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: Time Limit Exceeded)

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

namespace MyProject
{
    class Program
    {
        static void Main(string[] args)
        {
#if Local
            var sw = new Stopwatch();
            sw.Start();
            Console.SetIn(new StreamReader("input.in"));
#endif

            var solver = new Solver();
            solver.Solve();
#if Local
            Console.WriteLine($"{sw.ElapsedMilliseconds}ms");
#endif
        }
    }

    class Solver
    {
        private Dictionary<string, int> cache = new Dictionary<string, int>();

        public int PopCount(int x)
        {
            return Convert.ToString(x, 2).Count(elem => elem == '1');
        }

        public int func(string x)
        {
            if (cache.ContainsKey(x))
            {
                return cache[x];
            }
            else
            {
                var ret = 0;
                var work = GetValue(x);
                while (work != 0)
                {
                    work = work % PopCount(work);
                    ret++;
                }

                cache[x] = ret;
                return ret;
            }
        }

        public int GetValue(string x)
        {
            var ret = 0;

            foreach (var c in x)
            {
                ret = ret << 1;
                if (c == '1')
                {
                    ret += 1;
                }
            }

            return ret;
        }

        public void Solve()
        {
            var N = int.Parse(Console.ReadLine());
            var X = Console.ReadLine();
            foreach (var i in Enumerable.Range(0, N))
            {
                var val = string.Empty;
                foreach (var j in Enumerable.Range(0, N))
                {
                    if (i == j)
                    {
                        val += X[j] == '0' ? '1' : '0';
                    }
                    else
                    {
                        val += X[j];
                    }
                }

                Console.WriteLine(func(val));
            }
        }
    }
}