


"""simple docstring"""






def 			A__		(					_UpperCAmelCase    :      int				,    _UpperCAmelCase    :      int				,    _UpperCAmelCase    :      Union[str, Any]				,    _UpperCAmelCase    :      List[Any]	)							->	Dict:

	'''simple docstring'''


	if start == end:
		return start if lst[start] > item else start + 1
	if start > end:
		return start

	snake_case__  :						Tuple		=      (start + end) // 2
	if lst[mid] < item:
		return binary_search(_UpperCAmelCase				,    _UpperCAmelCase				,    mid + 1				,    _UpperCAmelCase	)
	elif lst[mid] > item:
		return binary_search(_UpperCAmelCase				,    _UpperCAmelCase				,    _UpperCAmelCase				,    mid - 1	)
	else:
		return mid



def 			A__		(					_UpperCAmelCase    :      Dict	)							->	Dict:

	'''simple docstring'''


	snake_case__  :						Optional[Any]		=      len(_UpperCAmelCase	)

	for index in range(1				,    _UpperCAmelCase	):
		snake_case__  :						Union[str, Any]		=      lst[index]
		snake_case__  :						Union[str, Any]		=      binary_search(_UpperCAmelCase				,    _UpperCAmelCase				,    0				,    index - 1	)
		snake_case__  :						Tuple		=      lst[:pos] + [value] + lst[pos:index] + lst[index + 1 :]

	return lst



def 			A__		(					_UpperCAmelCase    :      str				,    _UpperCAmelCase    :      Optional[Any]	)							->	str:

	'''simple docstring'''


	if not left:
		return right

	if not right:
		return left

	if left[0] < right[0]:
		return [left[0], *merge(left[1:]				,    _UpperCAmelCase	)]

	return [right[0], *merge(_UpperCAmelCase				,    right[1:]	)]



def 			A__		(					_UpperCAmelCase    :      Any	)							->	str:

	'''simple docstring'''


	snake_case__  :						Tuple		=      len(_UpperCAmelCase	)
	snake_case__,					snake_case__  :						Optional[Any]		=      [], []
	snake_case__  :						Union[str, Any]		=      [lst[0]]
	snake_case__  :						Optional[int]		=      []
	snake_case__  :						Tuple		=      1
	while i < length:
		if lst[i] < lst[i - 1]:
			runs.append(_UpperCAmelCase	)
			snake_case__  :						Union[str, Any]		=      [lst[i]]
		else:
			new_run.append(lst[i]	)
		i += 1
	runs.append(_UpperCAmelCase	)

	for run in runs:
		sorted_runs.append(insertion_sort(_UpperCAmelCase	)	)
	for run in sorted_runs:
		snake_case__  :						str		=      merge(_UpperCAmelCase				,    _UpperCAmelCase	)

	return sorted_array



def 			A__		(					)							->	Tuple:

	'''simple docstring'''


	snake_case__  :						str		=      [5, 9, 10, 3, -4, 5, 1_78, 92, 46, -18, 0, 7]
	snake_case__  :						Dict		=      tim_sort(_UpperCAmelCase	)
	print(_UpperCAmelCase	)


if __name__ == "__main__":
					main()












from functools import lru_cache







@lru_cache
def      A__				(       SCREAMING_SNAKE_CASE__)  ->       int:
  if not isinstance(SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__):
    raise TypeError("""k must be an integer.""")
  if k < 0:
    raise ValueError("""k integer must be greater or equal to zero.""")
  if k == 0:
    return 0
  elif k == 1:
    return 1
  else:
    return fibonacci(k - 1) + fibonacci(k - 2)







def      A__				(       SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__)  ->       int:
  __snake_case:		str					    =		len(SCREAMING_SNAKE_CASE__)
  # Find m such that F_m >= n where F_i is the i_th fibonacci number.
  __snake_case:		Optional[int]					    =		0
  while True:
    if fibonacci(SCREAMING_SNAKE_CASE__) >= len_list:
      __snake_case:		str					    =		i
      break
    i += 1
  __snake_case:		Tuple					    =		0
  while fibb_k > 0:
    __snake_case:		str					    =		min(
        offset + fibonacci(fibb_k - 1)   ,   len_list - 1)  # Prevent out of range
    __snake_case:		List[Any]					    =		arr[index_k]
    if item_k_1 == val:
      return index_k
    elif val < item_k_1:
      fibb_k -= 1
    elif val > item_k_1:
      offset += fibonacci(fibb_k - 1)
      fibb_k -= 2
  else:
    return -1


if __name__ == "__main__":
      import doctest

      doctest.testmod()






