




"""simple docstring"""






from abc import ABC, abstractmethod
from typing import List, Optional






class lowerCAmelCase__			(  lowercase		):






			'''simple docstring'''



			def __init__(						self						):
							# test for the above condition
							self.test()
			def       A_					(						self						):
							_lowerCamelCase  :    int					=					0
							_lowerCamelCase  :    Optional[Any]					=					False
							while not completed:
											if counter == 1:
															self.reset()
											_lowerCamelCase  :    Any					=					self.advance()
											if not self.does_advance(lowercase						):
															raise Exception(
															    'Custom Constraint is not defined correctly. self.does_advance(self.advance()) must be true.'						)

											_lowerCamelCase  :    Tuple					=					self.update(lowercase						)
											counter += 1

											if counter > 10000:
															raise Exception('update() does not fulfill the constraint.'						)

							if self.remaining() != 0:
											raise Exception('Custom Constraint is not defined correctly.'						)
			@abstractmethod
			def       A_					(						self						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)
			@abstractmethod
			def       A_					(						self						,   lowercase						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)
			@abstractmethod
			def       A_					(						self						,   lowercase						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)
			@abstractmethod
			def       A_					(						self						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)
			@abstractmethod
			def       A_					(						self						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)






			@abstractmethod
			def       A_					(						self						,   lowercase=False						):
							raise NotImplementedError(
							    F'''{self.__class__} is an abstract class. Only classes inheriting this class can be called.'''						)






class lowerCAmelCase__			(  lowercase		):






			'''simple docstring'''



			def __init__(						self						,   lowercase						):
							super(lowercase						,   self						).__init__()

							if not isinstance(lowercase						,   lowercase						) or len(lowercase						) == 0:
											raise ValueError(F'''`token_ids` has to be a non-empty list, but is {token_ids}.'''						)
							if any((not isinstance(lowercase						,   lowercase						) or token_id < 0) for token_id in token_ids						):
											raise ValueError(F'''Each list in `token_ids` has to be a list of positive integers, but is {token_ids}.'''						)

							_lowerCamelCase  :    Any					=					token_ids

							_lowerCamelCase  :    Optional[int]					=					len(self.token_ids						)
							_lowerCamelCase  :    Optional[int]					=					-1  # the index of the currently fulfilled step
							_lowerCamelCase  :    Optional[Any]					=					False
			def       A_					(						self						):
							if self.completed:
											return None
							return self.token_ids[self.fulfilled_idx + 1]
			def       A_					(						self						,   lowercase						):
							if not isinstance(lowercase						,   lowercase						):
											raise ValueError(F'''`token_id` has to be an `int`, but is {token_id} of type {type(lowercase						)}'''						)

							if self.completed:
											return False

							return token_id == self.token_ids[self.fulfilled_idx + 1]
			def       A_					(						self						,   lowercase						):
							if not isinstance(lowercase						,   lowercase						):
											raise ValueError(F'''`token_id` has to be an `int`, but is {token_id} of type {type(lowercase						)}'''						)

							_lowerCamelCase  :    List[Any]					=					False
							_lowerCamelCase  :    int					=					False
							_lowerCamelCase  :    Any					=					False

							if self.does_advance(lowercase						):
											self.fulfilled_idx += 1
											_lowerCamelCase  :    int					=					True
											if self.fulfilled_idx == (self.seqlen - 1):
															_lowerCamelCase  :    Optional[int]					=					True
											_lowerCamelCase  :    Any					=					completed
							else:
											# failed to make progress.
											_lowerCamelCase  :    List[Any]					=					True
											self.reset()
							return stepped, completed, reset
			def       A_					(						self						):
							_lowerCamelCase  :    Union[str, Any]					=					False
							_lowerCamelCase  :    Optional[int]					=					0
			def       A_					(						self						):
							return self.seqlen - (self.fulfilled_idx + 1)






			def       A_					(						self						,   lowercase=False						):
							_lowerCamelCase  :    int					=					PhrasalConstraint(self.token_ids						)

							if stateful:
											_lowerCamelCase  :    List[str]					=					self.seqlen
											_lowerCamelCase  :    Optional[int]					=					self.fulfilled_idx
											_lowerCamelCase  :    int					=					self.completed

							return new_constraint






class lowerCAmelCase__			:






			'''simple docstring'''



			def __init__(						self						,   lowercase						,   lowercase=True						):
							_lowerCamelCase  :    Optional[Any]					=					max([len(lowercase						) for one in nested_token_ids]						)

							_lowerCamelCase  :    Tuple					=					{}
							for token_ids in nested_token_ids:
											_lowerCamelCase  :    str					=					root
											for tidx, token_id in enumerate(lowercase						):
															if token_id not in level:
																			_lowerCamelCase  :    Optional[int]					=					{}

															_lowerCamelCase  :    int					=					level[token_id]

							if no_subsets and self.has_subsets(lowercase						,   lowercase						):
											raise ValueError(
											    'Each list in `nested_token_ids` can\'t be a complete subset of another list, but is'
											    F''' {nested_token_ids}.'''						)

							_lowerCamelCase  :    str					=					root
			def       A_					(						self						,   lowercase						):
							_lowerCamelCase  :    List[str]					=					self.trie

							for current_token in current_seq:
											_lowerCamelCase  :    Dict					=					start[current_token]

							_lowerCamelCase  :    Optional[int]					=					list(start.keys()						)

							return next_tokens
			def       A_					(						self						,   lowercase						):
							_lowerCamelCase  :    int					=					self.next_tokens(lowercase						)

							return len(lowercase						) == 0
			def       A_					(						self						,   lowercase						):
							_lowerCamelCase  :    str					=					list(root.values()						)
							if len(lowercase						) == 0:
											return 1
							else:
											return sum([self.count_leaves(lowercase						) for nn in next_nodes]						)






			def       A_					(						self						,   lowercase						,   lowercase						):
							_lowerCamelCase  :    Tuple					=					self.count_leaves(lowercase						)
							return len(lowercase						) != leaf_count






class lowerCAmelCase__			(  lowercase		):






			'''simple docstring'''



			def __init__(						self						,   lowercase						):
							super(lowercase						,   self						).__init__()

							if not isinstance(lowercase						,   lowercase						) or len(lowercase						) == 0:
											raise ValueError(F'''`nested_token_ids` has to be a non-empty list, but is {nested_token_ids}.'''						)
							if any(not isinstance(lowercase						,   lowercase						) for token_ids in nested_token_ids						):
											raise ValueError(F'''`nested_token_ids` has to be a list of lists, but is {nested_token_ids}.'''						)
							if any(
							    any((not isinstance(lowercase						,   lowercase						) or token_id < 0) for token_id in token_ids						)
							    for token_ids in nested_token_ids						):
											raise ValueError(
											    F'''Each list in `nested_token_ids` has to be a list of positive integers, but is {nested_token_ids}.'''						)

							_lowerCamelCase  :    Union[str, Any]					=					DisjunctiveTrie(lowercase						)
							_lowerCamelCase  :    int					=					nested_token_ids

							_lowerCamelCase  :    int					=					self.trie.max_height
							_lowerCamelCase  :    str					=					[]
							_lowerCamelCase  :    Union[str, Any]					=					False
			def       A_					(						self						):
							_lowerCamelCase  :    List[Any]					=					self.trie.next_tokens(self.current_seq						)

							if len(lowercase						) == 0:
											return None
							else:
											return token_list
			def       A_					(						self						,   lowercase						):
							if not isinstance(lowercase						,   lowercase						):
											raise ValueError(F'''`token_id` is supposed to be type `int`, but is {token_id} of type {type(lowercase						)}'''						)

							_lowerCamelCase  :    Optional[Any]					=					self.trie.next_tokens(self.current_seq						)

							return token_id in next_tokens
			def       A_					(						self						,   lowercase						):
							if not isinstance(lowercase						,   lowercase						):
											raise ValueError(F'''`token_id` is supposed to be type `int`, but is {token_id} of type {type(lowercase						)}'''						)

							_lowerCamelCase  :    Any					=					False
							_lowerCamelCase  :    Tuple					=					False
							_lowerCamelCase  :    Union[str, Any]					=					False

							if self.does_advance(lowercase						):
											self.current_seq.append(lowercase						)
											_lowerCamelCase  :    Any					=					True
							else:
											_lowerCamelCase  :    Optional[int]					=					True
											self.reset()

							_lowerCamelCase  :    Union[str, Any]					=					self.trie.reached_leaf(self.current_seq						)
							_lowerCamelCase  :    List[Any]					=					completed

							return stepped, completed, reset
			def       A_					(						self						):
							_lowerCamelCase  :    Dict					=					False
							_lowerCamelCase  :    List[str]					=					[]
			def       A_					(						self						):
							if self.completed:
											# since this can be completed without reaching max height
											return 0
							else:
											return self.seqlen - len(self.current_seq						)






			def       A_					(						self						,   lowercase=False						):
							_lowerCamelCase  :    List[Any]					=					DisjunctiveConstraint(self.token_ids						)

							if stateful:
											_lowerCamelCase  :    Any					=					self.seqlen
											_lowerCamelCase  :    Dict					=					self.current_seq
											_lowerCamelCase  :    Tuple					=					self.completed

							return new_constraint






class lowerCAmelCase__			:






			'''simple docstring'''



			def __init__(						self						,   lowercase						):
							_lowerCamelCase  :    Tuple					=					constraints

							# max # of steps required to fulfill a given constraint
							_lowerCamelCase  :    Tuple					=					max([c.seqlen for c in constraints]						)
							_lowerCamelCase  :    Tuple					=					len(lowercase						)
							_lowerCamelCase  :    List[Any]					=					False

							self.init_state()
			def       A_					(						self						):
							_lowerCamelCase  :    int					=					[]
							_lowerCamelCase  :    Dict					=					None
							_lowerCamelCase  :    Tuple					=					[constraint.copy(stateful=lowercase						) for constraint in self.constraints]
			def       A_					(						self						):
							_lowerCamelCase  :    Dict					=					0
							if self.inprogress_constraint:
											# extra points for having a constraint mid-fulfilled
											add += self.max_seqlen - self.inprogress_constraint.remaining()

							return (len(self.complete_constraints						) * self.max_seqlen) + add
			def       A_					(						self						):
							_lowerCamelCase  :    Any					=					[]
							if self.inprogress_constraint is None:
											for constraint in self.pending_constraints:  # "pending" == "unfulfilled yet"
															_lowerCamelCase  :    Optional[int]					=					constraint.advance()
															if isinstance(lowercase						,   lowercase						):
																			token_list.append(lowercase						)
															elif isinstance(lowercase						,   lowercase						):
																			token_list.extend(lowercase						)
							else:
											_lowerCamelCase  :    Optional[Any]					=					self.inprogress_constraint.advance()
											if isinstance(lowercase						,   lowercase						):
															token_list.append(lowercase						)
											elif isinstance(lowercase						,   lowercase						):
															token_list.extend(lowercase						)

							if len(lowercase						) == 0:
											return None
							else:
											return token_list
			def       A_					(						self						,   lowercase						):
							self.init_state()

							if token_ids is not None:
											for token in token_ids:
															# completes or steps **one** constraint
															_lowerCamelCase  :    str					=					self.add(lowercase						)

															# the entire list of constraints are fulfilled
															if self.completed:
																			break
			def       A_					(						self						,   lowercase						):
							if not isinstance(lowercase						,   lowercase						):
											raise ValueError(F'''`token_id` should be an `int`, but is `{token_id}`.'''						)

							_lowerCamelCase  :    str					=					False, False

							if self.completed:
											_lowerCamelCase  :    List[Any]					=					True
											_lowerCamelCase  :    Optional[int]					=					False
											return complete, stepped

							if self.inprogress_constraint is not None:
											# In the middle of fulfilling a constraint. If the `token_id` *does* makes an incremental progress to current
											# job, simply update the state

											_lowerCamelCase  :    Optional[Any]					=					self.inprogress_constraint.update(lowercase						)
											if reset:
															# 1. If the next token breaks the progress, then we must restart.
															#     e.g. constraint = "I love pies" and sequence so far is "I love" but `token_id` == "books".

															#     But that doesn't mean we self.init_state(), since we only reset the state for this particular
															#     constraint, not the full list of constraints.

															self.pending_constraints.append(self.inprogress_constraint.copy(stateful=lowercase						)						)
															_lowerCamelCase  :    int					=					None

											if complete:
															# 2. If the next token completes the constraint, move it to completed list, set
															#     inprogress to None. If there are no pending constraints either, then this full list of constraints
															#     is complete.

															self.complete_constraints.append(self.inprogress_constraint						)
															_lowerCamelCase  :    List[str]					=					None

															if len(self.pending_constraints						) == 0:
																			# we're done!
																			_lowerCamelCase  :    Dict					=					True

							else:
											# Not in the middle of fulfilling a constraint. So does this `token_id` helps us step towards any of our list
											# of constraints?

											for cidx, pending_constraint in enumerate(self.pending_constraints						):
															if pending_constraint.does_advance(lowercase						):
																			_lowerCamelCase  :    int					=					pending_constraint.update(lowercase						)

																			if not stepped:
																							raise Exception(
																							    '`constraint.update(token_id)` is not yielding incremental progress, '
																							    'even though `constraint.does_advance(token_id)` is true.'						)

																			if complete:
																							self.complete_constraints.append(lowercase						)
																							_lowerCamelCase  :    Dict					=					None

																			if not complete and stepped:
																							_lowerCamelCase  :    Dict					=					pending_constraint

																			if complete or stepped:
																							# If we made any progress at all, then it's at least not a "pending constraint".

																							_lowerCamelCase  :    List[Any]					=					(
																							    self.pending_constraints[:cidx] + self.pending_constraints[cidx + 1 :]
																							)

																							if len(self.pending_constraints						) == 0 and self.inprogress_constraint is None:
																											# If there's no longer any pending after this and no inprogress either, then we must be
																											# complete.

																											_lowerCamelCase  :    Dict					=					True

																							break  # prevent accidentally stepping through multiple constraints with just one token.

							return complete, stepped






			def       A_					(						self						,   lowercase=True						):
							_lowerCamelCase  :    Optional[int]					=					ConstraintListState(self.constraints						)  # we actually never though self.constraints objects
							# throughout this process. So it's at initialization state.

							if stateful:
											_lowerCamelCase  :    Union[str, Any]					=					[
											    constraint.copy(stateful=lowercase						) for constraint in self.complete_constraints
											]
											if self.inprogress_constraint is not None:
															_lowerCamelCase  :    Dict					=					self.inprogress_constraint.copy(stateful=lowercase						)
											_lowerCamelCase  :    Optional[int]					=					[constraint.copy() for constraint in self.pending_constraints]

							return new_state





"""simple docstring"""






def   _snake_case    (       lowercase__ = 1000	):
				_lowerCamelCase,				_lowerCamelCase  :    Optional[int]					=					1, 1
				_lowerCamelCase  :    int					=					[]
				for i in range(1							,					n + 1	):
								_lowerCamelCase  :    Tuple					=					prev_numerator + 2 * prev_denominator
								_lowerCamelCase  :    str					=					prev_numerator + prev_denominator
								if len(str(lowercase__	)	) > len(str(lowercase__	)	):
												result.append(lowercase__	)
								_lowerCamelCase  :    Dict					=					numerator
								_lowerCamelCase  :    Optional[int]					=					denominator

				return len(lowercase__	)


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