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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import sys
sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline

n = int(input())
X = int(input(),2)

from collections import defaultdict
d = defaultdict(lambda: -1)

Xmemo = defaultdict(lambda: -1)
Xmemo[0]=0

def dfs(X):
    if Xmemo[X]==-1:
        if d[X]==-1:
            popcount = bin(X).count("1")
            d[X]=popcount
        else:
            popcount = d[X]

        if popcount == 1:
            Xmemo[X]=1
            return 1

        Xmemo[X] = dfs(X%popcount)+1

    return Xmemo[X]

ans = [0]*n

Xpopcount = bin(X).count("1")

for i in reversed(range(n)):
    neX = X^(1<<i)

    if X & (1 << i):
        d[neX] = Xpopcount-1
    else:
        d[neX] = Xpopcount+1
    print(dfs(neX))