text: '''This module implements specialized container datatypes providing
  alternatives to Python's general purpose built-in containers, dict,
  list, set, and tuple.
  
  * namedtuple   factory function for creating tuple subclasses with named fields
  * deque        list-like container with fast appends and pops on either end
  * ChainMap     dict-like class for creating a single view of multiple mappings
  * Counter      dict subclass for counting hashable objects
  * OrderedDict  dict subclass that remembers the order entries were added
  * defaultdict  dict subclass that calls a factory function to supply missing values
  * UserDict     wrapper around dictionary objects for easier dict subclassing
  * UserList     wrapper around list objects for easier list subclassing
  * UserString   wrapper around string objects for easier string subclassing
  
  '''
  
  __all__ = ['deque', 'defaultdict', 'namedtuple', 'UserDict', 'UserList',
              'UserString', 'Counter', 'OrderedDict', 'ChainMap']
  
  import _collections_abc
  from operator import itemgetter as _itemgetter, eq as _eq
  from keyword import iskeyword as _iskeyword
  import sys as _sys
  import heapq as _heapq
  from _weakref import proxy as _proxy
  from itertools import repeat as _repeat, chain as _chain, starmap as _starmap
  from reprlib import recursive_repr as _recursive_repr
  
  try:
      from _collections import deque
  except ImportError:
      pass
  else:
      _collections_abc.MutableSequence.register(deque)
  
  try:
      from _collections import defaultdict
  except ImportError:
      pass
  
  
  def __getattr__(name):
      # For backwards compatibility, continue to make the collections ABCs
      # through Python 3.6 available through the collections module.
      # Note, no new collections ABCs were added in Python 3.7
      if name in _collections_abc.__all__:
          obj = getattr(_collections_abc, name)
          import warnings
          warnings.warn("Using or importing the ABCs from 'collections' instead "
                        "of from 'collections.abc' is deprecated, "
                        "and in 3.8 it will stop working",
                        DeprecationWarning, stacklevel=2)
          globals()[name] = obj
          return obj
      raise AttributeError(f'module {__name__!r} has no attribute {name!r}')
  
  ################################################################################
  ### OrderedDict
  ################################################################################
  
  class _OrderedDictKeysView(_collections_abc.KeysView):
  
      def __reversed__(self):
          yield from reversed(self._mapping)
  
  class _OrderedDictItemsView(_collections_abc.ItemsView):
  
      def __reversed__(self):
          for key in reversed(self._mapping):
              yield (key, self._mapping[key])
  
  class _OrderedDictValuesView(_collections_abc.ValuesView):
  
      def __reversed__(self):
          for key in reversed(self._mapping):
              yield self._mapping[key]
  
  class _Link(object):
      __slots__ = 'prev', 'next', 'key', '__weakref__'
  
  class OrderedDict(dict):
      'Dictionary that remembers insertion order'
      # An inherited dict maps keys to values.
      # The inherited dict provides __getitem__, __len__, __contains__, and get.
      # The remaining methods are order-aware.
      # Big-O running times for all methods are the same as regular dictionaries.
  
      # The internal self.__map dict maps keys to links in a doubly linked list.
      # The circular doubly linked list starts and ends with a sentinel element.
      # The sentinel element never gets deleted (this simplifies the algorithm).
      # The sentinel is in self.__hardroot with a weakref proxy in self.__root.
      # The prev links are weakref proxies (to prevent circular references).
      # Individual links are kept alive by the hard reference in self.__map.
      # Those hard references disappear when a key is deleted from an OrderedDict.
  
      def __init__(*args, **kwds):
          '''Initialize an ordered dictionary.  The signature is the same as
          regular dictionaries.  Keyword argument order is preserved.
          '''
          if not args:
              raise TypeError("descriptor '__init__' of 'OrderedDict' object "
                              "needs an argument")
          self, *args = args
          if len(args) > 1:
              raise TypeError('expected at most 1 arguments, got %d' % len(args))
          try:
              self.__root
          except AttributeError:
              self.__hardroot = _Link()
              self.__root = root = _proxy(self.__hardroot)
              root.prev = root.next = root
              self.__map = {}
          self.__update(*args, **kwds)
  
      def __setitem__(self, key, value,
                      dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
          'od.__setitem__(i, y) <==> od[i]=y'
          # Setting a new item creates a new link at the end of the linked list,
          # and the inherited dictionary is updated with the new key/value pair.
          if key not in self:
              self.__map[key] = link = Link()
              root = self.__root
              last = root.prev
              link.prev, link.next, link.key = last, root, key
              last.next = link
              root.prev = proxy(link)
          dict_setitem(self, key, value)
  
      def __delitem__(self, key, dict_delitem=dict.__delitem__):
          'od.__delitem__(y) <==> del od[y]'
          # Deleting an existing item uses self.__map to find the link which gets
          # removed by updating the links in the predecessor and successor nodes.
          dict_delitem(self, key)
          link = self.__map.pop(key)
          link_prev = link.prev
          link_next = link.next
          link_prev.next = link_next
          link_next.prev = link_prev
          link.prev = None
          link.next = None
  
      def __iter__(self):
          'od.__iter__() <==> iter(od)'
          # Traverse the linked list in order.
          root = self.__root
          curr = root.next
          while curr is not root:
              yield curr.key
              curr = curr.next
  
      def __reversed__(self):
          'od.__reversed__() <==> reversed(od)'
          # Traverse the linked list in reverse order.
          root = self.__root
          curr = root.prev
          while curr is not root:
              yield curr.key
              curr = curr.prev
  
      def clear(self):
          'od.clear() -> None.  Remove all items from od.'
          root = self.__root
          root.prev = root.next = root
          self.__map.clear()
          dict.clear(self)
  
      def popitem(self, last=True):
          '''Remove and return a (key, value) pair from the dictionary.
  
          Pairs are returned in LIFO order if last is true or FIFO order if false.
          '''
          if not self:
              raise KeyError('dictionary is empty')
          root = self.__root
          if last:
              link = root.prev
              link_prev = link.prev
              link_prev.next = root
              root.prev = link_prev
          else:
              link = root.next
              link_next = link.next
              root.next = link_next
              link_next.prev = root
          key = link.key
          del self.__map[key]
          value = dict.pop(self, key)
          return key, value
  
      def move_to_end(self, key, last=True):
          '''Move an existing element to the end (or beginning if last is false).
  
          Raise KeyError if the element does not exist.
          '''
          link = self.__map[key]
          link_prev = link.prev
          link_next = link.next
          soft_link = link_next.prev
          link_prev.next = link_next
          link_next.prev = link_prev
          root = self.__root
          if last:
              last = root.prev
              link.prev = last
              link.next = root
              root.prev = soft_link
              last.next = link
          else:
              first = root.next
              link.prev = root
              link.next = first
              first.prev = soft_link
              root.next = link
  
      def __sizeof__(self):
          sizeof = _sys.getsizeof
          n = len(self) + 1                       # number of links including root
          size = sizeof(self.__dict__)            # instance dictionary
          size += sizeof(self.__map) * 2          # internal dict and inherited dict
          size += sizeof(self.__hardroot) * n     # link objects
          size += sizeof(self.__root) * n         # proxy objects
          return size
  
      update = __update = _collections_abc.MutableMapping.update
  
      def keys(self):
          "D.keys() -> a set-like object providing a view on D's keys"
          return _OrderedDictKeysView(self)
  
      def items(self):
          "D.items() -> a set-like object providing a view on D's items"
          return _OrderedDictItemsView(self)
  
      def values(self):
          "D.values() -> an object providing a view on D's values"
          return _OrderedDictValuesView(self)
  
      __ne__ = _collections_abc.MutableMapping.__ne__
  
      __marker = object()
  
      def pop(self, key, default=__marker):
          '''od.pop(k[,d]) -> v, remove specified key and return the corresponding
          value.  If key is not found, d is returned if given, otherwise KeyError
          is raised.
  
          '''
          if key in self:
              result = self[key]
              del self[key]
              return result
          if default is self.__marker:
              raise KeyError(key)
          return default
  
      def setdefault(self, key, default=None):
          '''Insert key with a value of default if key is not in the dictionary.
  
          Return the value for key if key is in the dictionary, else default.
          '''
          if key in self:
              return self[key]
          self[key] = default
          return default
  
      @_recursive_repr()
      def __repr__(self):
          'od.__repr__() <==> repr(od)'
          if not self:
              return '%s()' % (self.__class__.__name__,)
          return '%s(%r)' % (self.__class__.__name__, list(self.items()))
  
      def __reduce__(self):
          'Return state information for pickling'
          inst_dict = vars(self).copy()
          for k in vars(OrderedDict()):
              inst_dict.pop(k, None)
          return self.__class__, (), inst_dict or None, None, iter(self.items())
  
      def copy(self):
          'od.copy() -> a shallow copy of od'
          return self.__class__(self)
  
      @classmethod
      def fromkeys(cls, iterable, value=None):
          '''Create a new ordered dictionary with keys from iterable and values set to value.
          '''
          self = cls()
          for key in iterable:
              self[key] = value
          return self
  
      def __eq__(self, other):
          '''od.__eq__(y) <==> od==y.  Comparison to another OD is order-sensitive
          while comparison to a regular mapping is order-insensitive.
  
          '''
          if isinstance(other, OrderedDict):
              return dict.__eq__(self, other) and all(map(_eq, self, other))
          return dict.__eq__(self, other)
  
  
  try:
      from _collections import OrderedDict
  except ImportError:
      # Leave the pure Python version in place.
      pass
  
  
  ################################################################################
  ### namedtuple
  ################################################################################
  
  try:
      from _collections import _tuplegetter
  except ImportError:
      _tuplegetter = lambda index, doc: property(_itemgetter(index), doc=doc)
  
  def namedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
      """Returns a new subclass of tuple with named fields.
  
      >>> Point = namedtuple('Point', ['x', 'y'])
      >>> Point.__doc__                   # docstring for the new class
      'Point(x, y)'
      >>> p = Point(11, y=22)             # instantiate with positional args or keywords
      >>> p[0] + p[1]                     # indexable like a plain tuple
      33
      >>> x, y = p                        # unpack like a regular tuple
      >>> x, y
      (11, 22)
      >>> p.x + p.y                       # fields also accessible by name
      33
      >>> d = p._asdict()                 # convert to a dictionary
      >>> d['x']
      11
      >>> Point(**d)                      # convert from a dictionary
      Point(x=11, y=22)
      >>> p._replace(x=100)               # _replace() is like str.replace() but targets named fields
      Point(x=100, y=22)
  
      """
  
      # Validate the field names.  At the user's option, either generate an error
      # message or automatically replace the field name with a valid name.
      if isinstance(field_names, str):
          field_names = field_names.replace(',', ' ').split()
      field_names = list(map(str, field_names))
      typename = _sys.intern(str(typename))
  
      if rename:
          seen = set()
          for index, name in enumerate(field_names):
              if (not name.isidentifier()
                  or _iskeyword(name)
                  or name.startswith('_')
                  or name in seen):
                  field_names[index] = f'_{index}'
              seen.add(name)
  
      for name in [typename] + field_names:
          if type(name) is not str:
              raise TypeError('Type names and field names must be strings')
          if not name.isidentifier():
              raise ValueError('Type names and field names must be valid '
                               f'identifiers: {name!r}')
          if _iskeyword(name):
              raise ValueError('Type names and field names cannot be a '
                               f'keyword: {name!r}')
  
      seen = set()
      for name in field_names:
          if name.startswith('_') and not rename:
              raise ValueError('Field names cannot start with an underscore: '
                               f'{name!r}')
          if name in seen:
              raise ValueError(f'Encountered duplicate field name: {name!r}')
          seen.add(name)
  
      field_defaults = {}
      if defaults is not None:
          defaults = tuple(defaults)
          if len(defaults) > len(field_names):
              raise TypeError('Got more default values than field names')
          field_defaults = dict(reversed(list(zip(reversed(field_names),
                                                  reversed(defaults)))))
  
      # Variables used in the methods and docstrings
      field_names = tuple(map(_sys.intern, field_names))
      num_fields = len(field_names)
      arg_list = repr(field_names).replace("'", "")[1:-1]
      repr_fmt = '(' + ', '.join(f'{name}=%r' for name in field_names) + ')'
      tuple_new = tuple.__new__
      _dict, _tuple, _len, _map, _zip = dict, tuple, len, map, zip
  
      # Create all the named tuple methods to be added to the class namespace
  
      s = f'def __new__(_cls, {arg_list}): return _tuple_new(_cls, ({arg_list}))'
      namespace = {'_tuple_new': tuple_new, '__name__': f'namedtuple_{typename}'}
      # Note: exec() has the side-effect of interning the field names
      exec(s, namespace)
      __new__ = namespace['__new__']
      __new__.__doc__ = f'Create new instance of {typename}({arg_list})'
      if defaults is not None:
          __new__.__defaults__ = defaults
  
      @classmethod
      def _make(cls, iterable):
          result = tuple_new(cls, iterable)
          if _len(result) != num_fields:
              raise TypeError(f'Expected {num_fields} arguments, got {len(result)}')
          return result
  
      _make.__func__.__doc__ = (f'Make a new {typename} object from a sequence '
                                'or iterable')
  
      def _replace(_self, **kwds):
          result = _self._make(_map(kwds.pop, field_names, _self))
          if kwds:
              raise ValueError(f'Got unexpected field names: {list(kwds)!r}')
          return result
  
      _replace.__doc__ = (f'Return a new {typename} object replacing specified '
                          'fields with new values')
  
      def __repr__(self):
          'Return a nicely formatted representation string'
          return self.__class__.__name__ + repr_fmt % self
  
      def _asdict(self):
          'Return a new dict which maps field names to their values.'
          return _dict(_zip(self._fields, self))
  
      def __getnewargs__(self):
          'Return self as a plain tuple.  Used by copy and pickle.'
          return _tuple(self)
  
      # Modify function metadata to help with introspection and debugging
      for method in (__new__, _make.__func__, _replace,
                     __repr__, _asdict, __getnewargs__):
          method.__qualname__ = f'{typename}.{method.__name__}'
  
      # Build-up the class namespace dictionary
      # and use type() to build the result class
      class_namespace = {
          '__doc__': f'{typename}({arg_list})',
          '__slots__': (),
          '_fields': field_names,
          '_field_defaults': field_defaults,
          # alternate spelling for backward compatiblity
          '_fields_defaults': field_defaults,
          '__new__': __new__,
          '_make': _make,
          '_replace': _replace,
          '__repr__': __repr__,
          '_asdict': _asdict,
          '__getnewargs__': __getnewargs__,
      }
      for index, name in enumerate(field_names):
          doc = _sys.intern(f'Alias for field number {index}')
          class_namespace[name] = _tuplegetter(index, doc)
  
      result = type(typename, (tuple,), class_namespace)
  
      # For pickling to work, the __module__ variable needs to be set to the frame
      # where the named tuple is created.  Bypass this step in environments where
      # sys._getframe is not defined (Jython for example) or sys._getframe is not
      # defined for arguments greater than 0 (IronPython), or where the user has
      # specified a particular module.
      if module is None:
          try:
              module = _sys._getframe(1).f_globals.get('__name__', '__main__')
          except (AttributeError, ValueError):
              pass
      if module is not None:
          result.__module__ = module
  
      return result
  
  
  ########################################################################
  ###  Counter
  ########################################################################
  
  def _count_elements(mapping, iterable):
      'Tally elements from the iterable.'
      mapping_get = mapping.get
      for elem in iterable:
          mapping[elem] = mapping_get(elem, 0) + 1
  
  try:                                    # Load C helper function if available
      from _collections import _count_elements
  except ImportError:
      pass
  
  class Counter(dict):
      '''Dict subclass for counting hashable items.  Sometimes called a bag
      or multiset.  Elements are stored as dictionary keys and their counts
      are stored as dictionary values.
  
      >>> c = Counter('abcdeabcdabcaba')  # count elements from a string
  
      >>> c.most_common(3)                # three most common elements
      [('a', 5), ('b', 4), ('c', 3)]
      >>> sorted(c)                       # list all unique elements
      ['a', 'b', 'c', 'd', 'e']
      >>> ''.join(sorted(c.elements()))   # list elements with repetitions
      'aaaaabbbbcccdde'
      >>> sum(c.values())                 # total of all counts
      15
  
      >>> c['a']                          # count of letter 'a'
      5
      >>> for elem in 'shazam':           # update counts from an iterable
      ...     c[elem] += 1                # by adding 1 to each element's count
      >>> c['a']                          # now there are seven 'a'
      7
      >>> del c['b']                      # remove all 'b'
      >>> c['b']                          # now there are zero 'b'
      0
  
      >>> d = Counter('simsalabim')       # make another counter
      >>> c.update(d)                     # add in the second counter
      >>> c['a']                          # now there are nine 'a'
      9
  
      >>> c.clear()                       # empty the counter
      >>> c
      Counter()
  
      Note:  If a count is set to zero or reduced to zero, it will remain
      in the counter until the entry is deleted or the counter is cleared:
  
      >>> c = Counter('aaabbc')
      >>> c['b'] -= 2                     # reduce the count of 'b' by two
      >>> c.most_common()                 # 'b' is still in, but its count is zero
      [('a', 3), ('c', 1), ('b', 0)]
  
      '''
      # References:
      #   http://en.wikipedia.org/wiki/Multiset
      #   http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
      #   http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
      #   http://code.activestate.com/recipes/259174/
      #   Knuth, TAOCP Vol. II section 4.6.3
  
      def __init__(*args, **kwds):
          '''Create a new, empty Counter object.  And if given, count elements
          from an input iterable.  Or, initialize the count from another mapping
          of elements to their counts.
  
          >>> c = Counter()                           # a new, empty counter
          >>> c = Counter('gallahad')                 # a new counter from an iterable
          >>> c = Counter({'a': 4, 'b': 2})           # a new counter from a mapping
          >>> c = Counter(a=4, b=2)                   # a new counter from keyword args
  
          '''
          if not args:
              raise TypeError("descriptor '__init__' of 'Counter' object "
                              "needs an argument")
          self, *args = args
          if len(args) > 1:
              raise TypeError('expected at most 1 arguments, got %d' % len(args))
          super(Counter, self).__init__()
          self.update(*args, **kwds)
  
      def __missing__(self, key):
          'The count of elements not in the Counter is zero.'
          # Needed so that self[missing_item] does not raise KeyError
          return 0
  
      def most_common(self, n=None):
          '''List the n most common elements and their counts from the most
          common to the least.  If n is None, then list all element counts.
  
          >>> Counter('abracadabra').most_common(3)
          [('a', 5), ('b', 2), ('r', 2)]
  
          '''
          # Emulate Bag.sortedByCount from Smalltalk
          if n is None:
              return sorted(self.items(), key=_itemgetter(1), reverse=True)
          return _heapq.nlargest(n, self.items(), key=_itemgetter(1))
  
      def elements(self):
          '''Iterator over elements repeating each as many times as its count.
  
          >>> c = Counter('ABCABC')
          >>> sorted(c.elements())
          ['A', 'A', 'B', 'B', 'C', 'C']
  
          # Knuth's example for prime factors of 1836:  2**2 * 3**3 * 17**1
          >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
          >>> product = 1
          >>> for factor in prime_factors.elements():     # loop over factors
          ...     product *= factor                       # and multiply them
          >>> product
          1836
  
          Note, if an element's count has been set to zero or is a negative
          number, elements() will ignore it.
  
          '''
          # Emulate Bag.do from Smalltalk and Multiset.begin from C++.
          return _chain.from_iterable(_starmap(_repeat, self.items()))
  
      # Override dict methods where necessary
  
      @classmethod
      def fromkeys(cls, iterable, v=None):
          # There is no equivalent method for counters because the semantics
          # would be ambiguous in cases such as Counter.fromkeys('aaabbc', v=2).
          # Initializing counters to zero values isn't necessary because zero
          # is already the default value for counter lookups.  Initializing
          # to one is easily accomplished with Counter(set(iterable)).  For
          # more exotic cases, create a dictionary first using a dictionary
          # comprehension or dict.fromkeys().
          raise NotImplementedError(
              'Counter.fromkeys() is undefined.  Use Counter(iterable) instead.')
  
      def update(*args, **kwds):
          '''Like dict.update() but add counts instead of replacing them.
  
          Source can be an iterable, a dictionary, or another Counter instance.
  
          >>> c = Counter('which')
          >>> c.update('witch')           # add elements from another iterable
          >>> d = Counter('watch')
          >>> c.update(d)                 # add elements from another counter
          >>> c['h']                      # four 'h' in which, witch, and watch
          4
  
          '''
          # The regular dict.update() operation makes no sense here because the
          # replace behavior results in the some of original untouched counts
          # being mixed-in with all of the other counts for a mismash that
          # doesn't have a straight-forward interpretation in most counting
          # contexts.  Instead, we implement straight-addition.  Both the inputs
          # and outputs are allowed to contain zero and negative counts.
  
          if not args:
              raise TypeError("descriptor 'update' of 'Counter' object "
                              "needs an argument")
          self, *args = args
          if len(args) > 1:
              raise TypeError('expected at most 1 arguments, got %d' % len(args))
          iterable = args[0] if args else None
          if iterable is not None:
              if isinstance(iterable, _collections_abc.Mapping):
                  if self:
                      self_get = self.get
                      for elem, count in iterable.items():
                          self[elem] = count + self_get(elem, 0)
                  else:
                      super(Counter, self).update(iterable) # fast path when counter is empty
              else:
                  _count_elements(self, iterable)
          if kwds:
              self.update(kwds)
  
      def subtract(*args, **kwds):
          '''Like dict.update() but subtracts counts instead of replacing them.
          Counts can be reduced below zero.  Both the inputs and outputs are
          allowed to contain zero and negative counts.
  
          Source can be an iterable, a dictionary, or another Counter instance.
  
          >>> c = Counter('which')
          >>> c.subtract('witch')             # subtract elements from another iterable
          >>> c.subtract(Counter('watch'))    # subtract elements from another counter
          >>> c['h']                          # 2 in which, minus 1 in witch, minus 1 in watch
          0
          >>> c['w']                          # 1 in which, minus 1 in witch, minus 1 in watch
          -1
  
          '''
          if not args:
              raise TypeError("descriptor 'subtract' of 'Counter' object "
                              "needs an argument")
          self, *args = args
          if len(args) > 1:
              raise TypeError('expected at most 1 arguments, got %d' % len(args))
          iterable = args[0] if args else None
          if iterable is not None:
              self_get = self.get
              if isinstance(iterable, _collections_abc.Mapping):
                  for elem, count in iterable.items():
                      self[elem] = self_get(elem, 0) - count
              else:
                  for elem in iterable:
                      self[elem] = self_get(elem, 0) - 1
          if kwds:
              self.subtract(kwds)
  
      def copy(self):
          'Return a shallow copy.'
          return self.__class__(self)
  
      def __reduce__(self):
          return self.__class__, (dict(self),)
  
      def __delitem__(self, elem):
          'Like dict.__delitem__() but does not raise KeyError for missing values.'
          if elem in self:
              super().__delitem__(elem)
  
      def __repr__(self):
          if not self:
              return '%s()' % self.__class__.__name__
          try:
              items = ', '.join(map('%r: %r'.__mod__, self.most_common()))
              return '%s({%s})' % (self.__class__.__name__, items)
          except TypeError:
              # handle case where values are not orderable
              return '{0}({1!r})'.format(self.__class__.__name__, dict(self))
  
      # Multiset-style mathematical operations discussed in:
      #       Knuth TAOCP Volume II section 4.6.3 exercise 19
      #       and at http://en.wikipedia.org/wiki/Multiset
      #
      # Outputs guaranteed to only include positive counts.
      #
      # To strip negative and zero counts, add-in an empty counter:
      #       c += Counter()
  
      def __add__(self, other):
          '''Add counts from two counters.
  
          >>> Counter('abbb') + Counter('bcc')
          Counter({'b': 4, 'c': 2, 'a': 1})
  
          '''
          if not isinstance(other, Counter):
              return NotImplemented
          result = Counter()
          for elem, count in self.items():
              newcount = count + other[elem]
              if newcount > 0:
                  result[elem] = newcount
          for elem, count in other.items():
              if elem not in self and count > 0:
                  result[elem] = count
          return result
  
      def __sub__(self, other):
          ''' Subtract count, but keep only results with positive counts.
  
          >>> Counter('abbbc') - Counter('bccd')
          Counter({'b': 2, 'a': 1})
  
          '''
          if not isinstance(other, Counter):
              return NotImplemented
          result = Counter()
          for elem, count in self.items():
              newcount = count - other[elem]
              if newcount > 0:
                  result[elem] = newcount
          for elem, count in other.items():
              if elem not in self and count < 0:
                  result[elem] = 0 - count
          return result
  
      def __or__(self, other):
          '''Union is the maximum of value in either of the input counters.
  
          >>> Counter('abbb') | Counter('bcc')
          Counter({'b': 3, 'c': 2, 'a': 1})
  
          '''
          if not isinstance(other, Counter):
              return NotImplemented
          result = Counter()
          for elem, count in self.items():
              other_count = other[elem]
              newcount = other_count if count < other_count else count
              if newcount > 0:
                  result[elem] = newcount
          for elem, count in other.items():
              if elem not in self and count > 0:
                  result[elem] = count
          return result
  
      def __and__(self, other):
          ''' Intersection is the minimum of corresponding counts.
  
          >>> Counter('abbb') & Counter('bcc')
          Counter({'b': 1})
  
          '''
          if not isinstance(other, Counter):
              return NotImplemented
          result = Counter()
          for elem, count in self.items():
              other_count = other[elem]
              newcount = count if count < other_count else other_count
              if newcount > 0:
                  result[elem] = newcount
          return result
  
      def __pos__(self):
          'Adds an empty counter, effectively stripping negative and zero counts'
          result = Counter()
          for elem, count in self.items():
              if count > 0:
                  result[elem] = count
          return result
  
      def __neg__(self):
          '''Subtracts from an empty counter.  Strips positive and zero counts,
          and flips the sign on negative counts.
  
          '''
          result = Counter()
          for elem, count in self.items():
              if count < 0:
                  result[elem] = 0 - count
          return result
  
      def _keep_positive(self):
          '''Internal method to strip elements with a negative or zero count'''
          nonpositive = [elem for elem, count in self.items() if not count > 0]
          for elem in nonpositive:
              del self[elem]
          return self
  
      def __iadd__(self, other):
          '''Inplace add from another counter, keeping only positive counts.
  
          >>> c = Counter('abbb')
          >>> c += Counter('bcc')
          >>> c
          Counter({'b': 4, 'c': 2, 'a': 1})
  
          '''
          for elem, count in other.items():
              self[elem] += count
          return self._keep_positive()
  
      def __isub__(self, other):
          '''Inplace subtract counter, but keep only results with positive counts.
  
          >>> c = Counter('abbbc')
          >>> c -= Counter('bccd')
          >>> c
          Counter({'b': 2, 'a': 1})
  
          '''
          for elem, count in other.items():
              self[elem] -= count
          return self._keep_positive()
  
      def __ior__(self, other):
          '''Inplace union is the maximum of value from either counter.
  
          >>> c = Counter('abbb')
          >>> c |= Counter('bcc')
          >>> c
          Counter({'b': 3, 'c': 2, 'a': 1})
  
          '''
          for elem, other_count in other.items():
              count = self[elem]
              if other_count > count:
                  self[elem] = other_count
          return self._keep_positive()
  
      def __iand__(self, other):
          '''Inplace intersection is the minimum of corresponding counts.
  
          >>> c = Counter('abbb')
          >>> c &= Counter('bcc')
          >>> c
          Counter({'b': 1})
  
          '''
          for elem, count in self.items():
              other_count = other[elem]
              if other_count < count:
                  self[elem] = other_count
          return self._keep_positive()
  
  
  ########################################################################
  ###  ChainMap
  ########################################################################
  
  class ChainMap(_collections_abc.MutableMapping):
      ''' A ChainMap groups multiple dicts (or other mappings) together
      to create a single, updateable view.
  
      The underlying mappings are stored in a list.  That list is public and can
      be accessed or updated using the *maps* attribute.  There is no other
      state.
  
      Lookups search the underlying mappings successively until a key is found.
      In contrast, writes, updates, and deletions only operate on the first
      mapping.
  
      '''
  
      def __init__(self, *maps):
          '''Initialize a ChainMap by setting *maps* to the given mappings.
          If no mappings are provided, a single empty dictionary is used.
  
          '''
          self.maps = list(maps) or [{}]          # always at least one map
  
      def __missing__(self, key):
          raise KeyError(key)
  
      def __getitem__(self, key):
          for mapping in self.maps:
              try:
                  return mapping[key]             # can't use 'key in mapping' with defaultdict
              except KeyError:
                  pass
          return self.__missing__(key)            # support subclasses that define __missing__
  
      def get(self, key, default=None):
          return self[key] if key in self else default
  
      def __len__(self):
          return len(set().union(*self.maps))     # reuses stored hash values if possible
  
      def __iter__(self):
          d = {}
          for mapping in reversed(self.maps):
              d.update(mapping)                   # reuses stored hash values if possible
          return iter(d)
  
      def __contains__(self, key):
          return any(key in m for m in self.maps)
  
      def __bool__(self):
          return any(self.maps)
  
      @_recursive_repr()
      def __repr__(self):
          return f'{self.__class__.__name__}({", ".join(map(repr, self.maps))})'
  
      @classmethod
      def fromkeys(cls, iterable, *args):
          'Create a ChainMap with a single dict created from the iterable.'
          return cls(dict.fromkeys(iterable, *args))
  
      def copy(self):
          'New ChainMap or subclass with a new copy of maps[0] and refs to maps[1:]'
          return self.__class__(self.maps[0].copy(), *self.maps[1:])
  
      __copy__ = copy
  
      def new_child(self, m=None):                # like Django's Context.push()
          '''New ChainMap with a new map followed by all previous maps.
          If no map is provided, an empty dict is used.
          '''
          if m is None:
              m = {}
          return self.__class__(m, *self.maps)
  
      @property
      def parents(self):                          # like Django's Context.pop()
          'New ChainMap from maps[1:].'
          return self.__class__(*self.maps[1:])
  
      def __setitem__(self, key, value):
          self.maps[0][key] = value
  
      def __delitem__(self, key):
          try:
              del self.maps[0][key]
          except KeyError:
              raise KeyError('Key not found in the first mapping: {!r}'.format(key))
  
      def popitem(self):
          'Remove and return an item pair from maps[0]. Raise KeyError is maps[0] is empty.'
          try:
              return self.maps[0].popitem()
          except KeyError:
              raise KeyError('No keys found in the first mapping.')
  
      def pop(self, key, *args):
          'Remove *key* from maps[0] and return its value. Raise KeyError if *key* not in maps[0].'
          try:
              return self.maps[0].pop(key, *args)
          except KeyError:
              raise KeyError('Key not found in the first mapping: {!r}'.format(key))
  
      def clear(self):
          'Clear maps[0], leaving maps[1:] intact.'
          self.maps[0].clear()
  
  
  ################################################################################
  ### UserDict
  ################################################################################
  
  class UserDict(_collections_abc.MutableMapping):
  
      # Start by filling-out the abstract methods
      def __init__(*args, **kwargs):
          if not args:
              raise TypeError("descriptor '__init__' of 'UserDict' object "
                              "needs an argument")
          self, *args = args
          if len(args) > 1:
              raise TypeError('expected at most 1 arguments, got %d' % len(args))
          if args:
              dict = args[0]
          elif 'dict' in kwargs:
              dict = kwargs.pop('dict')
              import warnings
              warnings.warn("Passing 'dict' as keyword argument is deprecated",
                            DeprecationWarning, stacklevel=2)
          else:
              dict = None
          self.data = {}
          if dict is not None:
              self.update(dict)
          if kwargs:
              self.update(kwargs)
      __init__.__text_signature__ = '($self, dict=None, /, **kwargs)'
  
      def __len__(self): return len(self.data)
      def __getitem__(self, key):
          if key in self.data:
              return self.data[key]
          if hasattr(self.__class__, "__missing__"):
              return self.__class__.__missing__(self, key)
          raise KeyError(key)
      def __setitem__(self, key, item): self.data[key] = item
      def __delitem__(self, key): del self.data[key]
      def __iter__(self):
          return iter(self.data)
  
      # Modify __contains__ to work correctly when __missing__ is present
      def __contains__(self, key):
          return key in self.data
  
      # Now, add the methods in dicts but not in MutableMapping
      def __repr__(self): return repr(self.data)
      def __copy__(self):
          inst = self.__class__.__new__(self.__class__)
          inst.__dict__.update(self.__dict__)
          # Create a copy and avoid triggering descriptors
          inst.__dict__["data"] = self.__dict__["data"].copy()
          return inst
  
      def copy(self):
          if self.__class__ is UserDict:
              return UserDict(self.data.copy())
          import copy
          data = self.data
          try:
              self.data = {}
              c = copy.copy(self)
          finally:
              self.data = data
          c.update(self)
          return c
  
      @classmethod
      def fromkeys(cls, iterable, value=None):
          d = cls()
          for key in iterable:
              d[key] = value
          return d
  
  
  
  ################################################################################
  ### UserList
  ################################################################################
  
  class UserList(_collections_abc.MutableSequence):
      """A more or less complete user-defined wrapper around list objects."""
      def __init__(self, initlist=None):
          self.data = []
          if initlist is not None:
              # XXX should this accept an arbitrary sequence?
              if type(initlist) == type(self.data):
                  self.data[:] = initlist
              elif isinstance(initlist, UserList):
                  self.data[:] = initlist.data[:]
              else:
                  self.data = list(initlist)
      def __repr__(self): return repr(self.data)
      def __lt__(self, other): return self.data <  self.__cast(other)
      def __le__(self, other): return self.data <= self.__cast(other)
      def __eq__(self, other): return self.data == self.__cast(other)
      def __gt__(self, other): return self.data >  self.__cast(other)
      def __ge__(self, other): return self.data >= self.__cast(other)
      def __cast(self, other):
          return other.data if isinstance(other, UserList) else other
      def __contains__(self, item): return item in self.data
      def __len__(self): return len(self.data)
      def __getitem__(self, i):
          if isinstance(i, slice):
              return self.__class__(self.data[i])
          else:
              return self.data[i]
      def __setitem__(self, i, item): self.data[i] = item
      def __delitem__(self, i): del self.data[i]
      def __add__(self, other):
          if isinstance(other, UserList):
              return self.__class__(self.data + other.data)
          elif isinstance(other, type(self.data)):
              return self.__class__(self.data + other)
          return self.__class__(self.data + list(other))
      def __radd__(self, other):
          if isinstance(other, UserList):
              return self.__class__(other.data + self.data)
          elif isinstance(other, type(self.data)):
              return self.__class__(other + self.data)
          return self.__class__(list(other) + self.data)
      def __iadd__(self, other):
          if isinstance(other, UserList):
              self.data += other.data
          elif isinstance(other, type(self.data)):
              self.data += other
          else:
              self.data += list(other)
          return self
      def __mul__(self, n):
          return self.__class__(self.data*n)
      __rmul__ = __mul__
      def __imul__(self, n):
          self.data *= n
          return self
      def __copy__(self):
          inst = self.__class__.__new__(self.__class__)
          inst.__dict__.update(self.__dict__)
          # Create a copy and avoid triggering descriptors
          inst.__dict__["data"] = self.__dict__["data"][:]
          return inst
      def append(self, item): self.data.append(item)
      def insert(self, i, item): self.data.insert(i, item)
      def pop(self, i=-1): return self.data.pop(i)
      def remove(self, item): self.data.remove(item)
      def clear(self): self.data.clear()
      def copy(self): return self.__class__(self)
      def count(self, item): return self.data.count(item)
      def index(self, item, *args): return self.data.index(item, *args)
      def reverse(self): self.data.reverse()
      def sort(self, *args, **kwds): self.data.sort(*args, **kwds)
      def extend(self, other):
          if isinstance(other, UserList):
              self.data.extend(other.data)
          else:
              self.data.extend(other)
  
  
  
  ################################################################################
  ### UserString
  ################################################################################
  
  class UserString(_collections_abc.Sequence):
      def __init__(self, seq):
          if isinstance(seq, str):
              self.data = seq
          elif isinstance(seq, UserString):
              self.data = seq.data[:]
          else:
              self.data = str(seq)
      def __str__(self): return str(self.data)
      def __repr__(self): return repr(self.data)
      def __int__(self): return int(self.data)
      def __float__(self): return float(self.data)
      def __complex__(self): return complex(self.data)
      def __hash__(self): return hash(self.data)
      def __getnewargs__(self):
          return (self.data[:],)
  
      def __eq__(self, string):
          if isinstance(string, UserString):
              return self.data == string.data
          return self.data == string
      def __lt__(self, string):
          if isinstance(string, UserString):
              return self.data < string.data
          return self.data < string
      def __le__(self, string):
          if isinstance(string, UserString):
              return self.data <= string.data
          return self.data <= string
      def __gt__(self, string):
          if isinstance(string, UserString):
              return self.data > string.data
          return self.data > string
      def __ge__(self, string):
          if isinstance(string, UserString):
              return self.data >= string.data
          return self.data >= string
  
      def __contains__(self, char):
          if isinstance(char, UserString):
              char = char.data
          return char in self.data
  
      def __len__(self): return len(self.data)
      def __getitem__(self, index): return self.__class__(self.data[index])
      def __add__(self, other):
          if isinstance(other, UserString):
              return self.__class__(self.data + other.data)
          elif isinstance(other, str):
              return self.__class__(self.data + other)
          return self.__class__(self.data + str(other))
      def __radd__(self, other):
          if isinstance(other, str):
              return self.__class__(other + self.data)
          return self.__class__(str(other) + self.data)
      def __mul__(self, n):
          return self.__class__(self.data*n)
      __rmul__ = __mul__
      def __mod__(self, args):
          return self.__class__(self.data % args)
      def __rmod__(self, format):
          return self.__class__(format % args)
  
      # the following methods are defined in alphabetical order:
      def capitalize(self): return self.__class__(self.data.capitalize())
      def casefold(self):
          return self.__class__(self.data.casefold())
      def center(self, width, *args):
          return self.__class__(self.data.center(width, *args))
      def count(self, sub, start=0, end=_sys.maxsize):
          if isinstance(sub, UserString):
              sub = sub.data
          return self.data.count(sub, start, end)
      def encode(self, encoding=None, errors=None): # XXX improve this?
          if encoding:
              if errors:
                  return self.__class__(self.data.encode(encoding, errors))
              return self.__class__(self.data.encode(encoding))
          return self.__class__(self.data.encode())
      def endswith(self, suffix, start=0, end=_sys.maxsize):
          return self.data.endswith(suffix, start, end)
      def expandtabs(self, tabsize=8):
          return self.__class__(self.data.expandtabs(tabsize))
      def find(self, sub, start=0, end=_sys.maxsize):
          if isinstance(sub, UserString):
              sub = sub.data
          return self.data.find(sub, start, end)
      def format(self, *args, **kwds):
          return self.data.format(*args, **kwds)
      def format_map(self, mapping):
          return self.data.format_map(mapping)
      def index(self, sub, start=0, end=_sys.maxsize):
          return self.data.index(sub, start, end)
      def isalpha(self): return self.data.isalpha()
      def isalnum(self): return self.data.isalnum()
      def isascii(self): return self.data.isascii()
      def isdecimal(self): return self.data.isdecimal()
      def isdigit(self): return self.data.isdigit()
      def isidentifier(self): return self.data.isidentifier()
      def islower(self): return self.data.islower()
      def isnumeric(self): return self.data.isnumeric()
      def isprintable(self): return self.data.isprintable()
      def isspace(self): return self.data.isspace()
      def istitle(self): return self.data.istitle()
      def isupper(self): return self.data.isupper()
      def join(self, seq): return self.data.join(seq)
      def ljust(self, width, *args):
          return self.__class__(self.data.ljust(width, *args))
      def lower(self): return self.__class__(self.data.lower())
      def lstrip(self, chars=None): return self.__class__(self.data.lstrip(chars))
      maketrans = str.maketrans
      def partition(self, sep):
          return self.data.partition(sep)
      def replace(self, old, new, maxsplit=-1):
          if isinstance(old, UserString):
              old = old.data
          if isinstance(new, UserString):
              new = new.data
          return self.__class__(self.data.replace(old, new, maxsplit))
      def rfind(self, sub, start=0, end=_sys.maxsize):
          if isinstance(sub, UserString):
              sub = sub.data
          return self.data.rfind(sub, start, end)
      def rindex(self, sub, start=0, end=_sys.maxsize):
          return self.data.rindex(sub, start, end)
      def rjust(self, width, *args):
          return self.__class__(self.data.rjust(width, *args))
      def rpartition(self, sep):
          return self.data.rpartition(sep)
      def rstrip(self, chars=None):
          return self.__class__(self.data.rstrip(chars))
      def split(self, sep=None, maxsplit=-1):
          return self.data.split(sep, maxsplit)
      def rsplit(self, sep=None, maxsplit=-1):
          return self.data.rsplit(sep, maxsplit)
      def splitlines(self, keepends=False): return self.data.splitlines(keepends)
      def startswith(self, prefix, start=0, end=_sys.maxsize):
          return self.data.startswith(prefix, start, end)
      def strip(self, chars=None): return self.__class__(self.data.strip(chars))
      def swapcase(self): return self.__class__(self.data.swapcase())
      def title(self): return self.__class__(self.data.title())
      def translate(self, *args):
          return self.__class__(self.data.translate(*args))
      def upper(self): return self.__class__(self.data.upper())
      def zfill(self, width): return self.__class__(self.data.zfill(width))
  