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 Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
use std::cmp::*;
use std::collections::*;
use std::ops::*;
use std::io::{Write, BufWriter};

const MOD: usize = 998244353;
use mod_int::ModInt;

// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

fn solve() {
    input! {
        n: usize,
        s: chars,
    }
    let x: Vec<usize> = s.into_iter().map(|c| if c=='0' {0} else {1}).collect();
    let mut sum = ModInt(0);
    let mut ans = ModInt(0);
    let mut book: HashMap<usize, usize> = HashMap::new();
    for i in 1..n+1{
        sum = sum * 2 + ModInt(x[i-1]);
        let block = i;
        if n%i!=0 {continue;}
        let num = n/i;
        if num%2==0 {continue;}
        let mut v = 1;
        for j in 0..n{
            let tmp = x[j];
            let mut val = x[j%block];
            if j/block%2==1 {
                val = 1 - val;
            }
            if tmp > val { break;}
            if tmp < val { v = 0; break;}
        }
        let mut cnt = sum + v;
        if let Some(&r) = book.get(&block){
            cnt -= r;
        }
        // println!("{} {} {}", sum.0, i, cnt.0);
        ans += cnt * block * 2;
        let mut y = block;
        while y <= n {
            y += block;
            let mut tmp = 0;
            if let Some(&e) = book.get(&y) {
                tmp += e; 
            }
            book.insert(y, cnt.0 + tmp);
        }
    }
    println!("{}", ans.0);

}

pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};
 
    type Num = usize;
 
    #[derive(Clone, Copy, Debug)]
    pub struct ModInt<T: Copy + Clone>(pub T);
 
    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }
 
    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: Num) -> ModInt<Num> {
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }
 
    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }
 
    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }
 
    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }
 
    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }
 
    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }
 
    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: Num) -> ModInt<Num> {
            self * ModInt(rhs).pow(MOD - 2)
        }
    }
 
    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self / rhs.0
        }
    }
 
    impl DivAssign<Num> for ModInt<Num> {
        fn div_assign(&mut self, rhs: Num) {
            *self = *self / rhs
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self / rhs
        }
    }
 
    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
 
        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }
    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
 
        fn mul(self, rhs: Num) -> ModInt<Num> {
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }
 
    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }
 
    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }
 
    impl ModInt<Num> {
        pub fn pow(self, e: usize) -> ModInt<Num> {
            let mut result = ModInt(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }
    }
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}