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 Scala (2.11.7) (judge verdict: Accepted)

object Main {
  import java.io.{BufferedReader, InputStream, InputStreamReader}
  import java.util.StringTokenizer
  import scala.reflect.ClassTag

  def main(args: Array[String]): Unit = {
    val out = new java.io.PrintWriter(System.out)
    new Main(out, new InputReader(System.in)).solve()
    out.flush()
  }

  private[this] val oj = System.getenv("ATCODER_DEBUG") == null
  @inline private final def DEBUG(f: => Unit): Unit = {
    if (!oj){ f }
  }
  @inline private final def debug(as: Array[Boolean]): Unit = if (!oj){ debug(as.map(x => if(x) "1" else "0").mkString) }
  @inline private final def debug(as: Array[Int]): Unit = if (!oj){ debug(as.mkString(" ")) }
  @inline private final def debug(as: Array[Long]): Unit =if (!oj){ debug(as.mkString(" ")) }
  @inline private final def debugDim(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m.length) { i =>
      debug(m(i))
    }
  }
  @inline private final def debugDimFlip(m: Array[Array[Long]]): Unit = if (!oj){
    REP(m(0).length) { j =>
      REP(m.length) { i =>
        System.err.print(m(i)(j))
        System.err.print(" ")
      }
      System.err.println()
    }
  }
  @inline private final def debug(s: => String): Unit = DEBUG {
    System.err.println(s)
  }
  @inline private final def debugL(num: => Long): Unit = DEBUG {
    System.err.println(num)
  }
  private def isDebug[A](debug: => A, online: => A): A = {
    if (oj) online else debug
  }

  class InputReader(val stream: InputStream) {
    private[this] val reader = new BufferedReader(new InputStreamReader(stream), 32768)
    private[this] var tokenizer: StringTokenizer = _

    @inline private[this] final def next(): String = {
      while (tokenizer == null || !tokenizer.hasMoreTokens)
        tokenizer = new StringTokenizer(reader.readLine)
      tokenizer.nextToken
    }

    def nextInt(): Int = Integer.parseInt(next())
    def nextLong(): Long = java.lang.Long.parseLong(next())
    def nextChar(): Char = next().charAt(0)

    def ni(): Int = nextInt()
    def nl(): Long = nextLong()
    def nc(): Char = nextChar()
    def ns(): String = next()
    def ns(n: Int): Array[Char] = ns().toCharArray
    def na(n: Int, offset: Int = 0): Array[Int] = map(n)(_ => ni() + offset)
    def na2(n: Int, offset: Int = 0): (Array[Int], Array[Int]) = {
      val A1, A2 = Array.ofDim[Int](n)
      REP(n) { i =>
        A1(i) = ni() + offset
        A2(i) = ni() + offset
      }
      (A1, A2)
    }
    def nm(n: Int, m: Int): Array[Array[Int]] = {
      val A = Array.ofDim[Int](n, m)
      REP(n) { i =>
        REP(m) { j =>
          A(i)(j) = ni()
        }
      }
      A
    }
    def nal(n: Int): Array[Long] = map(n)(_ => nl())
    def nm_c(n: Int, m: Int): Array[Array[Char]] = map(n) (_ => ns(m))
  }

  def REP(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = offset
    val N = n + offset
    while(i < N) { f(i); i += 1 }
  }
  def REP_r(n: Int, offset: Int = 0)(f: Int => Unit): Unit = {
    var i = n - 1 + offset
    while(i >= offset) { f(i); i -= 1 }
  }
  def TO(from: Int, to: Int)(f: Int => Unit): Unit = {
    REP(to - from + 1, from)(f)
  }
  def map[@specialized A: ClassTag](n: Int, offset: Int = 0)(f: Int => A): Array[A] = {
    val res = Array.ofDim[A](n)
    REP(n)(i => res(i) = f(i + offset))
    res
  }

  def sumL(as: Array[Int]): Long = {
    var s = 0L
    REP(as.length)(i => s += as(i))
    s
  }
  def cumSum(as: Array[Int]): Array[Long] = {
    val cum = Array.ofDim[Long](as.length + 1)
    REP(as.length) { i =>
      cum(i + 1) = cum(i) + as(i)
    }
    cum
  }
}

object Workspace {
  import Main._

  /**
    * O(√x)
    */
  def divisors(x: Int): Array[Int] = {
    import scala.collection.mutable.ArrayBuffer
    val pre = ArrayBuffer[Int]()
    val post = ArrayBuffer[Int]()
    REP(math.sqrt(x).toInt, 1) { d =>
      if (x % d == 0) {
        pre += d
        if (d * d != x) post += x / d
      }
    }

    val res = Array.ofDim[Int](pre.length + post.length)
    REP(pre.length)(i => res(i) = pre(i))
    val preLen = pre.length
    val postLen = post.length
    REP(postLen)(i => res(i + preLen) = post(postLen - 1 - i))
    res
  }
}

class Main(out: java.io.PrintWriter, sc: Main.InputReader) {
  import sc._
  import Main._
  import java.util.Arrays.sort

  import scala.collection.mutable
  import math.{abs, max, min}
  import mutable.ArrayBuffer
  import Workspace._

  // toIntとか+7とかするならvalにしろ
  final private[this] val MOD = 998244353

  val pow2 = Array.ofDim[Long](2e5.toInt)
  pow2(0) = 1
  REP(pow2.length - 1)(i => pow2(i + 1) = pow2(i) * 2 % MOD)

  def solve(): Unit = {
    val n = ni()
    val x = ns(n).map(_-'0')
    val d = divisors(2 * n)
    val d_rev = Array.fill[Int](4e5.toInt + 1)(-1)
    REP(d.length) { i =>
      d_rev(d(i)) = i
    }
    val cnt = Array.ofDim[Long](d.length) // 約数毎の出現数

    // 先頭dを固定した時残りに~d, d, ~d, dのように並べることができるか
    def isJust(d: Int): Boolean = {
      REP(n / d - 1, 1) { i =>
        val xor = i&1
        REP(d) { j =>
          // d*i+jまで一致していて、後はlessの場合
          if (x(d * i + j) == 1 && x(j) == xor) return true

          if (x(j) != (x(d * i + j)^xor)) return false
        }
      }
      true
    }

    def cnt2n = {
      var res = 1L
      REP(n) { i =>
        if (x(i) == 1) {
          res += pow2(n - 1 - i)
        }
      }
      res % MOD
    }

    REP(d.length) { i =>
      val dd = d(i)
      if (dd % 2 == 0 && 2 * n / dd % 2 == 1 && dd <= n) {
        REP(dd / 2) { j =>
          if (x(j) == 1) {
            // lessを数える
            debug(s"less $dd $j")
            cnt(i) += pow2(dd / 2 - 1 - j)
          }
        }
        if (isJust(dd / 2)) {
          debug(s"just $dd")
          cnt(i) += 1
        }
        cnt(i) %= MOD
      }

      // 約数包除
      var k = dd * 2
      while(k <= 2 * n) {
        if (d_rev(k) != -1) cnt(d_rev(k)) = (MOD + cnt(d_rev(k)) - cnt(i)) % MOD
        k += dd
      }
    }
    cnt(d_rev(2 * n)) = (cnt(d_rev(2 * n)) + cnt2n) % MOD
    debug(d)
    debug(cnt)
    var ans = 0L
    REP(d.length) { i =>
      ans = (ans + d(i).toLong * cnt(i)) % MOD
    }
    out.println(ans)
  }
}