Problem:
Score : 800 points

Problem Statement

Given are integers N and X. For each integer k between 0 and X (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo 998244353.

Let us repeat the following operation on the integer k. Operation: if the integer is currently odd, subtract 1 from it and divide it by 2; otherwise, divide it by 2 and add 2^{N-1} to it. How many operations need to be performed until k returns to its original value? (The answer is considered to be 0 if k never returns to its original value.)

Constraints

1 \leq N \leq 2\times 10^5

0 \leq X < 2^N

X is given in binary and has exactly N digits. (In case X has less than N digits, it is given with leading zeroes.)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print the sum of the answers to the questions for the integers between 0 and X (inclusive), modulo 998244353.

Sample Input 1

3
111

Sample Output 1

40

For example, for k=3, the operation changes k as follows: 1,0,4,6,7,3. Therefore the answer for k=3 is 6.

Sample Input 2

6
110101

Sample Output 2

616

Sample Input 3

30
001110011011011101010111011100

Sample Output 3

549320998

Solution in C# (Mono 4.6.2.0) (judge verdict: Runtime Error)

using System;
using System.Collections.Generic;

public class Program
{
    private int N;
    HashSet<int>[] Edge;
    private int[,] Dist;


    private int[] D;

    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        Edge = new HashSet<int>[N];
        Dist = new int[N, N];
        for (int i = 0; i < N; i++)
        {
            Edge[i] = new HashSet<int>();
            for (int j = 0; j < N; j++)
            {
                Dist[i, j] = i == j ? 0 : 1 << 20;
            }
        }

        for (int i = 0; i < N; i++)
        {
            string s = sc.Next();
            for (int j = 0; j < N; j++)
            {
                if (s[j] == '1')
                {
                    Dist[i, j] = 1;
                    Edge[i].Add(j);
                }
            }
        }

        for (int k = 0; k < N; k++)
        {
            for (int i = 0; i < N; i++)
            {
                for (int j = 0; j < N; j++)
                {
                    Dist[i, j] = Math.Min(Dist[i, j], Dist[i, k] + Dist[k, j]);
                }
            }
        }

        D = new int[N];
        for (int i = 0; i < N; i++)
        {
            D[i] = -1;
        }

        if (!Search(0, 0))
        {
            Console.WriteLine(-1);
            return;
        }

        int max = 0;
        for (int i = 0; i < N - 1; i++)
        {
            for (int j = i + 1; j < N; j++)
            {
                max = Math.Max(max, Dist[i, j]);
            }
        }

        Console.WriteLine(max+1);
    }

    private bool Search(int pos, int d)
    {
        if (D[pos] != -1)
        {
            if (D[pos] % 2 == d % 2)
            {
                return true;
            }
            else
            {
                return false;
            }
        }
        else
        {
            D[pos] = d;
            foreach (int i in Edge[pos])
            {
                if (!Search(i, d + 1))
                {
                    return false;
                }
            }
        }

        return true;
    }

    public static void Main(string[] args)
    {
        new Program().Solve();
    }
}

class HashMap<K, V> : Dictionary<K, V>
{
    new public V this[K i]
    {
        get
        {
            V v;
            return TryGetValue(i, out v) ? v : base[i] = default(V);
        }
        set { base[i] = value; }
    }
}

class UnionFind
{
    private int[] Par, Rank;

    public UnionFind(int max)
    {
        Par = new int[max];
        Rank = new int[max];
        for (int i = 0; i < max; i++)
        {
            Par[i] = i;
        }
    }

    public int Find(int n)
    {
        if (Par[n] == n)
        {
            return n;
        }
        else
        {
            return Find(Par[n]);
        }
    }

    public bool Same(int a, int b)
    {
        return Find(a) == Find(b);
    }

    public void Union(int a, int b)
    {
        a = Find(a);
        b = Find(b);
        if (a == b) return;
        if (Rank[a] < Rank[b])
        {
            Par[a] = b;
        }
        else
        {
            Par[b] = a;
            if (Rank[a] == Rank[b]) Rank[a]++;
        }
    }
}

class Scanner
{
    public Scanner()
    {
        _pos = 0;
        _line = new string[0];
    }

    const char Separator = ' ';
    private int _pos;
    private string[] _line;

    #region スペース区切りで取得

    public string Next()
    {
        if (_pos >= _line.Length)
        {
            _line = Console.ReadLine().Split(Separator);
            _pos = 0;
        }

        return _line[_pos++];
    }

    public int NextInt()
    {
        return int.Parse(Next());
    }

    public long NextLong()
    {
        return long.Parse(Next());
    }

    public double NextDouble()
    {
        return double.Parse(Next());
    }

    #endregion

    #region 型変換

    private int[] ToIntArray(string[] array)
    {
        var result = new int[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = int.Parse(array[i]);
        }

        return result;
    }

    private long[] ToLongArray(string[] array)
    {
        var result = new long[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = long.Parse(array[i]);
        }

        return result;
    }

    private double[] ToDoubleArray(string[] array)
    {
        var result = new double[array.Length];
        for (int i = 0; i < array.Length; i++)
        {
            result[i] = double.Parse(array[i]);
        }

        return result;
    }

    #endregion

    #region 配列取得

    public string[] Array()
    {
        if (_pos >= _line.Length)
            _line = Console.ReadLine().Split(Separator);

        _pos = _line.Length;
        return _line;
    }

    public int[] IntArray()
    {
        return ToIntArray(Array());
    }

    public long[] LongArray()
    {
        return ToLongArray(Array());
    }

    public double[] DoubleArray()
    {
        return ToDoubleArray(Array());
    }

    #endregion
}