Problem:
Score : 600 points

Problem Statement

Given are a prime number p and a sequence of p integers a_0, \ldots, a_{p-1} consisting of zeros and ones.

Find a polynomial of degree at most p-1, f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + \ldots + b_0, satisfying the following conditions:

For each i (0 \leq i \leq p-1), b_i is an integer such that 0 \leq b_i \leq p-1.

For each i (0 \leq i \leq p-1), f(i) \equiv a_i \pmod p.

Constraints

2 \leq p \leq 2999

p is a prime number.

0 \leq a_i \leq 1

Input

Input is given from Standard Input in the following format:

p
a_0 a_1 \ldots a_{p-1}

Output

Print b_0, b_1, \ldots, b_{p-1} of a polynomial f(x) satisfying the conditions, in this order, with spaces in between.

It can be proved that a solution always exists. If multiple solutions exist, any of them will be accepted.

Sample Input 1

2
1 0

Sample Output 1

1 1

f(x) = x + 1 satisfies the conditions, as follows:

f(0) = 0 + 1 = 1 \equiv 1 \pmod 2

f(1) = 1 + 1 = 2 \equiv 0 \pmod 2

Sample Input 2

3
0 0 0

Sample Output 2

0 0 0

f(x) = 0 is also valid.

Sample Input 3

5
0 1 0 1 0

Sample Output 3

0 2 0 1 3

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

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

namespace Program
{
    public static class ABC137F
    {
        static public void Solve()
        {
            var p = NN;
            var aList = NNList(p);
            Mod._mod = p;
            var ans = new Mod[p];
            for (var i = 0; i < p; i++)
            {
                if (aList[i] == 1)
                {
                    var mul = new Mod[p];
                    mul[0] = 1;
                    ans[0] -= 1;
                    for (var j = 1; j < p; j++)
                    {
                        mul[j] = mul[j - 1] * -i;
                        ans[j] -= mul[j] * Mod.Comb(p - 1, j);
                    }
                    ans[p - 1] += 1;
                }
            }
            Console.WriteLine(string.Join(" ", ans.Reverse()));

        }
        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>();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void RevSort<T>(T[] l) where T : IComparable { Array.Sort(l, (x, y) => y.CompareTo(x)); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void RevSort<T>(T[] l, Comparison<T> comp) where T : IComparable { Array.Sort(l, (x, y) => comp(y, x)); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = 0; while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static Mat<T> Pow<T>(Mat<T> x, long y) => Mat<T>.Pow(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static T Pow<T>(T x, long y) { T a = (dynamic)1; while (y != 0) { if ((y & 1) == 1) a *= (dynamic)x; x *= (dynamic)x; y >>= 1; } return a; }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void _B(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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]; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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)]; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static Func<TR> Lambda<TR>(Func<Func<TR>, TR> f) { Func<TR> t = () => default(TR); return t = () => f(t); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        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); }
        class PQ<T> where T : IComparable
        {
            List<T> h; Comparison<T> c; public T Peek => h[0]; public int Count => h.Count;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, Comparison<T> c, bool asc = true) { h = new List<T>(cap); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<T> c, bool asc = true) { h = new List<T>(); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<TK> c, bool asc = true) { q = new PQ<Tuple<TK, TV>>((x, y) => c(x.Item1, y.Item1), asc); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Push(TK k, TV v) => q.Push(Tuple.Create(k, v));
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<TK, TV> Pop() => q.Pop();
        }
        public class UF
        {
            long[] d;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public UF(long s) { d = Repeat(-1L, s).ToArray(); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool IsSame(long x, long y) => Root(x) == Root(y);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public long Root(long x) => d[x] < 0 ? x : d[x] = Root(d[x]);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public long Count(long x) => -d[Root(d[x])];
        }
        struct Mod : IEquatable<object>
        {
            static public long _mod = 1000000007; long _val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mod(long x) { if (x < _mod && x >= 0) _val = x; else if ((_val = x % _mod) < 0) _val += _mod; }
            static public implicit operator Mod(long x) => new Mod(x);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator long(Mod x) => x._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator +(Mod x, Mod y) { var t = x._val + y._val; return t >= _mod ? new Mod { _val = t - _mod } : new Mod { _val = t }; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator -(Mod x, Mod y) { var t = x._val - y._val; return t < 0 ? new Mod { _val = t + _mod } : new Mod { _val = t }; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator *(Mod x, Mod y) => x._val * y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator /(Mod x, Mod y) => x._val * Inverse(y._val);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator ==(Mod x, Mod y) => x._val == y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator !=(Mod x, Mod y) => x._val != y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            bool IEquatable<object>.Equals(object obj) => obj == null ? false : Equals((Mod)obj);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override bool Equals(object obj) => obj == null ? false : Equals((Mod)obj);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Equals(Mod obj) => obj == null ? false : _val == obj._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override int GetHashCode() => _val.GetHashCode();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override string ToString() => _val.ToString();
            static List<Mod> _fact = new List<Mod>() { 1 };
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static void B(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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]; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mat(T[,] v) { m = (T[,])v.Clone(); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; } }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tree(List<long>[] p_, long r_) { N = p_.Length; p = p_; r = r_; lca = false; euler = false; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<long, long>[] BFSLeaf() => BFSRoot().Reverse().ToArray();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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 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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public BT(Comparison<T> _c) { c = _c; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public BT() : this((x, y) => x.CompareTo(y)) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            bool R(Node n) => n != null && !n.b;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            bool B(Node n) => n != null && n.b;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            Node RtLR(Node n) { n.l = RtL(n.l); return RtR(n); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            Node RtRL(Node n) { n.r = RtR(n.r); return RtL(n); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Add(T x) { r = A(r, x); r.b = true; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Remove(T x) { r = Rm(r, x); if (r != null) r.b = true; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); } }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Have(T x) { var t = FindUpper(x); return t.Item1 && t.Item2.CompareTo(x) == 0; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<bool, T> FindUpper(T x, bool findSame = true) { var v = FU(r, x, findSame); return v == null ? Tuple.Create(false, default(T)) : v; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            Tuple<bool, T> FU(Node n, T x, bool s) { if (n == null) return null; var r = c(x, n.v); if (r < 0) { var v = FU(n.l, x, s); return v == null ? Tuple.Create(true, n.v) : v; } if (r > 0 || !s && r == 0) return FU(n.r, x, s); return Tuple.Create(true, n.v); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<bool, T> FindLower(T x, bool findSame = true) { var v = FL(r, x, findSame); return v == null ? Tuple.Create(false, default(T)) : v; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            Tuple<bool, T> FL(Node n, T x, bool s) { if (n == null) return null; var r = c(x, n.v); if (r < 0 || !s && r == 0) return FL(n.l, x, s); if (r > 0) { var v = FL(n.r, x, s); return v == null ? Tuple.Create(true, n.v) : v; } return Tuple.Create(true, n.v); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Min() { Node n = r, p = null; while (n != null) { p = n; n = n.l; } return p == null ? default(T) : p.v; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Max() { Node n = r, p = null; while (n != null) { p = n; n = n.r; } return p == null ? default(T) : p.v; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Any() => r != null;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public int Count() => r?.c ?? 0;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public IEnumerable<T> List() => L(r);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Dict(Func<K, V> _d) { d = _d; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; } }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Deque(int cap = 16) { b = new T[c = cap]; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            int gi(int i) { if (i >= c) throw new Exception(); var r = o + i; return r >= c ? r - c : r; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void PushFront(T x) { if (Count == c) e(); if (--o < 0) o += b.Length; b[o] = x; ++Count; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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];
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void PushBack(T x) { if (Count == c) e(); var i = o + Count++; b[i >= c ? i - c : i] = x; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T PopBack() { if (Count == 0) throw new Exception(); return b[gi(--Count)]; }
            public T Back => b[gi(Count - 1)];
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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 ImplicitTreep<T, E> where T : IEquatable<T> where E : IEquatable<E>
        {
            class Node
            {
                public E value;
                public T acc;
                public E lazy;
                public uint priority;
                public int cnt;
                public bool rev;
                public Node l;
                public Node r;
                [MethodImpl(MethodImplOptions.AggressiveInlining)]
                public Node(E val, E ei, T ti)
                {
                    priority = xorshift;
                    value = val;
                    acc = ti;
                    lazy = ei;
                    cnt = 1;
                    rev = false;
                    l = null;
                    r = null;
                }
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            Node CreateNode(E val) => new Node(val, ei, ti);

            T ti;
            E ei;
            Func<T, T, T> f;
            Func<T, E, T> g;
            Func<E, E, E> h;
            Node root = null;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public ImplicitTreep(T _ti, E _ei, Func<T, T, T> _f, Func<T, E, T> _g, Func<E, E, E> _h)
            {
                ti = _ti;
                ei = _ei;
                f = _f;
                g = _g;
                h = _h;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            int Count(Node n) => n?.cnt ?? 0;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            T Acc(Node n) => n == null ? ti : n.acc;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void UpdateCount(Node n)
            {
                if (n != null) n.cnt = 1 + Count(n.l) + Count(n.r);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void UpdateAcc(Node n)
            {
                if (n != null) n.acc = f(Acc(n.l), f(g(ti, n.value), Acc(n.r)));
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Pushup(Node n)
            {
                UpdateCount(n);
                UpdateAcc(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Pushdown(Node n)
            {
                if (n?.rev ?? false)
                {
                    n.rev = false;
                    var t = n.l; n.l = n.r; n.r = t;
                    if (n.l != null) n.l.rev ^= true;
                    if (n.r != null) n.r.rev ^= true;
                }
                if (n != null && !n.lazy.Equals(ei))
                {
                    Action<Node> apply = t =>
                    {
                        if (t == null) return;
                        t.lazy = h(t.lazy, n.lazy);
                        t.acc = g(t.acc, n.lazy);
                    };
                    apply(n.l);
                    apply(n.r);
                    n.value = h(n.value, n.lazy);
                    n.lazy = ei;
                }
                Pushup(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Split(Node n, long key, out Node l, out Node r)
            {
                if (n == null)
                {
                    l = r = null;
                    return;
                }
                Pushdown(n);
                var implicitKey = Count(n.l) + 1;
                if (key < implicitKey)
                {
                    Split(n.l, key, out l, out n.l);
                    r = n;
                }
                else
                {
                    Split(n.r, key - implicitKey, out n.r, out r);
                    l = n;
                }
                Pushup(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Insert(ref Node n, long key, Node x)
            {
                if (n == null)
                {
                    n = x;
                    return;
                }
                Pushdown(n);
                if (Count(n.l) == key || x.priority > n.priority)
                {
                    Split(n, key, out x.l, out x.r);
                    n = x;
                }
                else
                {
                    if (Count(n.l) > key)
                    {
                        Insert(ref n.l, key, x);
                    }
                    else
                    {
                        Insert(ref n.r, key - Count(n.l) - 1, x);
                    }
                }
                Pushup(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Merge(out Node n, Node l, Node r)
            {
                Pushdown(l);
                Pushdown(r);
                if (l == null || r == null)
                {
                    n = l != null ? l : r;
                }
                else if (l.priority > r.priority)
                {
                    Merge(out l.r, l.r, r);
                    n = l;
                }
                else
                {
                    Merge(out r.l, l, r.l);
                    n = r;
                }
                Pushup(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Erase(ref Node n, long key)
            {
                Pushdown(n);
                if (Count(n.l) == key)
                {
                    Merge(out n, n.l, n.r);
                }
                else
                {
                    if (Count(n.l) > key)
                    {
                        Erase(ref n.l, key);
                    }
                    else
                    {
                        Erase(ref n.r, key - Count(n.l) - 1);
                    }
                }
                Pushup(n);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Update(Node n, long l, long r, E x)
            {
                Node n1, n2, n3;
                Split(n, l, out n1, out n2);
                Split(n2, r - l, out n2, out n3);
                n2.lazy = h(n2.lazy, x);
                Merge(out n2, n2, n3);
                Merge(out n, n1, n2);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            T Query(Node n, long l, long r)
            {
                Node n1, n2, n3;
                Split(n, l, out n1, out n2);
                Split(n2, r - l, out n2, out n3);
                var ret = Acc(n2);
                Merge(out n2, n2, n3);
                Merge(out n, n1, n2);
                return ret;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            long Find(Node n, T x, long offset, bool left = true)
            {
                if (f(Acc(n), x).Equals(x))
                {
                    return -1;
                }
                else
                {
                    if (left)
                    {
                        if (n.l != null && !f(n.l.acc, x).Equals(x))
                        {
                            return Find(n.l, x, offset, left);
                        }
                        else
                        {
                            return (!f(g(ti, n.value), x).Equals(x)) ? offset + Count(n.l) : Find(n.r, x, offset + Count(n.l) + 1, left);
                        }
                    }
                    else
                    {
                        if (n.r != null && !f(n.r.acc, x).Equals(x))
                        {
                            return Find(n.r, x, offset + Count(n.l) + 1, left);
                        }
                        else
                        {
                            return (!f(g(ti, n.value), x).Equals(x)) ? offset + Count(n.l) : Find(n.l, x, offset, left);
                        }
                    }
                }
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Reverse(Node n, long l, long r)
            {
                if (l >= r) return;
                Node n1, n2, n3;
                Split(n, l, out n1, out n2);
                Split(n2, r - l, out n2, out n3);
                n2.rev ^= true;
                Merge(out n2, n2, n3);
                Merge(out n, n1, n2);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Rotate(Node n, long l, long m, long r)
            {
                Reverse(n, l, r);
                Reverse(n, l, l + r - m);
                Reverse(n, l + r - m, r);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Insert(long idx, E x) => Insert(ref root, idx, CreateNode(x));
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Add(E x) => Insert(Count(), x);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Update(long l, long r, E x) => Update(root, l, r, x);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Query(long l, long r) => Query(root, l, r);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public long Find(long l, long r, T x, bool left = true)
            {
                Node n1, n2, n3;
                Split(root, l, out n1, out n2);
                Split(n2, r - l, out n2, out n3);
                var ret = Find(n2, x, l, left);
                Merge(out n2, n2, n3);
                Merge(out root, n1, n2);
                return ret < 0 ? r : ret;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Erase(long idx) => Erase(ref root, idx);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Reverse(long l, long r) => Reverse(root, l, r);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Rotate(long l, long m, long r) => Rotate(root, l, m, r);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public int Count() => Count(root);
            public E this[long idx]
            {
                get
                {
                    Node n1, n2, n3;
                    Split(root, idx + 1, out n1, out n2);
                    Split(n1, idx, out n1, out n3);
                    var ret = n3.value;
                    Merge(out n1, n1, n3);
                    Merge(out root, n1, n2);
                    return ret;
                }
                private set { }
            }
        }

        class SegTree<T>
        {
            int n; T ti; Func<T, T, T> f; T[] dat;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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();
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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]);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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]);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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> where T : IEquatable<T> where E : IEquatable<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;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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();
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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]);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            T Reflect(long i) => laz[i].Equals(ei) ? dat[i] : g(dat[i], laz[i]);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Thrust(long i) { for (var j = height; j > 0; j--) Eval(i >> j); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            void Recalc(long i) { while ((i >>= 1) > 0) dat[i] = f(Reflect((i << 1) | 0), Reflect((i << 1) | 1)); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            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); }
            }
        }
    }
}