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)


def g(n):
    c = 0
    while n > 0:
        p = 0
        n2 = n
        while n2 > 0:
            if n2%2 == 1:
                p += 1

            n2 //= 2

        n %= p
        c += 1

    return c


def h(b, m):
    x = 1
    for _ in range(b):
        x = x*2%m

    return x


def f(N, X):
    b = sum(1 if x == '1' else 0 for x in X)

    Xplus = 0
    Xminus = 0

    for x in X:
        Xplus = (Xplus*2 + (1 if x == '1' else 0))%(b+1)
        
        if b > 1:
            Xminus = (Xminus*2 + (1 if x == '1' else 0))%(b-1)

    for i in range(N):
        if X[i] == '0':
            X2 = (Xplus + h(N-i-1, b+1))%(b+1)
            print(g(X2) + 1)
        elif b > 1:
            X2 = (Xminus - h(N-i-1, b-1))%(b-1)
            print(g(X2) + 1)
        else:
            print(0)


N = int(input())
X = input()
f(N, X)