


"""simple docstring"""
from __future__ import annotations






def 						lowercase				(	_snake_case		: list[int | float]	,  _snake_case		: int	,  _snake_case		: int	)	->int | float:
	"""simple docstring"""





	if len(_snake_case	) == 0:
		raise ValueError('''find_min() arg is an empty sequence'''	)
	if (
	    left >= len(_snake_case	)
	    or left < -len(_snake_case	)
	    or right >= len(_snake_case	)
	    or right < -len(_snake_case	)
	):
		raise IndexError('''list index out of range'''	)
	if left == right:
		return nums[left]
	__snake_case      :		List[Any]         =							(left + right) >> 1  # the middle
	__snake_case      :		Dict         =							find_min(_snake_case	,  _snake_case	,  _snake_case	)  # find min in range[left, mid]
	__snake_case      :		Tuple         =							find_min(_snake_case	,  mid + 1	,  _snake_case	)  # 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)






from __future__ import annotations







def     A__	(       __A ,			__A						):







    '''simple docstring'''
    _lowerCamelCase  :		List[str]			    =    [None] * (len(__A						) + len(__A						))

    _lowerCamelCase  :		Optional[int]			    =    0  # pointer to current index for left Half
    _lowerCamelCase  :		List[str]			    =    0  # pointer to current index for the right Half
    _lowerCamelCase  :		List[str]			    =    0  # pointer to current index for the sorted array Half

    while pointera < len(__A						) and pointera < len(__A						):
        if left_half[pointera] < right_half[pointera]:
            _lowerCamelCase  :		Optional[Any]			    =    left_half[pointera]
            pointera += 1
            index += 1
        else:
            _lowerCamelCase  :		List[Any]			    =    right_half[pointera]
            pointera += 1
            index += 1
    while pointera < len(__A						):
        _lowerCamelCase  :		int			    =    left_half[pointera]
        pointera += 1
        index += 1

    while pointera < len(__A						):
        _lowerCamelCase  :		Optional[Any]			    =    right_half[pointera]
        pointera += 1
        index += 1

    return sorted_array







def     A__	(       __A						):







    '''simple docstring'''
    if len(__A						) <= 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  :		int			    =    0 + (len(__A						) - 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  :		List[str]			    =    array[:middle]
    _lowerCamelCase  :		Optional[int]			    =    array[middle:]

    return merge(merge_sort(__A						) ,			merge_sort(__A						)						)


if __name__ == "__main__":
    import doctest

    doctest.testmod()




