





from __future__ import annotations

from random import random
from typing import Generic, TypeVar

_snake_case           =			TypeVar("""KT""")
_snake_case           =			TypeVar("""VT""")





class  lowerCAmelCase							(       Generic[KT, VT] ):







  def __init__(      self  :int   ,							_lowercase  :KT | str = "root"   ,							_lowercase  :VT | None = None ):






     '''simple docstring'''





     lowercase__			       =  key
     lowercase__			       =  value
     lowercase__			       =  []







  def __repr__(      self  :Union[str, Any] ):






     '''simple docstring'''





     return f'''Node({self.key}: {self.value})'''







  @property
  def   UpperCAmelCase					(      self  :Optional[int] ):






     '''simple docstring'''





     return len(self.forward )





class  lowerCAmelCase							(       Generic[KT, VT] ):







  def __init__(      self  :Union[str, Any]   ,							_lowercase  :float = 0.5   ,							_lowercase  :int = 16 ):






     '''simple docstring'''





     lowercase__			       =  Node[KT, VT]()
     lowercase__			       =  0
     lowercase__			       =  p
     lowercase__			       =  max_level







  def __str__(      self  :Optional[int] ):






     '''simple docstring'''





     lowercase__			       =  list(self )

     if len(_lowercase ) == 0:
        return f'''SkipList(level={self.level})'''

     lowercase__			       =  max((len(str(_lowercase ) ) for item in items)   ,							default=4 )
     lowercase__			       =  max(_lowercase   ,							4 ) + 4

     lowercase__			       =  self.head
     lowercase__			       =  []

     lowercase__			       =  node.forward.copy()
     lines.append(f'''[{node.key}]'''.ljust(_lowercase   ,							"-" ) + "* " * len(_lowercase ) )
     lines.append(" " * label_size + "| " * len(_lowercase ) )

     while len(node.forward ) != 0:
        lowercase__			       =  node.forward[0]

        lines.append(
            f'''[{node.key}]'''.ljust(_lowercase   ,							"-" )
            + " ".join(str(n.key ) if n.key == node.key else "|" for n in forwards ) )
        lines.append(" " * label_size + "| " * len(_lowercase ) )
        lowercase__			       =  node.forward

     lines.append("None".ljust(_lowercase ) + "* " * len(_lowercase ) )
     return f'''SkipList(level={self.level})\n''' + "\n".join(_lowercase )







  def __iter__(      self  :List[Any] ):






     '''simple docstring'''





     lowercase__			       =  self.head

     while len(node.forward ) != 0:
        yield node.forward[0].key
        lowercase__			       =  node.forward[0]







  def   UpperCAmelCase					(      self  :str ):






     '''simple docstring'''





     lowercase__			       =  1
     while random() < self.p and level < self.max_level:
        level += 1

     return level







  def   UpperCAmelCase					(      self  :Dict   ,							_lowercase  :Tuple ):






     '''simple docstring'''





     lowercase__			       =  []

     lowercase__			       =  self.head

     for i in reversed(range(self.level ) ):
        # i < node.level - When node level is lesser than `i` decrement `i`.
        # node.forward[i].key < key - Jumping to node with key value higher
        #                             or equal to searched key would result
        #                             in skipping searched key.
        while i < node.level and node.forward[i].key < key:
           lowercase__			       =  node.forward[i]
        # Each leftmost node (relative to searched node) will potentially have to
        # be updated.
        update_vector.append(_lowercase )

     update_vector.reverse()  # Note that we were inserting values in reverse order.

     # len(node.forward) != 0 - If current node doesn't contain any further
     #                          references then searched key is not present.
     # node.forward[0].key == key - Next node key should be equal to search key
     #                              if key is present.
     if len(node.forward ) != 0 and node.forward[0].key == key:
        return node.forward[0], update_vector
     else:
        return None, update_vector







  def   UpperCAmelCase					(      self  :List[str]   ,							_lowercase  :KT ):






     '''simple docstring'''





     lowercase__       ,							lowercase__			       =  self._locate_node(_lowercase )

     if node is not None:
        for i, update_node in enumerate(_lowercase ):
           # Remove or replace all references to removed node.
           if update_node.level > i and update_node.forward[i].key == key:
              if node.level > i:
                 lowercase__			       =  node.forward[i]
              else:
                 lowercase__			       =  update_node.forward[:i]







  def   UpperCAmelCase					(      self  :Optional[Any]   ,							_lowercase  :KT   ,							_lowercase  :VT ):






     '''simple docstring'''





     lowercase__       ,							lowercase__			       =  self._locate_node(_lowercase )
     if node is not None:
        lowercase__			       =  value
     else:
        lowercase__			       =  self.random_level()

        if level > self.level:
           # After level increase we have to add additional nodes to head.
           for _ in range(self.level - 1   ,							_lowercase ):
              update_vector.append(self.head )
           lowercase__			       =  level

        lowercase__			       =  Node(_lowercase   ,							_lowercase )

        for i, update_node in enumerate(update_vector[:level] ):
           # Change references to pass through new node.
           if update_node.level > i:
              new_node.forward.append(update_node.forward[i] )

           if update_node.level < i + 1:
              update_node.forward.append(_lowercase )
           else:
              lowercase__			       =  new_node







  def   UpperCAmelCase					(      self  :Tuple   ,							_lowercase  :VT ):






     '''simple docstring'''





     lowercase__       ,							lowercase__			       =  self._locate_node(_lowercase )

     if node is not None:
        return node.value

     return None


def _A       (   ):
   lowercase__			       =  SkipList()
   skip_list.insert("Key1"  , 3	)
   skip_list.insert("Key2"  , 12	)
   skip_list.insert("Key3"  , 41	)
   skip_list.insert("Key4"  , -19	)

   lowercase__			       =  skip_list.head
   lowercase__			       =  {}
   while node.level != 0:
      lowercase__			       =  node.forward[0]
      lowercase__			       =  node.value

   assert len(__magic_name__	) == 4
   assert all_values["Key1"] == 3
   assert all_values["Key2"] == 12
   assert all_values["Key3"] == 41
   assert all_values["Key4"] == -19


def _A       (   ):
   lowercase__			       =  SkipList()
   skip_list.insert("Key1"  , 10	)
   skip_list.insert("Key1"  , 12	)

   skip_list.insert("Key5"  , 7	)
   skip_list.insert("Key7"  , 10	)
   skip_list.insert("Key10"  , 5	)

   skip_list.insert("Key7"  , 7	)
   skip_list.insert("Key5"  , 5	)
   skip_list.insert("Key10"  , 10	)

   lowercase__			       =  skip_list.head
   lowercase__			       =  {}
   while node.level != 0:
      lowercase__			       =  node.forward[0]
      lowercase__			       =  node.value

   if len(__magic_name__	) != 4:
      print()
   assert len(__magic_name__	) == 4
   assert all_values["Key1"] == 12
   assert all_values["Key7"] == 7
   assert all_values["Key5"] == 5
   assert all_values["Key10"] == 10


def _A       (   ):
   lowercase__			       =  SkipList()
   assert skip_list.find("Some key"	) is None


def _A       (   ):
   lowercase__			       =  SkipList()

   skip_list.insert("Key2"  , 20	)
   assert skip_list.find("Key2"	) == 20

   skip_list.insert("Some Key"  , 10	)
   skip_list.insert("Key2"  , 8	)
   skip_list.insert("V"  , 13	)

   assert skip_list.find("Y"	) is None
   assert skip_list.find("Key2"	) == 8
   assert skip_list.find("Some Key"	) == 10
   assert skip_list.find("V"	) == 13


def _A       (   ):
   lowercase__			       =  SkipList()
   skip_list.delete("Some key"	)

   assert len(skip_list.head.forward	) == 0


def _A       (   ):
   lowercase__			       =  SkipList()

   skip_list.insert("Key1"  , 12	)
   skip_list.insert("V"  , 13	)
   skip_list.insert("X"  , 14	)
   skip_list.insert("Key2"  , 15	)

   skip_list.delete("V"	)
   skip_list.delete("Key2"	)

   assert skip_list.find("V"	) is None
   assert skip_list.find("Key2"	) is None


def _A       (   ):
   lowercase__			       =  SkipList()

   skip_list.insert("Key1"  , 12	)
   skip_list.insert("V"  , 13	)
   skip_list.insert("X"  , 14	)
   skip_list.insert("Key2"  , 15	)

   skip_list.delete("V"	)
   assert skip_list.find("V"	) is None
   assert skip_list.find("X"	) == 14
   assert skip_list.find("Key1"	) == 12
   assert skip_list.find("Key2"	) == 15

   skip_list.delete("X"	)
   assert skip_list.find("V"	) is None
   assert skip_list.find("X"	) is None
   assert skip_list.find("Key1"	) == 12
   assert skip_list.find("Key2"	) == 15

   skip_list.delete("Key1"	)
   assert skip_list.find("V"	) is None
   assert skip_list.find("X"	) is None
   assert skip_list.find("Key1"	) is None
   assert skip_list.find("Key2"	) == 15

   skip_list.delete("Key2"	)
   assert skip_list.find("V"	) is None
   assert skip_list.find("X"	) is None
   assert skip_list.find("Key1"	) is None
   assert skip_list.find("Key2"	) is None


def _A       (   ):
   lowercase__			       =  SkipList()

   skip_list.insert("Key1"  , 12	)
   skip_list.insert("V"  , 13	)
   skip_list.insert("X"  , 142	)
   skip_list.insert("Key2"  , 15	)

   skip_list.delete("X"	)

   def traverse_keys(__magic_name__	):
      yield node.key
      for forward_node in node.forward:
         yield from traverse_keys(__magic_name__	)

   assert len(set(traverse_keys(skip_list.head	)	)	) == 4


def _A       (   ):
   def is_sorted(__magic_name__	):
      return all(next_item >= item for item, next_item in zip(__magic_name__  , lst[1:]	)	)

   lowercase__			       =  SkipList()
   for i in range(10	):
      skip_list.insert(__magic_name__  , __magic_name__	)
   assert is_sorted(list(__magic_name__	)	)
   skip_list.delete(5	)
   skip_list.delete(8	)
   skip_list.delete(2	)
   assert is_sorted(list(__magic_name__	)	)
   skip_list.insert(-12  , -12	)
   skip_list.insert(77  , 77	)
   assert is_sorted(list(__magic_name__	)	)


def _A       (   ):
   for _ in range(100	):
      # Repeat test 100 times due to the probabilistic nature of skip list
      # random values == random bugs
      test_insert()
      test_insert_overrides_existing_value()

      test_searching_empty_list_returns_none()
      test_search()

      test_deleting_item_from_empty_list_do_nothing()
      test_deleted_items_are_not_founded_by_find_method()
      test_delete_removes_only_given_key()
      test_delete_doesnt_leave_dead_nodes()

      test_iter_always_yields_sorted_values()


def _A       (   ):
   lowercase__			       =  SkipList()
   skip_list.insert(2  , "2"	)
   skip_list.insert(4  , "4"	)
   skip_list.insert(6  , "4"	)
   skip_list.insert(4  , "5"	)
   skip_list.insert(8  , "4"	)
   skip_list.insert(9  , "4"	)

   skip_list.delete(4	)

   print(__magic_name__	)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()













import math


def _A       (   __magic_name__	):
   lowercase__			       =  math.loga(math.sqrt(4 * positive_integer + 1	) / 2 + 1 / 2	)

   return exponent == int(__magic_name__	)


def _A       (   __magic_name__ = 1 / 1_2345	):
   lowercase__			       =  0
   lowercase__			       =  0

   lowercase__			       =  3
   while True:
      lowercase__			       =  (integer**2 - 1) / 4
      # if candidate is an integer, then there is a partition for k
      if partition_candidate == int(__magic_name__	):
         lowercase__			       =  int(__magic_name__	)
         total_partitions += 1
         if check_partition_perfect(__magic_name__	):
            perfect_partitions += 1
      if perfect_partitions > 0:
         if perfect_partitions / total_partitions < max_proportion:
            return int(__magic_name__	)
      integer += 1


if __name__ == "__main__":
  print(F"""{solution() = }""")






