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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

import java.util.Scanner;
     
    public class Main { //クラス名はMain
        public static void main(String[] args) {
            //コード
          int c1 = 0;	//2の何乗
          int ttl = 0;  //計算回数
          int moto = 0; //計算用変数
          int taihi = 0;//退避用変数
          Scanner sc = new Scanner(System.in);
    	   c1 = sc.nextInt();
    	   String c2 = sc.next();
          int dec = Integer.parseInt(c2, 2);
          //判定
          for(int i=1;i<=dec;i++) {
            moto = i;
            taihi = i;
            while(true) {
            //現在の整数が奇数なら、1を引いて 2で割る。そうでなければ、2 で割って 2N−1を足す。
          		if (Math.abs(i) % 2 == 0) {
                //偶数文字
                  taihi = (int) Math.pow((taihi / 2), (c1 - 1));
                  ttl++;
                  if(java.util.Objects.equals(taihi,moto)){
                  break;
                  }
            }
            else {
              	//奇数文字
                taihi = (taihi - 1 ) / 2;
              	ttl++;
               if(java.util.Objects.equals(taihi,moto)){
              	break;
               }
           }
            }
          }
          //計算表示
          System.out.println(ttl % 998244353);
        }
    }