Problem:
Score : 800 points

Problem Statement

Given are integers N and X. For each integer k between 0 and X (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo 998244353.

Let us repeat the following operation on the integer k. Operation: if the integer is currently odd, subtract 1 from it and divide it by 2; otherwise, divide it by 2 and add 2^{N-1} to it. How many operations need to be performed until k returns to its original value? (The answer is considered to be 0 if k never returns to its original value.)

Constraints

1 \leq N \leq 2\times 10^5

0 \leq X < 2^N

X is given in binary and has exactly N digits. (In case X has less than N digits, it is given with leading zeroes.)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print the sum of the answers to the questions for the integers between 0 and X (inclusive), modulo 998244353.

Sample Input 1

3
111

Sample Output 1

40

For example, for k=3, the operation changes k as follows: 1,0,4,6,7,3. Therefore the answer for k=3 is 6.

Sample Input 2

6
110101

Sample Output 2

616

Sample Input 3

30
001110011011011101010111011100

Sample Output 3

549320998

Solution in Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

n, x, MOD, f = gets.to_i, gets.chomp + '1', 998244353, [0]
n.times{|i| f << (f[-1]*2 + x[i].to_i) % MOD}
m, ans = Array.new(n,0), (f[n]+1) * 2*n
ha = Hash.new{|h,k| u = k; u += 1 while u < n && x[u-k] != x[u]; h[k] = f[k] + x[u].to_i}
(1..n).each{|i| t = i.gcd(n); (ans += (m[i] += ha[t]) * (i - 2*n); (i*2).step(n-1,i){|j| m[j] -= m[i]}) if i/t%2 == 0}
p ans % MOD