Problem:
Score: 600 points

Problem Statement

E869120 is initially standing at the origin (0, 0) in a two-dimensional plane.

He has N engines, which can be used as follows:

When E869120 uses the i-th engine, his X- and Y-coordinate change by x_i and y_i, respectively. In other words, if E869120 uses the i-th engine from coordinates (X, Y), he will move to the coordinates (X + x_i, Y + y_i).

E869120 can use these engines in any order, but each engine can be used at most once. He may also choose not to use some of the engines.

He wants to go as far as possible from the origin.
Let (X, Y) be his final coordinates. Find the maximum possible value of \sqrt{X^2 + Y^2}, the distance from the origin.

Constraints

1 \leq N \leq 100

-1 \ 000 \ 000 \leq x_i \leq 1 \ 000 \ 000

-1 \ 000 \ 000 \leq y_i \leq 1 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
x_2 y_2
: :
x_N y_N

Output

Print the maximum possible final distance from the origin, as a real value.
Your output is considered correct when the relative or absolute error from the true answer is at most 10^{-10}.

Sample Input 1

3
0 10
5 -5
-5 -5

Sample Output 1

10.000000000000000000000000000000000000000000000000

The final distance from the origin can be 10 if we use the engines in one of the following three ways:

Use Engine 1 to move to (0, 10).

Use Engine 2 to move to (5, -5), and then use Engine 3 to move to (0, -10).

Use Engine 3 to move to (-5, -5), and then use Engine 2 to move to (0, -10).

The distance cannot be greater than 10, so the maximum possible distance is 10.

Sample Input 2

5
1 1
1 0
0 1
-1 0
0 -1

Sample Output 2

2.828427124746190097603377448419396157139343750753

The maximum possible final distance is 2 \sqrt{2} = 2.82842....
One of the ways to achieve it is:

Use Engine 1 to move to (1, 1), and then use Engine 2 to move to (2, 1), and finally use Engine 3 to move to (2, 2).

Sample Input 3

5
1 1
2 2
3 3
4 4
5 5

Sample Output 3

21.213203435596425732025330863145471178545078130654

If we use all the engines in the order 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5, we will end up at (15, 15), with the distance 15 \sqrt{2} = 21.2132... from the origin.

Sample Input 4

3
0 0
0 1
1 0

Sample Output 4

1.414213562373095048801688724209698078569671875376

There can be useless engines with (x_i, y_i) = (0, 0).

Sample Input 5

1
90447 91000

Sample Output 5

128303.000000000000000000000000000000000000000000000000

Note that there can be only one engine.

Sample Input 6

2
96000 -72000
-72000 54000

Sample Output 6

120000.000000000000000000000000000000000000000000000000

There can be only two engines, too.

Sample Input 7

10
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20

Sample Output 7

148.660687473185055226120082139313966514489855137208

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 System.Runtime.CompilerServices;

namespace V
{
    partial class Solver
    {
        public void Solve()
        {
            var n = Read;
            var vs = new List<Vector>();
            foreach (var i in C.Loop(n))
            {
                var x = Read;
                var y = Read;
                if (x == 0 && y == 0)
                    continue;

                vs.Add(new Vector() { x = x, y = y, d = Math.Atan2(y, x) });
            }
            vs = vs.OrderBy(x => x.d).ToList();
            vs = vs.Concat(vs.Select(x => new Vector() { x = x.x, y = x.y, d = x.d + Math.PI * 2d })).ToList();

            var res = 0L;

            foreach (var v in vs)
            {
                var xx = 0L;
                var yy = 0L;
                foreach (var a in vs.Where(x => v.d <= x.d && x.d < v.d + Math.PI))
                {
                    xx += a.x;
                    yy += a.y;
                }

                res = Math.Max(res, xx * xx + yy * yy);
            }

            Wr(Math.Sqrt(res));
            //YesNo(SolveBool());
        }

        public class Vector
        {
            public long x;
            public long y;
            public double d;
        }

    }
}
namespace V
{
    class StartingPoint
    {
        static void Main(string[] args)
        {
            try
            {
                var streamReader = args.Any() ? new StreamReader(args[0]) : new StreamReader(Console.OpenStandardInput());
                var streamWriter = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
                var scanner = new Scanner(streamReader);
                var printer = new Printer(streamWriter);
                var solver = new Solver(scanner, printer);
                solver.Solve();
                streamWriter.Flush();
            }
            catch (Exception e)
            {
                Console.WriteLine(e);
                if (args.Any() == false)
                    throw e;
            }

            if (args.Any())
                Console.ReadKey();
        }
    }
    partial class Solver
    {
        public Solver(Scanner sc, Printer pr) { this.sc = sc; this.pr = pr; }
        private readonly Scanner sc;
        private readonly Printer pr;

        private IEnumerable<int> Loop(int n) => C.Loop(n);
        private IEnumerable<long> Loop(long n) => C.Loop(n);

        private int RdInt => sc.Int;
        private int ReadInt => sc.Int;
        private long Rd => sc.Long;
        private long Read => sc.Long;
        private long ReadLong => sc.Long;
        private double RdDouble => sc.Double;
        private double ReadDouble => sc.Double;
        private string Str => sc.Str;
        private string RdStr => sc.Str;
        private string ReadStr => sc.Str;
        private int[] ArrInt(int n) => sc.Ints(n);
        private int[] ArrInt(long n) => sc.Ints(n);
        private long[] Arr(int n) => sc.Longs(n);
        private long[] Arr(long n) => sc.Longs(n);
        private long[] ArrLong(int n) => sc.Longs(n);
        private long[] ArrLong(long n) => sc.Longs(n);
        private double[] ArrDouble(int n) => sc.Doubles(n);
        private double[] ArrDouble(long n) => sc.Doubles(n);
        private string[] ArrStr(int n) => sc.Strs(n);
        private string[] ArrStr(long n) => sc.Strs(n);

        private void Wr(string s) => pr.Write(s);
        private void Wr(object obj) => pr.Write(obj);
        private void Wr<T>(IEnumerable<T> ts) => pr.Write(ts);
        private void Wr(params object[] objs) => pr.Write(objs);
        private void Write(string s) => pr.Write(s);
        private void Write(object obj) => pr.Write(obj);
        private void Write<T>(IEnumerable<T> ts) => pr.Write(ts);
        private void Write(params object[] objs) => pr.Write(objs);
        private void YesNo(bool b) => Write(b ? "Yes" : "No");
        private void YESNO(bool b) => Write(b ? "YES" : "NO");
    }
    class Scanner
    {
        private readonly TextReader reader;
        public Scanner(TextReader reader) { this.reader = reader; }
        private Queue<string> strQueue = new Queue<string>();

        public string Str
        {
            get
            {
                if (strQueue.Count > 0)
                    return strQueue.Dequeue();

                string[] strs = null;
                do
                {
                    strs = reader.ReadLine().Split(new[] { ' ' }, StringSplitOptions.RemoveEmptyEntries);
                } while (strs.Any() == false);

                foreach (var s in strs.Skip(1))
                    strQueue.Enqueue(s);

                return strs[0];
            }
        }

        public int Int => int.Parse(this.Str);
        public long Long => long.Parse(this.Str);
        public double Double => double.Parse(this.Str);

        public static bool TypeEquals<T1, T2>() => typeof(T1).Equals(typeof(T2));
        public static T1 ChangeTypes<T1, T2>(T2 t2) => (T1)System.Convert.ChangeType(t2, typeof(T1));
        public static T1 Convert<T1>(string s) =>
            TypeEquals<T1, int>() ? ChangeTypes<T1, int>(int.Parse(s)) :
            TypeEquals<T1, long>() ? ChangeTypes<T1, long>(long.Parse(s)) :
            TypeEquals<T1, double>() ? ChangeTypes<T1, double>(int.Parse(s)) :
            TypeEquals<T1, char>() ? ChangeTypes<T1, char>(s[0]) : ChangeTypes<T1, string>(s);

        public Pair<TX, TY> P2<TX, TY>() => new Pair<TX, TY>(Convert<TX>(this.Str), Convert<TY>(this.Str));
        public Pair3<TX, TY, TZ> P3<TX, TY, TZ>() => new Pair3<TX, TY, TZ>(Convert<TX>(this.Str), Convert<TY>(this.Str), Convert<TZ>(this.Str));
        public Pair4<TX, TY, TZ, TW> P4<TX, TY, TZ, TW>() => new Pair4<TX, TY, TZ, TW>(Convert<TX>(this.Str), Convert<TY>(this.Str), Convert<TZ>(this.Str), Convert<TW>(this.Str));
    }
    static class ScannerExtension
    {
        public static int[] Ints(this Scanner scanner, int n) => scanner.Ints((long)n);
        public static int[] Ints(this Scanner scanner, long n) => scanner.ScanStrs<int>(n).ToArray();
        public static long[] Longs(this Scanner scanner, int n) => scanner.Longs((long)n);
        public static long[] Longs(this Scanner scanner, long n) => scanner.ScanStrs<long>(n).ToArray();
        public static double[] Doubles(this Scanner scanner, int n) => scanner.Doubles((long)n);
        public static double[] Doubles(this Scanner scanner, long n) => scanner.ScanStrs<double>(n).ToArray();
        public static string[] Strs(this Scanner scanner, int n) => scanner.Strs((long)n);
        public static string[] Strs(this Scanner scanner, long n) => scanner.ScanStrs<string>(n).ToArray();
        private static IEnumerable<T> ScanStrs<T>(this Scanner scanner, long n) { for (long i = 0; i < n; i++) yield return Scanner.Convert<T>(scanner.Str); }

        public static Pair<TX, TY>[] Pairs<TX, TY>(this Scanner scanner, int n) => scanner.Pairs<TX, TY>((long)n);
        public static Pair<TX, TY>[] Pairs<TX, TY>(this Scanner scanner, long n) => scanner.ScanPairs<TX, TY>(n).ToArray();
        public static Pair<long, long>[] Pairs(this Scanner scanner, int n) => scanner.Pairs((long)n);
        public static Pair<long, long>[] Pairs(this Scanner scanner, long n) => scanner.ScanPairs<long, long>(n).ToArray();
        public static Pair<int, int>[] PairsInt(this Scanner scanner, int n) => scanner.PairsInt((long)n);
        public static Pair<int, int>[] PairsInt(this Scanner scanner, long n) => scanner.ScanPairs<int, int>(n).ToArray();
        private static IEnumerable<Pair<TX, TY>> ScanPairs<TX, TY>(this Scanner scanner, long n) { for (long i = 0; i < n; i++) yield return scanner.P2<TX, TY>(); }

        public static Pair3<TX, TY, TZ>[] Pairs3<TX, TY, TZ>(this Scanner scanner, int n) => scanner.Pairs3<TX, TY, TZ>((long)n);
        public static Pair3<TX, TY, TZ>[] Pairs3<TX, TY, TZ>(this Scanner scanner, long n) => scanner.ScanPairs3<TX, TY, TZ>(n).ToArray();
        public static Pair3<long, long, long>[] Pairs3(this Scanner scanner, int n) => scanner.Pairs3((long)n);
        public static Pair3<long, long, long>[] Pairs3(this Scanner scanner, long n) => scanner.ScanPairs3<long, long, long>(n).ToArray();
        public static Pair3<int, int, int>[] Pairs3Int(this Scanner scanner, int n) => scanner.Pairs3Int((long)n);
        public static Pair3<int, int, int>[] Pairs3Int(this Scanner scanner, long n) => scanner.ScanPairs3<int, int, int>(n).ToArray();
        private static IEnumerable<Pair3<TX, TY, TZ>> ScanPairs3<TX, TY, TZ>(this Scanner scanner, long n) { for (long i = 0; i < n; i++) yield return scanner.P3<TX, TY, TZ>(); }

        public static Pair4<TX, TY, TZ, TW>[] Pairs4<TX, TY, TZ, TW>(this Scanner scanner, int n) => scanner.Pairs4<TX, TY, TZ, TW>((long)n);
        public static Pair4<TX, TY, TZ, TW>[] Pairs4<TX, TY, TZ, TW>(this Scanner scanner, long n) => scanner.ScanPairs4<TX, TY, TZ, TW>(n).ToArray();
        public static Pair4<long, long, long, long>[] Pairs4(this Scanner scanner, int n) => scanner.Pairs4((long)n);
        public static Pair4<long, long, long, long>[] Pairs4(this Scanner scanner, long n) => scanner.ScanPairs4<long, long, long, long>(n).ToArray();
        public static Pair4<int, int, int, int>[] Pairs4Int(this Scanner scanner, int n) => scanner.Pairs4Int((long)n);
        public static Pair4<int, int, int, int>[] Pairs4Int(this Scanner scanner, long n) => scanner.ScanPairs4<int, int, int, int>(n).ToArray();
        private static IEnumerable<Pair4<TX, TY, TZ, TW>> ScanPairs4<TX, TY, TZ, TW>(this Scanner scanner, long n) { for (long i = 0; i < n; i++) yield return scanner.P4<TX, TY, TZ, TW>(); }
    }
    class Pair<TX, TY> { public TX X { get; } public TY Y { get; } public Pair(TX x, TY y) { this.X = x; this.Y = y; } }
    class Pair : Pair<long, long> { public Pair(long x, long y) : base(x, y) { } }
    class PairInt : Pair<int, int> { public PairInt(int x, int y) : base(x, y) { } }
    class Pair3<TX, TY, TZ> { public TX X { get; } public TY Y { get; } public TZ Z { get; } public Pair3(TX x, TY y, TZ z) { this.X = x; this.Y = y; this.Z = z; } }
    class Pair3 : Pair3<long, long, long> { public Pair3(long x, long y, long z) : base(x, y, z) { } }
    class Pair3Int : Pair3<int, int, int> { public Pair3Int(int x, int y, int z) : base(x, y, z) { } }
    class Pair4<TX, TY, TZ, TW> { public TX X { get; } public TY Y { get; } public TZ Z { get; } public TW W { get; } public Pair4(TX x, TY y, TZ z, TW w) { this.X = x; this.Y = y; this.Z = z; this.W = w; } }
    class Pair4 : Pair4<long, long, long, long> { public Pair4(long x, long y, long z, long w) : base(x, y, z, w) { } }
    class Pair4Int : Pair4<int, int, int, int> { public Pair4Int(int x, int y, int z, int w) : base(x, y, z, w) { } }
    class Printer
    {
        private readonly TextWriter writer;
        public Printer(TextWriter writer) { this.writer = writer; }
        public void Write(string obj) { writer.WriteLine(obj); }
        public void Write(object obj) { writer.WriteLine(obj); }
        public void Write<T>(IEnumerable<T> ts) { writer.WriteLine(string.Join(" ", ts)); }
        public void Write(params object[] objs) { writer.WriteLine(string.Join(" ", objs)); }
    }
    static class Extension
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static bool SafeAdd<T>(this HashSet<T> ts, T t)
        {
            if (ts.Contains(t))
            {
                return false;
            }
            else
            {
                ts.Add(t);
                return false;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static bool SafeRemove<T>(this HashSet<T> ts, T t)
        {
            if (ts.Contains(t))
            {
                ts.Remove(t);
                return true;
            }
            else
            {
                return false;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void SafeSet<T>(this Dictionary<T, long> ts, T t, long value)
        {
            if (ts.ContainsKey(t))
                ts[t] = value;
            else
                ts.Add(t, value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void SafePlus<T>(this Dictionary<T, long> ts, T t, long value)
        {
            if (ts.ContainsKey(t))
                ts[t] += value;
            else
                ts.Add(t, value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void SafeSub<T>(this Dictionary<T, long> ts, T t, long value)
        {
            if (ts.ContainsKey(t))
                ts[t] -= value;
            else
                ts.Add(t, value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void SafeMult<T>(this Dictionary<T, long> ts, T t, long value)
        {
            if (ts.ContainsKey(t))
                ts[t] *= value;
            else
                ts.Add(t, value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void SafeDiv<T>(this Dictionary<T, long> ts, T t, long value)
        {
            if (ts.ContainsKey(t))
                ts[t] /= value;
            else
                ts.Add(t, value);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static HashSet<T> ToHashSet<T>(this IEnumerable<T> ts) => new HashSet<T>(ts.Distinct());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long ToDigit(this char c) => (long)(c - '0');
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long ToSmallAbcIndex(this char c) => (long)(c - 'a');
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long ToLargeAbcIndex(this char c) => (long)(c - 'A');
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Count<T1, T2>(this IGrouping<T1, T2> gs) => gs.LongCount();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static string ToStr(this IEnumerable<char> cs) => new string(cs.ToArray());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long ToLong(this IEnumerable<char> s)
        {
            var basis = 1L;
            var res = 0L;
            foreach (var c in s)
            {
                var d = c.ToSmallAbcIndex() + 1;
                res += d * basis;
                basis *= 27;
            }
            return res;
        }
        //[MethodImpl(MethodImplOptions.AggressiveInlining)]
        //public static bool TryMin<T>(this ref T current, T newer) where T : struct, IComparable<T>
        //{
        //    if (current.CompareTo(newer) <= 0)
        //        return false;

        //    current = newer;
        //    return true;
        //}
        //[MethodImpl(MethodImplOptions.AggressiveInlining)]
        //public static bool TryMax<T>(this ref T current, T newer) where T : struct, IComparable<T>
        //{
        //    if (current.CompareTo(newer) >= 0)
        //        return false;

        //    current = newer;
        //    return true;
        //}
    }
    class C
    {
        public class SegmentTree<T>
        {
            private readonly int valueCount;
            private readonly int baseCount;
            private readonly int baseIndex;
            private readonly T[] nodes;
            private readonly Func<T, T, T> func;
            private readonly T defaultValue;

            public SegmentTree(IReadOnlyList<T> ts, Func<T, T, T> func, T filling = default(T))
            {
                this.func = func;
                this.defaultValue = filling;
                valueCount = ts.Count;
                baseCount = 1;
                while (valueCount > baseCount)
                    baseCount <<= 1;
                nodes = new T[baseCount * 2 - 1];
                baseIndex = baseCount - 1;

                for (int i = 0; i < ts.Count; i++)
                    nodes[baseIndex + i] = ts[i];
                for (int i = ts.Count; i < baseCount; i++)
                    nodes[baseIndex + i] = filling;

                for (int i = baseIndex - 1; i >= 0; i--)
                    nodes[i] = func.Invoke(nodes[i * 2 + 1], nodes[i * 2 + 2]);
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Update(int index, T t)
            {
                var i = baseIndex + index;
                nodes[i] = t;
                while (i > 0)
                {
                    i -= 1;
                    i /= 2;
                    nodes[i] = func.Invoke(nodes[i * 2 + 1], nodes[i * 2 + 2]);
                }
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Query(int leftIndex, int rightNextIndex)
            {
                T left = defaultValue;
                T right = defaultValue;

                int l = leftIndex + baseCount - 1;
                int r = rightNextIndex + baseCount - 1;
                for (; l < r; l >>= 1, r >>= 1)
                {
                    if ((l & 1) == 0)
                    {
                        left = func.Invoke(left, nodes[l]);
                    }

                    if ((r & 1) == 0)
                    {
                        r--;
                        right = func.Invoke(right, nodes[r]);
                    }
                }

                return func.Invoke(left, right);
            }
        }
        public class SegmentTree
        {
            public static SegmentTree<int> Sum(IReadOnlyList<int> values) => new SegmentTree<int>(values, (x, y) => x + y);
            public static SegmentTree<long> Sum(IReadOnlyList<long> values) => new SegmentTree<long>(values, (x, y) => x + y);
            public static SegmentTree<double> Sum(IReadOnlyList<double> values) => new SegmentTree<double>(values, (x, y) => x + y);

            public static SegmentTree<int> Mult(IReadOnlyList<int> values) => new SegmentTree<int>(values, (x, y) => x * y, 1);
            public static SegmentTree<long> Mult(IReadOnlyList<long> values) => new SegmentTree<long>(values, (x, y) => x * y, 1);
            public static SegmentTree<double> Mult(IReadOnlyList<double> values) => new SegmentTree<double>(values, (x, y) => x * y, 1);

            public static SegmentTree<int> Min(IReadOnlyList<int> values) => new SegmentTree<int>(values, Math.Min, int.MaxValue);
            public static SegmentTree<long> Min(IReadOnlyList<long> values) => new SegmentTree<long>(values, Math.Min, long.MaxValue);
            public static SegmentTree<double> Min(IReadOnlyList<double> values) => new SegmentTree<double>(values, Math.Min, double.MaxValue);

            public static SegmentTree<int> Max(IReadOnlyList<int> values) => new SegmentTree<int>(values, Math.Max, int.MinValue);
            public static SegmentTree<long> Max(IReadOnlyList<long> values) => new SegmentTree<long>(values, Math.Max, long.MinValue);
            public static SegmentTree<double> Max(IReadOnlyList<double> values) => new SegmentTree<double>(values, Math.Max, double.MinValue);
        }

        public class UnionFind
        {
            private int[] parents;

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public UnionFind(int count)
            {
                parents = Enumerable.Repeat(-1, count).ToArray();
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool TryUnite(int x, int y)
            {
                var rootX = GetRoot(x);
                var rootY = GetRoot(y);

                if (rootX == rootY)
                    return false;

                if (parents[rootY] < parents[rootX])
                {
                    var temp = rootX;
                    rootX = rootY;
                    rootY = temp;
                }
                parents[rootX] += parents[rootY];
                parents[rootY] = rootX;

                return true;
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Find(int x, int y) => GetRoot(x) == GetRoot(y);

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public int GetRoot(int x)
            {
                while (parents[x] >= 0)
                    x = parents[x];
                return x;
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public long GetSize(int x) => -parents[GetRoot(x)];
        }
        public class IterTools
        {
            /// <summary>
            /// 組み合わせ（重複なし）
            /// n = 4, k = 3 => (0,1,2) (0,1,3) (0,2,3) (1,2,3)
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<long>> Combinations(long n, long k)
            {
                if (k <= 0)
                    yield break;

                long[] indices = new long[k];
                long pointer = 0;

                while (pointer >= 0)
                {
                    if (indices[pointer] < n)
                    {
                        if (pointer >= k - 1)
                        {
                            yield return indices;
                            indices[pointer]++;
                        }
                        else
                        {
                            indices[pointer + 1] = indices[pointer] + 1;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                            indices[pointer]++;
                    }
                }
            }

            /// <summary>
            /// 組み合わせ（重複なし）
            /// n = 4, k = 3 => (0,1,2) (0,1,3) (0,2,3) (1,2,3)
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<T>> Combinations<T>(T[] ts, long k)
            {
                if (k <= 0)
                    yield break;

                long[] indices = new long[k];
                T[] container = new T[k];
                long pointer = 0;
                long n = ts.LongLength;

                while (pointer >= 0)
                {
                    if (indices[pointer] < n)
                    {
                        container[pointer] = ts[indices[pointer]];

                        if (pointer >= k - 1)
                        {
                            yield return container;
                            indices[pointer]++;
                        }
                        else
                        {
                            indices[pointer + 1] = indices[pointer] + 1;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                            indices[pointer]++;
                    }
                }
            }

            /// <summary>
            /// 組み合わせ（重複あり）
            /// n = 3, k = 2 => (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) 
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<long>> CombinationsWithReplacement(long n, long k)
            {
                if (k <= 0)
                    yield break;

                long[] container = new long[k];
                long pointer = 0;

                while (pointer >= 0)
                {
                    if (container[pointer] < n)
                    {
                        if (pointer >= k - 1)
                        {
                            yield return container;
                            container[pointer]++;
                        }
                        else
                        {
                            container[pointer + 1] = 0;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                            container[pointer]++;
                    }
                }
            }

            /// <summary>
            /// 組み合わせ（重複あり）
            /// n = 3, k = 2 => (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) 
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<T>> CombinationsWithReplacement<T>(T[] ts, long k)
            {
                if (k <= 0)
                    yield break;

                long[] indices = new long[k];
                T[] container = new T[k];
                long pointer = 0;
                long n = ts.LongLength;

                while (pointer >= 0)
                {
                    if (indices[pointer] < n)
                    {
                        container[pointer] = ts[indices[pointer]];

                        if (pointer >= k - 1)
                        {
                            yield return container;
                            indices[pointer]++;
                        }
                        else
                        {
                            indices[pointer + 1] = 0;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                            indices[pointer]++;
                    }
                }
            }

            /// <summary>
            /// 順列
            /// n = 3, k = 2 => (0,1) (0,2) (1,0) (1,2) (2,0) (2,1) 
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<long>> Permutations(long n, long k)
            {
                if (k <= 0)
                    yield break;

                HashSet<long> enumed = new HashSet<long>();
                long[] container = new long[k];
                long pointer = 0;

                while (pointer >= 0)
                {
                    if (container[pointer] < n)
                    {
                        if (enumed.Contains(container[pointer]))
                        {
                            container[pointer]++;
                        }
                        else if (pointer >= k - 1)
                        {
                            yield return container;
                            container[pointer]++;
                        }
                        else
                        {
                            enumed.Add(container[pointer]);
                            container[pointer + 1] = 0;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                        {
                            enumed.Remove(container[pointer]);
                            container[pointer]++;
                        }
                    }
                }
            }

            /// <summary>
            /// 順列
            /// n = 3, k = 2 => (0,1) (0,2) (1,0) (1,2) (2,0) (2,1) 
            /// </summary>
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static IEnumerable<IReadOnlyList<T>> Permutations<T>(T[] ts, long k)
            {
                if (k <= 0)
                    yield break;

                HashSet<long> enumed = new HashSet<long>();
                long[] indices = new long[k];
                T[] container = new T[k];
                long pointer = 0;
                long n = ts.LongLength;

                while (pointer >= 0)
                {
                    if (indices[pointer] < n)
                    {
                        if (enumed.Contains(indices[pointer]))
                        {
                            indices[pointer]++;
                        }
                        else if (pointer >= k - 1)
                        {
                            container[pointer] = ts[indices[pointer]];
                            yield return container;
                            indices[pointer]++;
                        }
                        else
                        {
                            container[pointer] = ts[indices[pointer]];
                            enumed.Add(indices[pointer]);
                            indices[pointer + 1] = 0;
                            pointer++;
                        }
                    }
                    else
                    {
                        pointer--;

                        if (pointer >= 0)
                        {
                            enumed.Remove(indices[pointer]);
                            indices[pointer]++;
                        }
                    }
                }
            }
        }
        public class Tree
        {
            public Tree() { toNodes = new Dictionary<long, long[]>(); }
            public Tree(Scanner sc, long n, bool base1 = true, bool singleDirection = false) { Adjust(sc.Pairs(n), base1, singleDirection); }
            public Tree(Pair<long, long>[] edges, bool base1 = true, bool singleDirection = false) { Adjust(edges, base1, singleDirection); }
            public Tree(IEnumerable<long> ps, IEnumerable<long> qs, bool base1 = true, bool singleDirection = false) { Adjust(ps.Zip(qs, (p, q) => new Pair<long, long>(p, q)).ToArray(), base1, singleDirection); }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            private void Adjust(Pair<long, long>[] edges, bool base1, bool singleDirection)
            {
                var arrows = base1
                    ? edges.Select(x => new { from = x.X - 1, to = x.Y - 1 })
                    : edges.Select(x => new { from = x.X, to = x.Y });
                if (singleDirection == false)
                    arrows = arrows.Concat(arrows.Select(x => new { from = x.to, to = x.from }));
                toNodes = arrows.GroupBy(x => x.from).ToDictionary(x => x.Key, x => x.Select(xs => xs.to).ToArray());
            }

            private long[] empty = new long[0];
            private Dictionary<long, long[]> toNodes;
            public long[] To(long from)
            {
                long[] res = null;
                if (toNodes.TryGetValue(from, out res))
                    return res;
                else
                    return empty;
            }

            public IEnumerable<Pair<long, long>> GetRouteEdges(long from, long to)
            {
                return GetRouteEdgesImpl(from, to).Skip(1);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            private IEnumerable<Pair<long, long>> GetRouteEdgesImpl(long from, long to)
            {
                var routeNodes = GetRouteNodes(from, to);
                var current = -1L;
                foreach (var next in routeNodes)
                {
                    yield return new Pair<long, long>(current, next);
                    current = next;
                }

            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public IEnumerable<long> GetRouteNodes(long from, long to)
            {
                Stack<long> routeNodes = new Stack<long>();
                HashSet<long> checkedNodes = new HashSet<long>();

                GetRouteNodes(from, to, routeNodes, checkedNodes);

                return routeNodes.Reverse();
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            private bool GetRouteNodes(long current, long dest, Stack<long> routeNodes, HashSet<long> checkedNodes)
            {
                routeNodes.Push(current);
                checkedNodes.Add(current);

                if (current == dest)
                    return true;

                foreach (var next in toNodes[current])
                {
                    if (checkedNodes.Contains(next))
                        continue;

                    if (GetRouteNodes(next, dest, routeNodes, checkedNodes))
                        return true;
                }

                routeNodes.Pop();
                return false;
            }
            /// <summary>
            /// 木の直径（一番長い枝）を求める
            /// </summary>
            /// <returns>木の直径（一番長い枝）</returns>
            public long GetDiameter()
            {
                long firstFarthest = 0;
                long _1 = 0;
                GetDiameterImpl(-1, 0, 0, ref _1, ref firstFarthest);
                long maxDistance = 0;
                long _2 = 0;
                GetDiameterImpl(-1, firstFarthest, 0, ref maxDistance, ref _2);
                return maxDistance;
            }
            private void GetDiameterImpl(long from, long current, long distance, ref long maxDistance, ref long farthest)
            {
                if (distance > maxDistance)
                {
                    maxDistance = distance;
                    farthest = current;
                }

                foreach (var to in To(current))
                {
                    if (from == to)
                        continue;

                    GetDiameterImpl(current, to, distance + 1, ref maxDistance, ref farthest);
                }
            }
        }
        public class PriorityQueue<TKey, TState> where TKey : IComparable<TKey>
        {
            public int Count { get; private set; }
            private readonly Func<TState, TKey> keySelector;
            private readonly bool desc;
            private TState[] states = new TState[65536];
            private TKey[] keys = new TKey[65536];
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PriorityQueue(Func<TState, TKey> keySelector, bool desc = false) { this.keySelector = keySelector; this.desc = desc; }
            public TState Top
            {
                [MethodImpl(MethodImplOptions.AggressiveInlining)]
                get { ValidateNonEmpty(); return states[1]; }
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public TState Dequeue()
            {
                var top = Top;
                var item = states[Count];
                var key = keys[Count];
                Count--;
                int index = 1;

                while (true)
                {
                    if ((index << 1) >= Count)
                    {
                        if (index << 1 > Count)
                            break;

                        if (key.CompareTo(keys[index << 1]) <= 0 ^ desc)
                            break;

                        states[index] = states[index << 1];
                        keys[index] = keys[index << 1];
                        index <<= 1;
                    }
                    else
                    {
                        var nextIndex = keys[index << 1].CompareTo(keys[(index << 1) + 1]) <= 0 ^ desc
                            ? (index << 1)
                            : (index << 1) + 1;

                        if (key.CompareTo(keys[nextIndex]) <= 0 ^ desc)
                            break;

                        states[index] = states[nextIndex];
                        keys[index] = keys[nextIndex];
                        index = nextIndex;
                    }
                }

                states[index] = item;
                keys[index] = key;

                return top;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Enqueue(TState state)
            {
                var key = keySelector.Invoke(state);
                Count++;
                int index = Count;
                if (states.Length == Count)
                    Extend(states.Length * 2);

                while ((index >> 1) != 0)
                {
                    if (keys[index >> 1].CompareTo(key) <= 0 ^ desc)
                        break;

                    states[index] = states[index >> 1];
                    keys[index] = keys[index >> 1];
                    index >>= 1;
                }

                states[index] = state;
                keys[index] = key;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            private void Extend(int newSize)
            {
                TState[] newStates = new TState[newSize];
                TKey[] newKeys = new TKey[newSize];
                states.CopyTo(newStates, 0);
                keys.CopyTo(newKeys, 0);
                states = newStates;
                keys = newKeys;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            private void ValidateNonEmpty()
            {
                if (Count == 0)
                    throw new IndexOutOfRangeException();
            }
        }
        public class BinaryIndexTree
        {
            long length;
            long[] binaryIndexedTree;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public BinaryIndexTree(long length)
            {
                this.length = length;
                binaryIndexedTree = new long[length + 1];
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Add(long indexZeroBase, long additional)
            {
                // i += i & -i
                // 1が立っている最下位ビットを足す、の意味
                for (long i = indexZeroBase + 1; i <= length; i += i & -i)
                {
                    binaryIndexedTree[i] += additional;
                }
            }

            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public long Sum(long indexZeroBase)
            {
                long result = 0;

                // i += i & -i
                // 1が立っている最下位ビットを引く、の意味
                for (long i = indexZeroBase + 1; i > 0; i -= i & -i)
                {
                    result += binaryIndexedTree[i];
                }

                return result;
            }
        }
        public static class BinarySearch
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetCountLarger<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return listOrdered.Count - GetCountSmallerOrEqual(x, listOrdered);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetCountLargerOrEqual<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return listOrdered.Count - GetCountSmaller(x, listOrdered);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetCountSmaller<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return GetLastIndexLess(x, listOrdered) + 1;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetCountSmallerOrEqual<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return GetFirstIndexGreater(x, listOrdered);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetFirstIndexGreater<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return GetFirstIndexGreater(x, listOrdered, 0, listOrdered.Count - 1);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetFirstIndexGreater<T>(T x, IList<T> listOrdered, int low, int high) where T : IComparable
            {
                var count = listOrdered.Count;

                if (count == 0)
                    return low;

                if (listOrdered[high].CompareTo(x) <= 0)
                    return high + 1;

                while (low < high)
                {
                    var mid = (low + high) / 2;

                    if (listOrdered[mid].CompareTo(x) > 0)
                        high = mid;
                    else
                        low = mid + 1;
                }

                return low;
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetLastIndexLess<T>(T x, IList<T> listOrdered) where T : IComparable
            {
                return GetLastIndexLess(x, listOrdered, 0, listOrdered.Count - 1);
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static long GetLastIndexLess<T>(T x, IList<T> listOrdered, int low, int high) where T : IComparable
            {
                var count = listOrdered.Count;

                if (count == 0)
                    return low - 1;

                if (listOrdered[0].CompareTo(x) >= 0)
                    return low - 1;

                while (low < high)
                {
                    var mid = (low + high + 1) / 2;

                    if (listOrdered[mid].CompareTo(x) < 0)
                        low = mid;
                    else
                        high = mid - 1;
                }

                return low;
            }
        }
        public static class BellmanFord
        {
            public class Vertex
            {
                public long Distance { get; set; }

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

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

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

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

                reached.Add(origin);

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

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

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

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

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

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

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

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

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

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

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

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

                return vertices[dest].Distance;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Gcd(int a, int b) => Gcd((long)a, (long)b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Gcd(long a, long b)
        {
            if (a < b)
                return GcdImpl(b, a);
            else
                return GcdImpl(a, b);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long GcdImpl(long a, long b)
        {
            var remainder = a % b;
            if (remainder == 0)
                return b;
            else
                return GcdImpl(b, remainder);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Lcm(int a, int b) => Lcm((long)a, (long)b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Lcm(long a, long b)
        {
            return a / Gcd(a, b) * b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Pow(int n, int p) => Pow((long)n, (long)p);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static long Pow(long n, long p)
        {
            var res = 1L;
            for (long i = 0; i < p; i++)
                res *= n;
            return res;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Dictionary<long, long> Factorize(int n) => Factorize((long)n);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Dictionary<long, long> Factorize(long n)
        {
            var res = new Dictionary<long, long>();
            var r = n;
            for (long i = 2; i * i <= r; i++)
            {
                var c = 0L;
                while (r % i == 0)
                {
                    c++;
                    r /= i;
                }
                if (c > 0)
                    res.Add(i, c);
            }
            if (r > 1)
                res.Add(r, 1);
            return res;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static IEnumerable<long> Divisors(int n) => Divisors((long)n);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static IEnumerable<long> Divisors(long n)
        {
            var cache = new Stack<long>();
            for (long i = 1; i * i <= n; i++)
            {
                if (n % i == 0)
                {
                    yield return i;
                    cache.Push(i);
                }
            }

            var r = cache.Peek();
            if (r * r == n)
                cache.Pop();

            while (cache.Count > 0)
            {
                var i = cache.Pop();
                yield return n / i;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static IEnumerable<int> Loop(int n)
        {
            for (int i = 0; i < n; i++)
                yield return i;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static IEnumerable<long> Loop(long n)
        {
            for (long i = 0L; i < n; i++)
                yield return i;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static IEnumerable<T> Repeat<T>(T t, long n)
        {
            for (long i = 0L; i < n; i++)
                yield return t;
        }
    }
    struct Mint
    {
        public static long Divider { set { divider = value; } }
        private static long divider = 1000000007L;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void Set998244353() { divider = 998244353L; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void Set1000000009() { divider = 1000000009L; }

        public long Value { get; }
        public override string ToString() => this.Value.ToString();

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public Mint(long value) { this.Value = value; }

        //public static implicit operator Mint(long a) => new Mint(a % divider);
        //public static implicit operator Mint(int a) => new Mint(a % divider);
        //public static explicit operator long(Mint a) => a.Value;
        //public static explicit operator int(Mint a) => (int)a.Value;

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator +(Mint a, Mint b) => new Mint((a.Value + b.Value) % divider);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator +(Mint a, long b) => a + new Mint(b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator +(Mint a, int b) => a + new Mint(b);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator -(Mint a, Mint b)
        {
            var t = (a.Value - b.Value) % divider;
            if (t < 0L)
                t += divider;
            return new Mint(t);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator -(Mint a, long b) => a - new Mint(b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator -(Mint a, int b) => a - new Mint(b);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator *(Mint a, Mint b) => new Mint((a.Value * b.Value) % divider);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator *(Mint a, long b) => a * new Mint(b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator *(Mint a, int b) => a * new Mint(b);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator /(Mint a, Mint b) => new Mint((a.Value * InvImpl(b.Value)) % divider);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator /(Mint a, long b) => a / new Mint(b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint operator /(Mint a, int b) => a / new Mint(b);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public Mint Pow(long p) => new Mint(PowImpl(Value, p));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint Pow(long a, long p) => new Mint(PowImpl(a, p));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long PowImpl(long a, long p)
        {
            if (p == 0L)
                return 1L;

            if (p == 1L)
                return a;

            var halfP = p / 2L;
            var halfPowered = PowImpl(a, halfP);
            var powered = halfPowered * halfPowered % divider;
            return p % 2L == 0L ? powered : powered * a % divider;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint Inv(long a) => new Mint(InvImpl(a));
        private static readonly Dictionary<long, long> invCache = new Dictionary<long, long>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long InvImpl(long a)
        {
            long cache;
            if (invCache.TryGetValue(a, out cache))
                return cache;

            var result = PowImpl(a, divider - 2L);
            invCache.Add(a, result);
            return result;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint Fac(long a) => new Mint(FacImpl(a));
        private static long[] facCache = new long[262144];
        private static long cachedFac = 0L;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long FacImpl(long a)
        {
            if (a >= divider)
                return 0L;

            facCache[0] = 1L;
            if (facCache.LongLength <= a)
            {
                long newSize = facCache.LongLength;
                while (newSize <= a)
                {
                    newSize *= 2;
                }

                ExtendFacCache(newSize);
            }

            if (cachedFac < a)
            {
                var val = facCache[cachedFac];
                var start = cachedFac + 1L;
                for (long i = start; i <= a; i++)
                {
                    val = (val * i) % divider;
                    facCache[i] = val;
                }

                cachedFac = a;
            }

            return facCache[a];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static void ExtendFacCache(long newSize)
        {
            long[] newFacCache = new long[newSize];
            facCache.CopyTo(newFacCache, 0);
            facCache = newFacCache;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint Perm(long n, long r) => new Mint(PermImpl(n, r));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long PermImpl(long n, long r)
        {
            if (n < r)
                return 0L;

            if (r <= 0L)
                return 1L;

            var nn = FacImpl(n);
            var nr = FacImpl(n - r);
            return (nn * InvImpl(nr)) % divider;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Mint Comb(long n, long r) => new Mint(CombImpl(n, r));
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static long CombImpl(long n, long r)
        {
            if (n < r)
                return 0L;

            if (n == r)
                return 1L;

            if (n - r < r)
                return CombImpl(n, n - r);

            var nn = FacImpl(n);
            var nr = FacImpl(n - r);
            var rr = FacImpl(r);

            var nrrr = (nr * rr) % divider;
            return (nn * InvImpl(nrrr)) % divider;
        }
    }
}