class Set(Collection):

    """A set is a finite, iterable container.

    This class provides concrete generic implementations of all
    methods except for __contains__, __iter__ and __len__.

    To override the comparisons (presumably for speed, as the
    semantics are fixed), redefine __le__ and __ge__,
    then the other operations will automatically follow suit.
    """

    __slots__ = ()

    def __le__(self, other):
        if not isinstance(other, Set):
            return NotImplemented
        if len(self) > len(other):
            return False
        for elem in self:
            if elem not in other:
                return False
        return True

    def __lt__(self, other):
        if not isinstance(other, Set):
            return NotImplemented
        return len(self) < len(other) and self.__le__(other)

    def __gt__(self, other):
        if not isinstance(other, Set):
            return NotImplemented
        return len(self) > len(other) and self.__ge__(other)

    def __ge__(self, other):
        if not isinstance(other, Set):
            return NotImplemented
        if len(self) < len(other):
            return False
        for elem in other:
            if elem not in self:
                return False
        return True

    def __eq__(self, other):
        if not isinstance(other, Set):
            return NotImplemented
        return len(self) == len(other) and self.__le__(other)

    @classmethod
    def _from_iterable(cls, it):
        '''Construct an instance of the class from any iterable input.

        Must override this method if the class constructor signature
        does not accept an iterable for an input.
        '''
        return cls(it)

    def __and__(self, other):
        if not isinstance(other, Iterable):
            return NotImplemented
        return self._from_iterable(value for value in other if value in self)

    __rand__ = __and__

    def isdisjoint(self, other):
        'Return True if two sets have a null intersection.'
        for value in other:
            if value in self:
                return False
        return True

    def __or__(self, other):
        if not isinstance(other, Iterable):
            return NotImplemented
        chain = (e for s in (self, other) for e in s)
        return self._from_iterable(chain)

    __ror__ = __or__

    def __sub__(self, other):
        if not isinstance(other, Set):
            if not isinstance(other, Iterable):
                return NotImplemented
            other = self._from_iterable(other)
        return self._from_iterable(value for value in self
                                   if value not in other)

    def __rsub__(self, other):
        if not isinstance(other, Set):
            if not isinstance(other, Iterable):
                return NotImplemented
            other = self._from_iterable(other)
        return self._from_iterable(value for value in other
                                   if value not in self)

    def __xor__(self, other):
        if not isinstance(other, Set):
            if not isinstance(other, Iterable):
                return NotImplemented
            other = self._from_iterable(other)
        return (self - other) | (other - self)

    __rxor__ = __xor__

    def _hash(self):
        """Compute the hash value of a set.

        Note that we don't define __hash__: not all sets are hashable.
        But if you define a hashable set type, its __hash__ should
        call this function.

        This must be compatible __eq__.

        All sets ought to compare equal if they contain the same
        elements, regardless of how they are implemented, and
        regardless of the order of the elements; so there's not much
        freedom for __eq__ or __hash__.  We match the algorithm used
        by the built-in frozenset type.
        """
        MAX = sys.maxsize
        MASK = 2 * MAX + 1
        n = len(self)
        h = 1927868237 * (n + 1)
        h &= MASK
        for x in self:
            hx = hash(x)
            h ^= (hx ^ (hx << 16) ^ 89869747)  * 3644798167
            h &= MASK
        h = h * 69069 + 907133923
        h &= MASK
        if h > MAX:
            h -= MASK + 1
        if h == -1:
            h = 590923713
        return h

class MutableSet(Set):
    """A mutable set is a finite, iterable container.

    This class provides concrete generic implementations of all
    methods except for __contains__, __iter__, __len__,
    add(), and discard().

    To override the comparisons (presumably for speed, as the
    semantics are fixed), all you have to do is redefine __le__ and
    then the other operations will automatically follow suit.
    """

    __slots__ = ()

    @abstractmethod
    def add(self, value):
        """Add an element."""
        raise NotImplementedError

    @abstractmethod
    def discard(self, value):
        """Remove an element.  Do not raise an exception if absent."""
        raise NotImplementedError

    def remove(self, value):
        """Remove an element. If not a member, raise a KeyError."""
        if value not in self:
            raise KeyError(value)
        self.discard(value)

    def pop(self):
        """Return the popped value.  Raise KeyError if empty."""
        it = iter(self)
        try:
            value = next(it)
        except StopIteration:
            raise KeyError
        self.discard(value)
        return value

    def clear(self):
        """This is slow (creates N new iterators!) but effective."""
        try:
            while True:
                self.pop()
        except KeyError:
            pass

    def __ior__(self, it):
        for value in it:
            self.add(value)
        return self

    def __iand__(self, it):
        for value in (self - it):
            self.discard(value)
        return self

    def __ixor__(self, it):
        if it is self:
            self.clear()
        else:
            if not isinstance(it, Set):
                it = self._from_iterable(it)
            for value in it:
                if value in self:
                    self.discard(value)
                else:
                    self.add(value)
        return self

    def __isub__(self, it):
        if it is self:
            self.clear()
        else:
            for value in it:
                self.discard(value)
        return self

class MappingView(Sized):

    __slots__ = '_mapping',

    def __init__(self, mapping):
        self._mapping = mapping

    def __len__(self):
        return len(self._mapping)

    def __repr__(self):
        return '{0.__class__.__name__}({0._mapping!r})'.format(self)


class KeysView(MappingView, Set):

    __slots__ = ()

    @classmethod
    def _from_iterable(self, it):
        return set(it)

    def __contains__(self, key):
        return key in self._mapping

    def __iter__(self):
        yield from self._mapping


class ItemsView(MappingView, Set):

    __slots__ = ()

    @classmethod
    def _from_iterable(self, it):
        return set(it)

    def __contains__(self, item):
        key, value = item
        try:
            v = self._mapping[key]
        except KeyError:
            return False
        else:
            return v is value or v == value

    def __iter__(self):
        for key in self._mapping:
            yield (key, self._mapping[key])

class ValuesView(MappingView):

    __slots__ = ()

    def __contains__(self, value):
        for key in self._mapping:
            v = self._mapping[key]
            if v is value or v == value:
                return True
        return False

    def __iter__(self):
        for key in self._mapping:
            yield self._mapping[key]

class Mapping(Collection):

    __slots__ = ()

    """A Mapping is a generic container for associating key/value
    pairs.

    This class provides concrete generic implementations of all
    methods except for __getitem__, __iter__, and __len__.

    """

    @abstractmethod
    def __getitem__(self, key):
        raise KeyError

    def get(self, key, default=None):
        'D.get(k[,d]) -> D[k] if k in D, else d.  d defaults to None.'
        try:
            return self[key]
        except KeyError:
            return default

    def __contains__(self, key):
        try:
            self[key]
        except KeyError:
            return False
        else:
            return True

    def keys(self):
        "D.keys() -> a set-like object providing a view on D's keys"
        return KeysView(self)

    def items(self):
        "D.items() -> a set-like object providing a view on D's items"
        return ItemsView(self)

    def values(self):
        "D.values() -> an object providing a view on D's values"
        return ValuesView(self)

    def __eq__(self, other):
        if not isinstance(other, Mapping):
            return NotImplemented
        return dict(self.items()) == dict(other.items())

    __reversed__ = None

class MutableMapping(Mapping):

    __slots__ = ()

    """A MutableMapping is a generic container for associating
    key/value pairs.

    This class provides concrete generic implementations of all
    methods except for __getitem__, __setitem__, __delitem__,
    __iter__, and __len__.

    """

    @abstractmethod
    def __setitem__(self, key, value):
        raise KeyError

    @abstractmethod
    def __delitem__(self, key):
        raise KeyError

    __marker = object()

    def pop(self, key, default=__marker):
        '''D.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.
        '''
        try:
            value = self[key]
        except KeyError:
            if default is self.__marker:
                raise
            return default
        else:
            del self[key]
            return value

    def popitem(self):
        '''D.popitem() -> (k, v), remove and return some (key, value) pair
           as a 2-tuple; but raise KeyError if D is empty.
        '''
        try:
            key = next(iter(self))
        except StopIteration:
            raise KeyError
        value = self[key]
        del self[key]
        return key, value

    def clear(self):
        'D.clear() -> None.  Remove all items from D.'
        try:
            while True:
                self.popitem()
        except KeyError:
            pass

    def update(*args, **kwds):
        ''' D.update([E, ]**F) -> None.  Update D from mapping/iterable E and F.
            If E present and has a .keys() method, does:     for k in E: D[k] = E[k]
            If E present and lacks .keys() method, does:     for (k, v) in E: D[k] = v
            In either case, this is followed by: for k, v in F.items(): D[k] = v
        '''
        if not args:
            raise TypeError("descriptor 'update' of 'MutableMapping' object "
                            "needs an argument")
        self, *args = args
        if len(args) > 1:
            raise TypeError('update expected at most 1 arguments, got %d' %
                            len(args))
        if args:
            other = args[0]
            if isinstance(other, Mapping):
                for key in other:
                    self[key] = other[key]
            elif hasattr(other, "keys"):
                for key in other.keys():
                    self[key] = other[key]
            else:
                for key, value in other:
                    self[key] = value
        for key, value in kwds.items():
            self[key] = value

    def setdefault(self, key, default=None):
        'D.setdefault(k[,d]) -> D.get(k,d), also set D[k]=d if k not in D'
        try:
            return self[key]
        except KeyError:
            self[key] = default
        return default

class Sequence(Reversible, Collection):

    """All the operations on a read-only sequence.

    Concrete subclasses must override __new__ or __init__,
    __getitem__, and __len__.
    """

    __slots__ = ()

    @abstractmethod
    def __getitem__(self, index):
        raise IndexError

    def __iter__(self):
        i = 0
        try:
            while True:
                v = self[i]
                yield v
                i += 1
        except IndexError:
            return

    def __contains__(self, value):
        for v in self:
            if v is value or v == value:
                return True
        return False

    def __reversed__(self):
        for i in reversed(range(len(self))):
            yield self[i]

    def index(self, value, start=0, stop=None):
        '''S.index(value, [start, [stop]]) -> integer -- return first index of value.
           Raises ValueError if the value is not present.
        '''
        if start is not None and start < 0:
            start = max(len(self) + start, 0)
        if stop is not None and stop < 0:
            stop += len(self)

        i = start
        while stop is None or i < stop:
            try:
                if self[i] == value:
                    return i
            except IndexError:
                break
            i += 1
        raise ValueError

    def count(self, value):
        'S.count(value) -> integer -- return number of occurrences of value'
        return sum(1 for v in self if v == value)

class MutableSequence(Sequence):

    __slots__ = ()

    """All the operations on a read-write sequence.

    Concrete subclasses must provide __new__ or __init__,
    __getitem__, __setitem__, __delitem__, __len__, and insert().

    """

    @abstractmethod
    def __setitem__(self, index, value):
        raise IndexError

    @abstractmethod
    def __delitem__(self, index):
        raise IndexError

    @abstractmethod
    def insert(self, index, value):
        'S.insert(index, value) -- insert value before index'
        raise IndexError

    def append(self, value):
        'S.append(value) -- append value to the end of the sequence'
        self.insert(len(self), value)

    def clear(self):
        'S.clear() -> None -- remove all items from S'
        try:
            while True:
                self.pop()
        except IndexError:
            pass

    def reverse(self):
        'S.reverse() -- reverse *IN PLACE*'
        n = len(self)
        for i in range(n//2):
            self[i], self[n-i-1] = self[n-i-1], self[i]

    def extend(self, values):
        'S.extend(iterable) -- extend sequence by appending elements from the iterable'
        for v in values:
            self.append(v)

    def pop(self, index=-1):
        '''S.pop([index]) -> item -- remove and return item at index (default last).
           Raise IndexError if list is empty or index is out of range.
        '''
        v = self[index]
        del self[index]
        return v

    def remove(self, value):
        '''S.remove(value) -- remove first occurrence of value.
           Raise ValueError if the value is not present.
        '''
        del self[self.index(value)]

    def __iadd__(self, values):
        self.extend(values)
        return self