Problem:
Score : 700 points

Problem Statement

Given is a sequence of N digits a_1a_2\ldots a_N, where each element is 1, 2, or 3.
Let x_{i,j} defined as follows:

x_{1,j} := a_j \quad (1 \leq j \leq N)

x_{i,j} := | x_{i-1,j} - x_{i-1,j+1} | \quad (2 \leq i \leq N and 1 \leq j \leq N+1-i)

Find x_{N,1}.

Constraints

2 \leq N \leq 10^6

a_i = 1,2,3 (1 \leq i \leq N)

Input

Input is given from Standard Input in the following format:

N
a_1a_2\ldotsa_N

Output

Print x_{N,1}.

Sample Input 1

4
1231

Sample Output 1

1

x_{1,1},x_{1,2},x_{1,3},x_{1,4} are respectively 1,2,3,1.

x_{2,1},x_{2,2},x_{2,3} are respectively |1-2| = 1,|2-3| = 1,|3-1| = 2.

x_{3,1},x_{3,2} are respectively |1-1| = 0,|1-2| = 1.

Finally, x_{4,1} = |0-1| = 1, so the answer is 1.

Sample Input 2

10
2311312312

Sample Output 2

0

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 static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class AGC043B
    {
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var a = NS;
            var set = new LIB_Bitset((N - 1) * 2);
            var mask1 = new LIB_Bitset((N - 1) * 2);
            var tmp = a.Select(e => e - '0').ToArray();
            for (var i = 0; i < N - 1; i++)
            {
                var t = Abs(tmp[i] - tmp[i + 1]);
                if (t == 0)
                {
                    set[i * 2] = false;
                    set[i * 2 + 1] = false;
                }
                else if (t == 1)
                {
                    set[i * 2] = true;
                    set[i * 2 + 1] = false;
                }
                else
                {
                    set[i * 2] = false;
                    set[i * 2 + 1] = true;
                }
                mask1[i * 2] = true;
                mask1[i * 2 + 1] = false;
            }
            var mask2 = ~mask1;
            for (var i = 2; i < N; i++)
            {
                var aite = set >> 2;
                var bit1 = (aite & mask1) ^ (set & mask1);
                var flag1 = ((aite & mask2) >> 1) & (~aite & mask1) & ((~set & mask2) >> 1) & (~set & mask1);
                var flag2 = ((~aite & mask2) >> 1) & (~aite & mask1) & ((set & mask2) >> 1) & (~set & mask1);
                var bit2 = flag1 | flag2;
                set = bit1 | bit2;
            }
            if (!set[0] && !set[1]) Console.WriteLine(0);
            else if (set[0] && !set[1]) Console.WriteLine(1);
            else Console.WriteLine(2);
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } var t = new Thread(Solve, 134217728); t.Start(); t.Join(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => xorshift);
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_Bitset : IEquatable<LIB_Bitset>
    {
        long n;
        ulong[] ary;
        static readonly ulong[] ceil;
        static LIB_Bitset()
        {
            ceil = new ulong[64];
            ceil[0] = 0xffffffffffffffff;
            ceil[1] = 1;
            for (var i = 2; i < 64; i++)
            {
                ceil[i] = (ceil[i - 1] << 1) | 1;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Bitset(long size)
        {
            if (size <= 0) throw new Exception();
            n = size;
            ary = new ulong[((n - 1) >> 6) + 1];
        }
        public long Count => n;
        public long CountOne => ary.Sum(e => LIB_BitUtil.CountOne(e));
        public bool this[int idx]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return LIB_BitUtil.IsSet(ary[idx >> 6], idx & 63); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            set
            {
                if (value) ary[idx >> 6] |= LIB_BitUtil.BitMask[idx & 63];
                else ary[idx >> 6] &= ~LIB_BitUtil.BitMask[idx & 63];
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator &(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] &= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator |(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] |= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ^(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            for (var i = 0; i < y.ary.Length; i++) ret.ary[i] ^= y.ary[i];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator &(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] &= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator |(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] |= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ^(LIB_Bitset x, long y)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = x.ary[i];
            ret.ary[0] ^= (ulong)y;
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Flip()
        {
            for (var i = 0; i < ary.Length; i++) ary[i] = ~ary[i];
            ary[ary.Length - 1] &= ceil[n & 63];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator ~(LIB_Bitset x)
        {
            var ret = new LIB_Bitset(x.n);
            for (var i = 0; i < ret.ary.Length; i++) ret.ary[i] = ~x.ary[i];
            ret.ary[ret.ary.Length - 1] &= ceil[ret.n & 63];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void ShiftLeft(int num)
        {
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            for (var i = ary.Length - 1; i >= moveCnt; i--)
            {
                ary[i] = ary[i - moveCnt] << moveBit;
                if (moveBit > 0 && i > moveCnt) ary[i] |= ary[i - moveCnt - 1] >> (64 - moveBit);
            }
            ary[ary.Length - 1] &= ceil[n & 63];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator <<(LIB_Bitset x, int num)
        {
            var ret = new LIB_Bitset(x.n);
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            for (var i = ret.ary.Length - 1; i >= moveCnt; i--)
            {
                ret.ary[i] = x.ary[i - moveCnt] << moveBit;
                if (moveBit > 0 && i > moveCnt) ret.ary[i] |= x.ary[i - moveCnt - 1] >> (64 - moveBit);
            }
            ret.ary[ret.ary.Length - 1] &= ceil[ret.n & 63];
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void ShiftRight(int num)
        {
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var aryMax = ary.Length - moveCnt;
            for (var i = 0; i < aryMax; i++)
            {
                ary[i] = ary[i + moveCnt] >> moveBit;
                if (moveBit > 0 && i < aryMax - 1) ary[i] |= ary[i + moveCnt + 1] << (64 - moveBit);
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public LIB_Bitset operator >>(LIB_Bitset x, int num)
        {
            var ret = new LIB_Bitset(x.n);
            var moveCnt = num >> 6;
            var moveBit = num & 63;
            var aryMax = ret.ary.Length - moveCnt;
            for (var i = 0; i < aryMax; i++)
            {
                ret.ary[i] = x.ary[i + moveCnt] >> moveBit;
                if (moveBit > 0 && i < aryMax - 1) ret.ary[i] |= x.ary[i + moveCnt + 1] << (64 - moveBit);
            }
            return ret;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator ==(LIB_Bitset x, LIB_Bitset y)
        {
            if (x.n < y.n) { var t = x; x = y; y = t; }
            var i = 0;
            for (; i < y.ary.Length; i++) if (x.ary[i] != y.ary[i]) return false;
            for (; i < x.ary.Length; i++) if (x.ary[i] != 0) return false;
            return true;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool operator !=(LIB_Bitset x, LIB_Bitset y) => !(x == y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool Equals(LIB_Bitset x) => x == this;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object x) => x == null ? false : Equals((LIB_Bitset)x);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode()
        {
            var t = ary.Aggregate((a, x) => a ^ x);
            return (int)(((t >> 32) ^ t) & 0x00000000ffffffff);
        }
    }
    class LIB_FastIO
    {
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        public LIB_FastIODebug() { }
        public override string Scan() => NextString();
        public override long Long() => long.Parse(NextString());
        public override double Double() => double.Parse(NextString());
    }
    class LIB_BitUtil
    {
        public static readonly ulong[] BitMask;
        static LIB_BitUtil()
        {
            BitMask = new ulong[64];
            BitMask[0] = 1;
            for (var i = 1; i < 64; i++) BitMask[i] = BitMask[i - 1] << 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> ScanOne(long value)
        {
            for (; value > 0; value &= value - 1) yield return value & -value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long CountOne(ulong value)
        {
            value = (value & 0x5555555555555555) + ((value >> 1) & 0x5555555555555555);
            value = (value & 0x3333333333333333) + ((value >> 2) & 0x3333333333333333);
            value = (value & 0x0f0f0f0f0f0f0f0f) + ((value >> 4) & 0x0f0f0f0f0f0f0f0f);
            value = (value & 0x00ff00ff00ff00ff) + ((value >> 8) & 0x00ff00ff00ff00ff);
            value = (value & 0x0000ffff0000ffff) + ((value >> 16) & 0x0000ffff0000ffff);
            value = (value & 0x00000000ffffffff) + ((value >> 32) & 0x00000000ffffffff);
            return (long)value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long CountOne(long value) => CountOne((ulong)value);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(ulong value, int idx) => (value & BitMask[idx]) != 0;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(long value, int idx) => IsSet((ulong)value, idx);
    }
}