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 PHP (7.4.4) (judge verdict: Runtime Error)

<?php
  
fscanf(STDIN, "%d", $N);
fscanf(STDIN, "%s", $X);
$X_array = str_split($X);

$X_count = 0;
foreach ($X_array as $X) {
 	if ($X == 1) $X_count++; 
}

for($i = 0;$i<$N;$i++){
  $a = $X_array;
  $a_count = $X_count;
  if ($a[$i] == 0) {
    $a[$i] = 1;
    $a_count++;
  }else {
    $a[$i] = 0;
    $a_count--;
  }
 $a = implode($a);
 $a = bindec($a);
 
  $r_count = 0;
  for($j = 0;$j<1000;$j++){
    $stock_a = $a;
   	$a = $a%$a_count;
    $r_count++;
    if ($a == 0) {
      $stock[$stock_a] = $r_count;
      break; 
    }
    
    if (isset($stock[$a])) {
      $r_count = $stock[$a];
      break;
    }
    
    $b = decbin($a);
    $b_array = str_split($b);

    $b_count = 0;
    foreach ($b_array as $b) {
        if ($b == 1) $b_count++; 
    }
    $a_count =$b_count;
  }  
  echo($r_count . "\n");
 }