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 $998244353$.

3
111

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 $6$.

6
110101

616

30
001110011011011101010111011100

549320998