code: 
  
  
  """simple docstring"""
  
  
  
  
  def 						__a      (  a,		a,		a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if start == end:
                return start if lst[start] > item else start + 1
         if start > end:
                return start
  
         _a          =				(start + end) // 2
         if lst[mid] < item:
                return binary_search(a,		a,		mid + 1,		a					)
         elif lst[mid] > item:
                return binary_search(a,		a,		a,		mid - 1					)
         else:
                return mid
  
  
  
  def 						__a      (  a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         _a          =				len(a					)
  
         for index in range(1,		a					):
                _a          =				lst[index]
                _a          =				binary_search(a,		a,		0,		index - 1					)
                _a          =				lst[:pos] + [value] + lst[pos:index] + lst[index + 1 :]
  
         return lst
  
  
  
  def 						__a      (  a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if not left:
                return right
  
         if not right:
                return left
  
         if left[0] < right[0]:
                return [left[0], *merge(left[1:],		a					)]
  
         return [right[0], *merge(a,		right[1:]					)]
  
  
  
  def 						__a      (  a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         _a          =				len(a					)
         _a   , _a          =				[], []
         _a          =				[lst[0]]
         _a          =				[]
         _a          =				1
         while i < length:
                if lst[i] < lst[i - 1]:
                       runs.append(a					)
                       _a          =				[lst[i]]
                else:
                       new_run.append(lst[i]					)
                i += 1
         runs.append(a					)
  
         for run in runs:
                sorted_runs.append(insertion_sort(a					)					)
         for run in sorted_runs:
                _a          =				merge(a,		a					)
  
         return sorted_array
  
  
  
  def 						__a      (  ):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         _a          =				[5, 9, 1_0, 3, -4, 5, 1_7_8, 9_2, 4_6, -1_8, 0, 7]
         _a          =				tim_sort(a					)
         print(a					)
  
  
  if __name__ == "__main__":
     main()
  
style_context: 
  
  
  """simple docstring"""
  
  
  
  
  from __future__ import annotations
  
  __SCREAMING_SNAKE_CASE					=						[2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]
  
  
  
  
  
  class __snake_case		:
  
  
  
  
  
  
  
     """simple docstring"""
  
  
  
  
     def __init__(		self							:Optional[Any]				,  UpperCamelCase__							:int       ):
            _a          =				-1
            _a          =				-1
            _a          =				[-1] * length
            _a          =				None
            _a          =				None
  
  
  
  
     def __repr__(		self							:Optional[Any]       ):
            return f'Node(min_value={self.minn} max_value={self.maxx})'
  
  
  
  
  
  
  def 						__a      (  a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         _a          =				Node(len(a					)					)
         _a   , _a          =				min(a					), max(a					)
         # Leaf node case where the node contains only one unique value
         if root.minn == root.maxx:
                return root
         _a          =				(root.minn + root.maxx) // 2
  
         _a          =				[]
         _a          =				[]
  
         for index, num in enumerate(a					):
                if num <= pivot:
                       left_arr.append(a					)
                else:
                       right_arr.append(a					)
                _a          =				len(a					)
         _a          =				build_tree(a					)
         _a          =				build_tree(a					)
         return root
  
  
  
  def 						__a      (  a,		a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if index < 0 or node is None:
                return 0
         # Leaf node cases
         if node.minn == node.maxx:
                return index + 1 if node.minn == num else 0
         _a          =				(node.minn + node.maxx) // 2
         if num <= pivot:
                # go the left subtree and map index to the left subtree
                return rank_till_index(node.left,		a,		node.map_left[index] - 1					)
         else:
                # go to the right subtree and map index to the right subtree
                return rank_till_index(node.right,		a,		index - node.map_left[index]					)
  
  
  
  def 						__a      (  a,		a,		a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if start > end:
                return 0
         _a          =				rank_till_index(a,		a,		a					)
         _a          =				rank_till_index(a,		a,		start - 1					)
         return rank_till_end - rank_before_start
  
  
  
  def 						__a      (  a,		a,		a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if index > (end - start) or start > end or node is None:
                return -1
         # Leaf node case
         if node.minn == node.maxx:
                return node.minn
         # Number of elements in the left subtree in interval [start, end]
         _a          =				node.map_left[end] - (
             node.map_left[start - 1] if start else 0
         )
         if num_elements_in_left_tree > index:
                return quantile(
                    node.left,		a,		(node.map_left[start - 1] if start else 0),		node.map_left[end] - 1,		)
         else:
                return quantile(
                    node.right,		index - num_elements_in_left_tree,		start - (node.map_left[start - 1] if start else 0),		end - node.map_left[end],		)
  
  
  
  def 						__a      (  a,		a,		a,		a,		a					):
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
  
         if (
             start > end
             or node is None
             or start_num > end_num
             or node.minn > end_num
             or node.maxx < start_num
         ):
                return 0
         if start_num <= node.minn and node.maxx <= end_num:
                return end - start + 1
         _a          =				range_counting(
             node.left,		(node.map_left[start - 1] if start else 0),		node.map_left[end] - 1,		a,		a,		)
         _a          =				range_counting(
             node.right,		start - (node.map_left[start - 1] if start else 0),		end - node.map_left[end],		a,		a,		)
         return left + right
  
  
  if __name__ == "__main__":
     import doctest
  
     doctest.testmod()
  