Problem:
Score : 400 points

Problem Statement

Let \mathrm{popcount}(n) be the number of 1s in the binary representation of n.
For example, \mathrm{popcount}(3) = 2, \mathrm{popcount}(7) = 3, and \mathrm{popcount}(0) = 0.

Let f(n) be the number of times the following operation will be done when we repeat it until n becomes 0: "replace n with the remainder when n is divided by \mathrm{popcount}(n)." (It can be proved that, under the constraints of this problem, n always becomes 0 after a finite number of operations.)

For example, when n=7, it becomes 0 after two operations, as follows:

\mathrm{popcount}(7)=3, so we divide 7 by 3 and replace it with the remainder, 1.

\mathrm{popcount}(1)=1, so we divide 1 by 1 and replace it with the remainder, 0.

You are given an integer X with N digits in binary.
For each integer i such that 1 \leq i \leq N, let X_i be what X becomes when the i-th bit from the top is inverted.
Find f(X_1), f(X_2), \ldots, f(X_N).

Constraints

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

X is an integer with N digits in binary, possibly with leading zeros.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print N lines. The i-th line should contain the value f(X_i).

Sample Input 1

3
011

Sample Output 1

2
1
1

X_1 = 7, which will change as follows: 7 \rightarrow 1 \rightarrow 0. Thus, f(7) = 2.

X_2 = 1, which will change as follows: 1 \rightarrow 0. Thus, f(1) = 1.

X_3 = 2, which will change as follows: 2 \rightarrow 0. Thus, f(2) = 1.

Sample Input 2

23
00110111001011011001110

Sample Output 2

2
1
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
3

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

using System;
using System.Linq;
using System.Numerics;

using E = System.Linq.Enumerable;

public class Program
{
    public static void Main()
    {
        var varN = int.Parse(Console.ReadLine());
        var varX = Console.ReadLine().Trim().Select(x => (byte)(x - '0')).ToArray();

        var popcount = 0;
        int lastPop = -1;
        for (int i = 0; i < varX.Length; ++i)
        {
            popcount += varX[i];
            if (varX[i] != 0)
            {
                lastPop = i;
            }
        }

        var fs = new int[varX.Length];
        if (popcount == 0)
        {
            fs.AsSpan().Fill(1);
        }
        else if (popcount == 1)
        {
            fs.AsSpan().Fill(varX[^1] == 0 ? 1 : 2);
            fs[^1] = 2;
            fs[lastPop] = 0;
        }
        else
        {
            var pow2 = new (int Upper, int Lower)[varX.Length];
            var inc = popcount + 1;
            var dec = popcount - 1;
            pow2[0] = (1 % inc, 1 % dec);
            for (int i = 1; i < pow2.Length; ++i)
            {
                pow2[i].Upper = pow2[i - 1].Upper * 2 % inc;
                pow2[i].Lower = pow2[i - 1].Lower * 2 % dec;
            }
            Array.Reverse(pow2);

            var (upperX, lowerX) = (0, 0);
            for (int i = 0; i < varX.Length; ++i)
            {
                if (varX[i] == 0)
                {
                    continue;
                }

                upperX += pow2[i].Upper; upperX %= inc;
                lowerX += pow2[i].Lower; lowerX %= dec;
            }

            for (int i = 0; i < varX.Length; ++i)
            {
                var (t, m) = varX[i] switch
                {
                    0 => (upperX + pow2[i].Upper, inc),
                    _ => (lowerX - pow2[i].Lower + dec, dec),
                };
                if (t < 1)
                {
                    continue;
                }

                t %= m;
                ++fs[i];

                while (t > 0)
                {
                    t %= BitOperations.PopCount(t.AsUInt32());
                    ++fs[i];
                }
            }
        }

        Console.WriteLine(string.Join(Environment.NewLine, fs));
    }
}

static class ConversionExtension
{
    public static uint AsUInt32(this int signed) => BitConverter.ToUInt32(BitConverter.GetBytes(signed));
}