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 Haskell (GHC 8.8.3) (judge verdict: Time Limit Exceeded)

import Data.Array
import Data.Bits
import qualified Data.ByteString.Char8 as BS
import Data.List
import Data.Maybe
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM

readInt = fst . fromJust . BS.readInt

getInt = readInt <$> BS.getLine

getVU n = VU.reverse . VU.unfoldrN n BS.uncons <$> BS.getLine

simulate :: Integer -> Integer -> Integer
simulate count 0 = count
simulate count x = simulate (count + 1) (x `mod` (fromIntegral (popCount x)))

solve :: Int -> Integer -> Int -> Integer -> Integer -> Integer -> Integer -> Integer
solve b x n p0 p1 s0 s1 = do
  let x' = complementBit x b
  if x' == 0
    then 0
    else do
      let b' = testBit x' (fromIntegral b)
      let d = if b' then (2 ^ (fromIntegral b) `mod` p0 + s0) `mod` p0 else (- 2 ^ (fromIntegral b) `mod` p1 + s1) `mod` p1
      simulate 1 d

createBits :: VU.Vector Char -> Integer
createBits = VU.ifoldl' (\bit index value -> if value == '1' then setBit bit index else bit) zeroBits

main = do
  n <- getInt
  x <- getVU n
  let x' = createBits x
  let ppc = fromIntegral (popCount x')
  let p0 = ppc + 1
  let p1 = ppc - 1
  let s0 = x' `mod` p0
  let s1 = x' `mod` p1
  let bs = reverse [0 .. (fromIntegral n -1)]
  let x'' = map (\b -> solve b x' n p0 p1 s0 s1) bs
  mapM_ print x''