
                            dist[path.Item2] = dist[path.Item1] + path.Item3;
                            neg[path.Item2] = true;
                        }
                return Tuple.Create(dist, pred, neg);
            }
            public Tuple<long[], int?[]> Dijkstra(long s)
            {
                var dist = Repeat(long.MaxValue >> 2, n).ToArray();
                var pred = new int?[n];
                dist[s] = 0;
                var q = new PQ<long, int>();
                q.Push(0, (int)s);
                while (q.Count > 0)
                {
                    var u = q.Pop().Item2;
                    foreach (var path in vtxPath[u])
                    {
                        var v = path.Key;
                        var alt = dist[u] + path.Value;
                        if (dist[v] > alt)
                        {
                            dist[v] = alt;
                            pred[v] = u;
                            q.Push(alt, v);
                        }
                    }
                }
                return Tuple.Create(dist, pred);
            }
        }
        class BT<T> where T : IComparable
        {
            class Node { public Node l; public Node r; public T v; public bool b; public int c; }
            Comparison<T> c; Node r; bool ch; T lm;
            public BT(Comparison<T> _c) { c = _c; }
            public BT() : this((x, y) => x.CompareTo(y)) { }
            bool R(Node n) => n != null && !n.b;
            bool B(Node n) => n != null && n.b;
            long C(Node n) => n?.c ?? 0;
            Node RtL(Node n) { Node m = n.r, t = m.l; m.l = n; n.r = t; n.c -= m.c - (t?.c ?? 0); m.c += n.c - (t?.c ?? 0); return m; }
            Node RtR(Node n) { Node m = n.l, t = m.r; m.r = n; n.l = t; n.c -= m.c - (t?.c ?? 0); m.c += n.c - (t?.c ?? 0); return m; }
            Node RtLR(Node n) { n.l = RtL(n.l); return RtR(n); }
            Node RtRL(Node n) { n.r = RtR(n.r); return RtL(n); }
            public void Add(T x) { r = A(r, x); r.b = true; }
            Node A(Node n, T x) { if (n == null) { ch = true; return new Node() { v = x, c = 1 }; } if (c(x, n.v) < 0) n.l = A(n.l, x); else n.r = A(n.r, x); n.c++; return Bl(n); }
            Node Bl(Node n) { if (!ch) return n; if (!B(n)) return n; if (R(n.l) && R(n.l.l)) { n = RtR(n); n.l.b = true; } else if (R(n.l) && R(n.l.r)) { n = RtLR(n); n.l.b = true; } else if (R(n.r) && R(n.r.l)) { n = RtRL(n); n.r.b = true; } else if (R(n.r) && R(n.r.r)) { n = RtL(n); n.r.b = true; } else ch = false; return n; }
            public void Remove(T x) { r = Rm(r, x); if (r != null) r.b = true; }
            Node Rm(Node n, T x) { if (n == null) { ch = false; return n; } n.c--; var r = c(x, n.v); if (r < 0) { n.l = Rm(n.l, x); return BlL(n); } if (r > 0) { n.r = Rm(n.r, x); return BlR(n); } if (n.l == null) { ch = n.b; return n.r; } n.l = RmM(n.l); n.v = lm; return BlL(n); }
            Node RmM(Node n) { n.c--; if (n.r != null) { n.r = RmM(n.r); return BlR(n); } lm = n.v; ch = n.b; return n.l; }
            Node BlL(Node n) { if (!ch) return n; if (B(n.r) && R(n.r.l)) { var b = n.b; n = RtRL(n); n.b = b; n.l.b = true; ch = false; } else if (B(n.r) && R(n.r.r)) { var b = n.b; n = RtL(n); n.b = b; n.r.b = true; n.l.b = true; ch = false; } else if (B(n.r)) { ch = n.b; n.b = true; n.r.b = false; } else { n = RtL(n); n.b = true; n.l.b = false; n.l = BlL(n.l); ch = false; } return n; }
            Node BlR(Node n) { if (!ch) return n; if (B(n.l) && R(n.l.r)) { var b = n.b; n = RtLR(n); n.b = b; n.r.b = true; ch = false; } else if (B(n.l) && R(n.l.l)) { var b = n.b; n = RtR(n); n.b = b; n.l.b = true; n.r.b = true; ch = false; } else if (B(n.l)) { ch = n.b; n.b = true; n.l.b = false; } else { n = RtR(n); n.b = true; n.r.b = false; n.r = BlR(n.r); ch = false; } return n; }
            public T this[long i] { get { return At(r, i); } }
            T At(Node n, long i) { if (n == null) return default(T); if (n.l == null) if (i == 0) return n.v; else return At(n.r, i - 1); if (n.l.c == i) return n.v; if (n.l.c > i) return At(n.l, i); return At(n.r, i - n.l.c - 1); }
            public bool Have(T x) { var t = FindUpper(x); return t < C(r) && At(r, t).CompareTo(x) == 0; }
            public long FindUpper(T x, bool allowSame = true) => allowSame ? FL(r, x) + 1 : FU(r, x);
            long FU(Node n, T x) { if (n == null) return 0; var r = c(x, n.v); if (r < 0) return FU(n.l, x); return C(n.l) + 1 + FU(n.r, x); }
            public long FindLower(T x, bool allowSame = true) { var t = FL(r, x); if (allowSame) return t + 1 < C(r) && At(r, t + 1).CompareTo(x) == 0 ? t + 1 : t < 0 ? C(r) : t; return t < 0 ? C(r) : t; }
            long FL(Node n, T x) { if (n == null) return -1; var r = c(x, n.v); if (r > 0) return C(n.l) + 1 + FL(n.r, x); return FL(n.l, x); }
            public T Min() { Node n = r, p = null; while (n != null) { p = n; n = n.l; } return p == null ? default(T) : p.v; }
            public T Max() { Node n = r, p = null; while (n != null) { p = n; n = n.r; } return p == null ? default(T) : p.v; }
            public bool Any() => r != null;
            public long Count() => C(r);
            public IEnumerable<T> List() => L(r);
            IEnumerable<T> L(Node n) { if (n == null) yield break; foreach (var i in L(n.l)) yield return i; yield return n.v; foreach (var i in L(n.r)) yield return i; }
        }
        class Dict<K, V> : Dictionary<K, V>
        {
            Func<K, V> d;
            public Dict(Func<K, V> _d) { d = _d; }
            public Dict() : this(_ => default(V)) { }
            new public V this[K i] { get { V v; return TryGetValue(i, out v) ? v : base[i] = d(i); } set { base[i] = value; } }
        }
        class Deque<T>
        {
            T[] b; int o, c;
            public int Count;
            public T this[int i] { get { return b[gi(i)]; } set { b[gi(i)] = value; } }
            public Deque(int cap = 16) { b = new T[c = cap]; }
            int gi(int i) { if (i >= c) throw new Exception(); var r = o + i; return r >= c ? r - c : r; }
            public void PushFront(T x) { if (Count == c) e(); if (--o < 0) o += b.Length; b[o] = x; ++Count; }
            public T PopFront() { if (Count-- == 0) throw new Exception(); var r = b[o++]; if (o >= c) o -= c; return r; }
            public T Front => b[o];
            public void PushBack(T x) { if (Count == c) e(); var i = o + Count++; b[i >= c ? i - c : i] = x; }
            public T PopBack() { if (Count == 0) throw new Exception(); return b[gi(--Count)]; }
            public T Back => b[gi(Count - 1)];
            void e() { T[] nb = new T[c << 1]; if (o > c - Count) { var l = b.Length - o; Array.Copy(b, o, nb, 0, l); Array.Copy(b, 0, nb, l, Count - l); } else Array.Copy(b, o, nb, 0, Count); b = nb; o = 0; c <<= 1; }
            public void Insert(int i, T x) { if (i > Count) throw new Exception(); this.PushFront(x); for (int j = 0; j < i; j++) this[j] = this[j + 1]; this[i] = x; }
            public T RemoveAt(int i) { if (i < 0 || i >= Count) throw new Exception(); var r = this[i]; for (int j = i; j > 0; j--) this[j] = this[j - 1]; this.PopFront(); return r; }
        }
        class SegTree<T>
        {
            int n; T ti; Func<T, T, T> f; T[] dat;
            public SegTree(long _n, T _ti, Func<T, T, T> _f) { n = 1; while (n < _n) n <<= 1; ti = _ti; f = _f; dat = Repeat(ti, n << 1).ToArray(); for (var i = n - 1; i > 0; i--) dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]); }
            public SegTree(List<T> l, T _ti, Func<T, T, T> _f) : this(l.Count, _ti, _f) { for (var i = 0; i < l.Count; i++) dat[n + i] = l[i]; for (var i = n - 1; i > 0; i--) dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]); }
            public void Update(long i, T v) { dat[i += n] = v; while ((i >>= 1) > 0) dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]); }
            public T Query(long l, long r) { var vl = ti; var vr = ti; for (long li = n + l, ri = n + r; li < ri; li >>= 1, ri >>= 1) { if ((li & 1) == 1) vl = f(vl, dat[li++]); if ((ri & 1) == 1) vr = f(dat[--ri], vr); } return f(vl, vr); }
            public T this[long idx] { get { return dat[idx + n]; } set { Update(idx, value); } }
        }
        class LazySegTree<T, E>
        {
            int n, height; T ti; E ei; Func<T, T, T> f; Func<T, E, T> g; Func<E, E, E> h; T[] dat; E[] laz;
            public LazySegTree(long _n, T _ti, E _ei, Func<T, T, T> _f, Func<T, E, T> _g, Func<E, E, E> _h) { n = 1; height = 0; while (n < _n) { n <<= 1; ++height; } ti = _ti; ei = _ei; f = _f; g = _g; h = _h; dat = Repeat(ti, n << 1).ToArray(); laz = Repeat(ei, n << 1).ToArray(); for (var i = n - 1; i > 0; i--) dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]); }
            public LazySegTree(List<T> l, T _ti, E _ei, Func<T, T, T> _f, Func<T, E, T> _g, Func<E, E, E> _h) : this(l.Count, _ti, _ei, _f, _g, _h) { for (var i = 0; i < l.Count; i++) dat[n + i] = l[i]; for (var i = n - 1; i > 0; i--) dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]); }
            T Reflect(long i) => laz[i].Equals(ei) ? dat[i] : g(dat[i], laz[i]);
            void Eval(long i) { if (laz[i].Equals(ei)) return; laz[(i << 1) | 0] = h(laz[(i << 1) | 0], laz[i]); laz[(i << 1) | 1] = h(laz[(i << 1) | 1], laz[i]); dat[i] = Reflect(i); laz[i] = ei; }
            void Thrust(long i) { for (var j = height; j > 0; j--) Eval(i >> j); }
            void Recalc(long i) { while ((i >>= 1) > 0) dat[i] = f(Reflect((i << 1) | 0), Reflect((i << 1) | 1)); }
            public void Update(long l, long r, E v) { Thrust(l += n); Thrust(r += n - 1); for (long li = l, ri = r + 1; li < ri; li >>= 1, ri >>= 1) { if ((li & 1) == 1) { laz[li] = h(laz[li], v); ++li; } if ((ri & 1) == 1) { --ri; laz[ri] = h(laz[ri], v); } } Recalc(l); Recalc(r); }
            public T Query(long l, long r) { Thrust(l += n); Thrust(r += n - 1); var vl = ti; var vr = ti; for (long li = l, ri = r + 1; li < ri; li >>= 1, ri >>= 1) { if ((li & 1) == 1) vl = f(vl, Reflect(li++)); if ((ri & 1) == 1) vr = f(Reflect(--ri), vr); } return f(vl, vr); }
            public T this[long idx] { get { Thrust(idx += n); return dat[idx] = Reflect(idx); } set { Thrust(idx += n); dat[idx] = value; laz[idx] = ei; Recalc(idx); } }
        }
    }
}