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 PyPy3 (7.3.0) (judge verdict: Runtime Error)

import sys
read = sys.stdin.read
from collections import defaultdict
def main():
    n = int(input())
    x = list(input())
    if n == 1:
        print(0)
        sys.exit()

    x10 = int("".join(x), 2)
    xcount1 = x.count('1')
    modx_plus = x10 % (xcount1 + 1)
    modx_minus = x10 % (xcount1 - 1)
    memo = defaultdict(int)
    for i1 in range(n):
        if x[i1] == '1':
            mod_e = (pow(2, (n - 1 - i1)) % (xcount1 - 1))
            x3 = modx_minus - mod_e
            if x3 < 0:
                x3 += (xcount1 - 1)
            elif x3 > (xcount1 - 1):
                x3 -= (xcount1 - 1)
            elif x3 == (xcount1 - 1):
                x3 = 0
        else:
            mod_e = (pow(2, (n - 1 - i1)) % (xcount1 + 1))
            x3 = modx_plus + mod_e
            if x3 < 0:
                x3 += (xcount1 + 1)
            elif x3 > (xcount1 + 1):
                x3 -= (xcount1 + 1)
            elif x3 == (xcount1 + 1):
                x3 = 0
        r = 1
        while x3:
            r += 1
            if memo[x3] == 0:
                p = bin(x3).count('1')
                memo[x3] = p
                x3 = x3 % p
            else:
                x3 = x3 % memo[x3]
        print(r)

if __name__ == '__main__':
    main()