Problem:
Score: 500 points

Problem Statement

N people are standing in a queue, numbered 1, 2, 3, ..., N from front to back. Each person wears a hat, which is red, blue, or green.

The person numbered i says:

"In front of me, exactly A_i people are wearing hats with the same color as mine."

Assuming that all these statements are correct, find the number of possible combinations of colors of the N people's hats.

Since the count can be enormous, compute it modulo 1000000007.

Constraints

1 \leq N \leq 100000

0 \leq A_i \leq N-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_N

Output

Print the number of possible combinations of colors of the N people's hats, modulo 1000000007.

Sample Input 1

6
0 1 2 3 4 5

Sample Output 1

3

We have three possible combinations, as follows:

Red, Red, Red, Red, Red, Red

Blue, Blue, Blue, Blue, Blue, Blue

Green, Green, Green, Green, Green, Green

Sample Input 2

3
0 0 0

Sample Output 2

6

Sample Input 3

54
0 0 1 0 1 2 1 2 3 2 3 3 4 4 5 4 6 5 7 8 5 6 6 7 7 8 8 9 9 10 10 11 9 12 10 13 14 11 11 12 12 13 13 14 14 15 15 15 16 16 16 17 17 17

Sample Output 3

115295190

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;

namespace CompetitionPrograming2
{
    public sealed class Program : BaseProgram
    {
        public static void Main(string[] _) { using var c = new SetConsole(); Solve(); }
        public static void Solve()
        {
            var n = GetNumber<int>();
            var array = GetArray<int>();
            var dic = new Dictionary<int, int>
            {
                { 0, 3 }
            };
            var ans = new Mint(1);
            foreach (var item in array)
            {
                if (dic.ContainsKey(item))
                {
                    ans *= dic[item];
                    dic[item]--;
                }
                else
                {
                    ans *= 0;
                }               
                if (dic.ContainsKey(item + 1))
                {
                    dic[item + 1]++;
                }
                else
                {
                    dic.Add(item + 1, 1);
                }
            }
            Write(ans);
        }
    }
    public abstract class BaseProgram
    {
        protected readonly static long divisor = 1000000007;
        protected static Action<object> Write => Console.WriteLine;
        protected static string GetString()
        {
            var str = Console.ReadLine();
            if (str is null) { throw new NullReferenceException("標準入力がnullです Solveメソッド内の解答コードが間違っています"); }
            return str;
        }
        protected static T GetNumber<T>() where T : IComparable, IComparable<T>, IConvertible, IEquatable<T>, IFormattable
        {
            var str = GetString();
            return Activator.CreateInstance<T>() switch
            {
                int _ => (T)(object)str.ToInt(),
                long _ => (T)(object)str.ToLong(),
                double _ => (T)(object)str.ToDouble(),
                decimal _ => (T)(object)str.ToDecimal(),
                BigInteger _ => (T)(object)str.ToBigInteger(),
                _ => throw new NotSupportedException(),
            };
        }
        protected static T[] GetArray<T>() where T : IComparable, IComparable<T>, IEquatable<T> => ToArray<T>(GetString());
        protected static List<T> GetList<T>() where T : IComparable, IComparable<T>, IEquatable<T> => ToList<T>(GetString());
        protected static void Swap<T>(ref T item1, ref T item2) => (item2, item1) = (item1, item2);
        protected static IEnumerable<char> AtoZ()
        {
            for (char i = 'a'; i <= 'z'; i++)
            {
                yield return i;
            }
        }
        private static T[] ToArray<T>(string str) where T : IComparable, IComparable<T>, IEquatable<T>
        {
            return ConvertEnumerator<T>(str).ToArray();
        }
        private static List<T> ToList<T>(string str) where T : IComparable, IComparable<T>, IEquatable<T>
        {
            return ConvertEnumerator<T>(str).ToList();
        }
        private static IEnumerable<T> ConvertEnumerator<T>(string str) where T : IComparable, IComparable<T>, IEquatable<T>
        {
            return Activator.CreateInstance<T>() switch
            {
                byte _ => str.Split().Select(byte.Parse).Cast<T>(),
                int _ => str.Split().Select(int.Parse).Cast<T>(),
                long _ => str.Split().Select(long.Parse).Cast<T>(),
                double _ => str.Split().Select(double.Parse).Cast<T>(),
                decimal _ => str.Split().Select(decimal.Parse).Cast<T>(),
                BigInteger _ => str.Split().Select(BigInteger.Parse).Cast<T>(),
                _ => throw new NotSupportedException()
            };
        }
        protected sealed class SetConsole : IDisposable
        {
            readonly StreamWriter sw = new StreamWriter(Console.OpenStandardOutput());
            public SetConsole()
            {
                sw.AutoFlush = false;
                Console.SetOut(sw);
            }
            public void Dispose()
            {
                Console.Out.Flush();
                sw.AutoFlush = true;
                Console.SetOut(sw);
            }
        }
    }
    public static class ExtentionsLibrary
    {
        public static long Choose(this long n, long a, long divisor = 1000000007)
        {
            long numerator = 1, denominator = 1;
            for (int i = 0; i < a; i++)
            {
                numerator *= n - i;
                numerator %= divisor;
                denominator *= i + 1;
                denominator %= divisor;
            }
            return numerator / denominator;
        }
        public static long Choose(this int n, int a, int divisor = 1000000007)
        {
            long numerator = 1, denominator = 1;
            for (int i = 0; i < a; i++)
            {
                numerator *= n - i;
                numerator %= divisor;
                denominator *= i + 1;
                denominator %= divisor;
            }
            return numerator / denominator;
        }
        /// <summary>
        /// 階乗を求める
        /// </summary>
        /// <param name="num"></param>
        /// <returns></returns>
        public static long Factorial(this long num)
        {
            var result = 1L;
            for (int i = 1; i <= num; i++)
            {
                result *= i;
            }
            return result;
        }
        private static T[] NextPermutation<T>(this IEnumerable<T> collection, IComparer<T> cmp) where T : IComparable
        {
            var array = collection.ToArray();
            int? exchangingNumIndex = null;
            for (int i = 1; i < array.Length; i++)
            {
                if (cmp.Compare(array[i - 1], array[i]) >= 0) { continue; }

                exchangingNumIndex = i - 1;
            }

            if (exchangingNumIndex is null) { return null; }

            for (int j = array.Length - 1; j >= 0; j--)
            {
                if (cmp.Compare(array[(int)exchangingNumIndex], array[j]) >= 0) { continue; }

                var tmp = array[j];
                array[j] = array[(int)exchangingNumIndex];
                array[(int)exchangingNumIndex] = tmp;
                array = array.Take((int)exchangingNumIndex + 1).Concat(array.Skip((int)exchangingNumIndex + 1).Reverse()).ToArray();
                break;
            }
            return array;
        }
        public static string[] NextPermutation(this IEnumerable<string> collection)
        {
            return NextPermutation(collection, StringComparer.Ordinal);
        }
        public static T[] NextPermutation<T>(this IEnumerable<T> collection) where T : IComparable
        {
            return NextPermutation(collection, Comparer<T>.Default);
        }
        public static IEnumerable<T[]> Permutations<T>(this IEnumerable<T> collection) where T : IComparable
        {
            var next = collection.NextPermutation();

            while (next != null)
            {
                yield return next;

                next = next.NextPermutation();
            }
        }
        public static long Gcd(this long a, long b)
        {
            if (a < b) { return Gcd(b, a); }
            long tmp;
            do
            {
                tmp = a % b;
                a = b;
                b = tmp;
            } while (tmp != 0);
            return a;
        }
        public static long Lcm(this long a, long b) => a * b / Gcd(a, b);
        /// <summary>
        /// 素因数分解を行う
        /// </summary>
        /// <param name="num"></param>
        /// <returns></returns>
        public static IEnumerable<(long num, int count)> Factorize(this long n)
        {
            yield return (1, 1);
            var rootN = Math.Sqrt(n);
            for (long i = 2; i <= rootN; i++)
            {
                if (n % i != 0) { continue; }

                var count = 1;
                n /= i;
                while (n % i == 0)
                {
                    n /= i;
                    count++;
                }
                yield return (i, count);
            }
            if (n != 1) { yield return (n, 1); }
        }
        public static int ToInt(this string str) => int.Parse(str);
        public static int ToInt(this char chr) => int.Parse(chr.ToString());
        public static long ToLong(this string str) => long.Parse(str);
        public static double ToDouble(this string str) => double.Parse(str);
        public static decimal ToDecimal(this string str) => decimal.Parse(str);
        public static BigInteger ToBigInteger(this string str) => BigInteger.Parse(str);
        public static DateTime ToDateTime(this string str) => DateTime.Parse(str);
        public static string StringJoin<T>(this IEnumerable<T> collection, string separator = "") => string.Join(separator, collection.Select(c => c.ToString()));
        public static string StringConcat(this IEnumerable<char> collection) => string.Concat(collection);
        public static int LowerBound<T>(this IReadOnlyList<T> a, T v) => LowerBound(a, v, Comparer<T>.Default);
        public static int LowerBound(this IReadOnlyList<string> str, string v) => LowerBound(str, v, StringComparer.Ordinal);
        public static int LowerBound<T>(this IReadOnlyList<T> a, T v, IComparer<T> cmp)
        {
            var l = 0;
            var r = a.Count - 1;
            while (l <= r)
            {
                var mid = l + (r - l) / 2;
                var result = cmp.Compare(a[mid], v);
                if (result == -1)
                {
                    l = mid + 1;
                }
                else
                {
                    r = mid - 1;
                }
            }
            return l;
        }
        public static int UpperBound(this IReadOnlyList<string> str, string v) => UpperBound(str, v, StringComparer.Ordinal);
        public static int UpperBound<T>(this IReadOnlyList<T> a, T v) => UpperBound(a, v, Comparer<T>.Default);
        public static int UpperBound<T>(this IReadOnlyList<T> a, T v, IComparer<T> cmp)
        {
            var l = 0;
            var r = a.Count - 1;
            while (l <= r)
            {
                var mid = l + (r - l) / 2;
                var res = cmp.Compare(a[mid], v);
                if (res <= 0) l = mid + 1;
                else r = mid - 1;
            }
            return l;
        }
        public static IOrderedEnumerable<TSource> OrderBy<TSource>(this IEnumerable<TSource> source, Func<TSource, string> keySelector)
        {
            return source.OrderBy(keySelector, StringComparer.Ordinal);
        }
        public static IOrderedEnumerable<TSource> OrderByDescending<TSource>(this IEnumerable<TSource> source, Func<TSource, string> keySelector)
        {
            return source.OrderByDescending(keySelector, StringComparer.Ordinal);
        }
        public static void Deconstruct<T>
        (
            this IList<T> self,
            out T first
        )
        {
            first = self[0];
        }

        public static void Deconstruct<T>
        (
            this IList<T> self,
            out T first,
            out T second
        )
        {
            first = self[0];
            second = self[1];
        }

        public static void Deconstruct<T>
        (
            this IList<T> self,
            out T first,
            out T second,
            out T third
        )
        {
            first = self[0];
            second = self[1];
            third = self[2];
        }

        public static void Deconstruct<T>
        (
            this IList<T> self,
            out T first,
            out T second,
            out T third,
            out T forth
        )
        {
            first = self[0];
            second = self[1];
            third = self[2];
            forth = self[3];
        }
        public static void Deconstruct<T>
        (
            this IList<T> self,
            out T first,
            out T second,
            out T third,
            out T forth,
            out T fifth
        )
        {
            first = self[0];
            second = self[1];
            third = self[2];
            forth = self[3];
            fifth = self[4];
        }
    }

    public struct Mint
    {
        private static readonly long MOD = (long)1e9 + 7;

        /// <summary>
        /// 0以上MOD未満の整数
        /// </summary>
        public long Value;

        public Mint(long val)
        {
            Value = val % MOD;
            if (Value < 0) Value += MOD;
        }

        private static Mint Ctor(long val) => new Mint() { Value = val };
        public static Mint operator +(Mint a, Mint b)
        {
            long res = a.Value + b.Value;
            if (res > MOD) res -= MOD;
            return Ctor(res);
        }
        public static Mint operator -(Mint a, Mint b)
        {
            long res = a.Value - b.Value;
            if (res < 0) res += MOD;
            return Ctor(res);
        }
        public static Mint operator *(Mint a, Mint b)
        {
            long res = a.Value * b.Value;
            if (res > MOD) res %= MOD;
            return Ctor(res);
        }
        public static Mint operator /(Mint a, Mint b) => a * Inv(b);
        public static Mint Pow(Mint a, long n)
        {
            if (n == 0) return new Mint(1);
            Mint b = Pow(a, n >> 1);
            b *= b;
            if ((n & 1) == 1) b *= a;
            return b;
        }
        public static Mint Inv(Mint n)
        {
            long a = n.Value;
            long b = MOD;

            long x = 1;
            long u = 0;
            while (b != 0)
            {
                long k = a / b;
                long _x = u;
                u = x - k * u;
                x = _x;

                long _a = a;
                a = b;
                b = _a - (k * b);
            }

            return new Mint(x);
        }
        public override string ToString() => Value.ToString();
        public static implicit operator Mint(long a) => new Mint(a);
        public static explicit operator long(Mint a) => a.Value;
        public static Mint Choose(long n, long k)
        {
            Mint numerator = 1, denominator = 1;
            for (int i = 0; i < k; i++)
            {
                numerator *= n - i;
                denominator *= i + 1;
            }
            return numerator / denominator;
        }
    }
    public enum Priority : byte
    {
        Smaller = 0,
        Larger = 1
    }
}