


"""simple docstring"""





from __future__ import annotations





def     lowercase__	(  lowerCamelCase     :  list[int | float]				,	lowerCamelCase     :  int				,	lowerCamelCase     :  int				)					->     int | float:
  if len(lowerCamelCase				) == 0:
    raise ValueError("find_min() arg is an empty sequence"				)
  if (
      left >= len(lowerCamelCase				)
      or left < -len(lowerCamelCase				)
      or right >= len(lowerCamelCase				)
      or right < -len(lowerCamelCase				)
  ):
    raise IndexError("list index out of range"				)
  if left == right:
    return nums[left]
  lowerCAmelCase__    :    Union[str, Any]								=					(left + right) >> 1  # the middle
  lowerCAmelCase__    :    str								=					find_min(lowerCamelCase				,	lowerCamelCase				,	lowerCamelCase				)  # find min in range[left, mid]
  lowerCAmelCase__    :    Optional[Any]								=					find_min(lowerCamelCase				,	mid + 1				,	lowerCamelCase				)  # find min in range[mid + 1, right]

  return left_min if left_min <= right_min else right_min


if __name__ == "__main__":
     import doctest

     doctest.testmod(verbose=True)




"""simple docstring"""





from __future__ import annotations





def     lowercase__	(  lowerCamelCase     :  list				,	lowerCamelCase     :  list				)					->     list:
  lowerCAmelCase__    :    Dict								=					[None] * (len(lowerCamelCase				) + len(lowerCamelCase				))

  lowerCAmelCase__    :    Optional[Any]								=					0  # pointer to current index for left Half
  lowerCAmelCase__    :    Any								=					0  # pointer to current index for the right Half
  lowerCAmelCase__    :    Dict								=					0  # pointer to current index for the sorted array Half

  while pointera < len(lowerCamelCase				) and pointera < len(lowerCamelCase				):
    if left_half[pointera] < right_half[pointera]:
      lowerCAmelCase__    :    Tuple								=					left_half[pointera]
      pointera += 1
      index += 1
    else:
      lowerCAmelCase__    :    int								=					right_half[pointera]
      pointera += 1
      index += 1
  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Tuple								=					left_half[pointera]
    pointera += 1
    index += 1

  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Optional[int]								=					right_half[pointera]
    pointera += 1
    index += 1

  return sorted_array





def     lowercase__	(  lowerCamelCase     :  list				)					->     list:
  if len(lowerCamelCase				) <= 1:
    return array
  # the actual formula to calculate the middle element = left + (right - left) // 2
  # this avoids integer overflow in case of large N
  lowerCAmelCase__    :    Optional[int]								=					0 + (len(lowerCamelCase				) - 0) // 2

  # Split the array into halves till the array length becomes equal to One
  # merge the arrays of single length returned by mergeSort function and
  # pass them into the merge arrays function which merges the array
  lowerCAmelCase__    :    Union[str, Any]								=					array[:middle]
  lowerCAmelCase__    :    List[str]								=					array[middle:]

  return merge(merge_sort(lowerCamelCase				)				,	merge_sort(lowerCamelCase				)				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()
