


"""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()




"""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()
