output_description: Print the number of different integers not exceeding X that can be written on
  the blackboard.
  
  * * *
input_description: Input is given from Standard Input in the following format:
  
      
      
      N X
      A_1
      :
      A_N
attempt: #!/usr/bin/env python3
  def divmod(f, g):
      assert g
      h = 0
      for i in reversed(range(f.bit_length() - g.bit_length() + 1)):
          if f & (1 << (g.bit_length() + i - 1)):
              f ^= g << i
              h ^= 1 << i
      return h, f
  
  
  def gcd(f, g):
      while g:
          q, r = divmod(f, g)
          f, g = g, r
      return f
  
  
  import functools
  
  
  def solve(n, x, a):
      # (g) = (a_1, ..., a_n) is a principal ideal since F_2[x] is a PID
      g = functools.reduce(gcd, a)
  
      # count h in F_2[x] s.t. h g <= x
      cnt = 0
      h = 0
      for k in reversed(range(x.bit_length() - g.bit_length() + 1)):
          bit = 1 << (g.bit_length() + k - 1)
          if x & bit:
              cnt += 1 << k
          if (x & bit) != (h & bit):
              h ^= g << k
      cnt += 1  # zero
      return cnt % 998244353
  
  
  def main():
      n, x = input().split()
      n = int(n)
      x = int(x, 2)
      a = [int(input(), 2) for _ in range(n)]
      print(solve(n, x, a))
  
  
  if __name__ == "__main__":
      main()
  
problem_description: Statement
  
  There are N non-negative integers written on a blackboard. The i-th integer is
  A_i.
  
  Takahashi can perform the following two kinds of operations any number of
  times in any order:
  
    * Select one integer written on the board (let this integer be X). Write 2X on the board, without erasing the selected integer.
    * Select two integers, possibly the same, written on the board (let these integers be X and Y). Write X XOR Y (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.
  
  How many different integers not exceeding X can be written on the blackboard?
  We will also count the integers that are initially written on the board. Since
  the answer can be extremely large, find the count modulo 998244353.
samples: [{"input": "3 111\n    1111\n    10111\n    10010", "output": "4\n    \n\nInitially, 15, 23 and 18 are written on the blackboard. Among the integers not\nexceeding 7, four integers, 0, 3, 5 and 6, can be written. For example, 6 can\nbe written as follows:\n\n  * Double 15 to write 30.\n  * Take XOR of 30 and 18 to write 12.\n  * Double 12 to write 24.\n  * Take XOR of 30 and 24 to write 6.\n\n* * *"}, {"input": "4 100100\n    1011\n    1110\n    110101\n    1010110", "output": "37\n    \n\n* * *"}, {"input": "4 111001100101001\n    10111110\n    1001000110\n    100000101\n    11110000011", "output": "1843\n    \n\n* * *"}, {"input": "1 111111111111111111111111111111111111111111111111111111111111111\n    1", "output": "466025955\n    \n\nBe sure to find the count modulo 998244353."}]