





'''simple docstring'''

def  a__    (    lowercase :					int				)		->		list:





     """simple docstring"""

     if bit_count < 0:
          raise ValueError('''The given input must be positive'''				)

     # get the generated string sequence
     _UpperCamelCase       =	gray_code_sequence_string(lowercase				)
     #
     # convert them to integers
     for i in range(len(lowercase				)				):
          _UpperCamelCase       =	int(sequence[i], 2				)

     return sequence







def  a__    (    lowercase :					int				)		->		list:





     """simple docstring"""

     if bit_count == 0:
          return ["0"]

     if bit_count == 1:
          return ["0", "1"]

     _UpperCamelCase       =	1 << bit_count  # defines the length of the sequence
     # 1<< n is equivalent to 2^n

     # recursive answer will generate answer for n-1 bits
     _UpperCamelCase       =	gray_code_sequence_string(bit_count - 1				)

     _UpperCamelCase       =	[]

     # append 0 to first half of the smaller sequence generated
     for i in range(seq_len // 2				):
          _UpperCamelCase       =	'''0''' + smaller_sequence[i]
          sequence.append(lowercase				)

     # append 1 to second half ... start from the end of the list
     for i in reversed(range(seq_len // 2				)				):
          _UpperCamelCase       =	'''1''' + smaller_sequence[i]
          sequence.append(lowercase				)

     return sequence


if __name__ == "__main__":
   import doctest

   doctest.testmod()










'''simple docstring'''

def  a__    (    lowercase :					int, lowercase :					int				)		->		int:





     """simple docstring"""

     return x if y == 0 else greatest_common_divisor(lowercase, x % y				)







def  a__    (    lowercase :					int, lowercase :					int				)		->		int:





     """simple docstring"""

     return (x * y) // greatest_common_divisor(lowercase, lowercase				)







def  a__    (    lowercase :					int = 20				)		->		int:





     """simple docstring"""

     _UpperCamelCase       =	1
     for i in range(1, n + 1				):
          _UpperCamelCase       =	lcm(lowercase, lowercase				)
     return g


if __name__ == "__main__":
   print(F"""{solution() = }""")



