





def A_	(		A__ = 10**12				)  ->		int:
       a__						:	Optional[Any] =				1
       a__						:	List[Any] =				0

       a__						:	Dict =				1
       a__						:	Union[str, Any] =				1

       while numerator <= 2 * min_total - 1:
              prev_numerator += 2 * numerator
              numerator += 2 * prev_numerator

              prev_denominator += 2 * denominator
              denominator += 2 * prev_denominator

       return (denominator + 1) // 2


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







"""simple docstring"""







from dataclasses import dataclass, field
from typing import TYPE_CHECKING, Any, ClassVar, Dict, List, Optional, Union

import pyarrow as pa


if TYPE_CHECKING:
							from .features import FeatureType
@dataclass
class        __A       :







		'''simple docstring'''





		lowerCAmelCase			:					List[str]
		lowerCAmelCase			:					Optional[str]   	=				None
		# Automatically constructed
		lowerCAmelCase			:					ClassVar[str]   	=				"dict"
		lowerCAmelCase			:					ClassVar[Any]   	=				None
		lowerCAmelCase			:					str   	=				field(default="Translation"     ,init=A_     ,repr=A_    )







		def __call__(	self						:  List[str]						)					->		Any:






				"""simple docstring"""



				return pa.struct({lang: pa.string() for lang in sorted(self.languages						)}						)







		def 					UpperCAmelCase    (	self						:  List[str]						)					->		Union["FeatureType", Dict[str, "FeatureType"]]:






				"""simple docstring"""



				from .features import Value

				return {k: Value('''string'''						) for k in sorted(self.languages						)}





@dataclass
class        __A       :







		'''simple docstring'''





		lowerCAmelCase			:					Optional[List]   	=				None
		lowerCAmelCase			:					Optional[int]   	=				None
		lowerCAmelCase			:					Optional[str]   	=				None
		# Automatically constructed
		lowerCAmelCase			:					ClassVar[str]   	=				"dict"
		lowerCAmelCase			:					ClassVar[Any]   	=				None
		lowerCAmelCase			:					str   	=				field(default="TranslationVariableLanguages"     ,init=A_     ,repr=A_    )







		def 					UpperCAmelCase    (	self						:  List[Any]						)					->		Optional[int]:






				"""simple docstring"""



				lowercase__   :    Optional[int]				=			sorted(set(self.languages						)						) if self.languages else None
				lowercase__   :    Dict				=			len(self.languages						) if self.languages else None







		def __call__(	self						:  List[Any]						)					->		List[Any]:






				"""simple docstring"""



				return pa.struct({'''language''': pa.list_(pa.string()						), '''translation''': pa.list_(pa.string()						)}						)







		def 					UpperCAmelCase    (	self						:  Dict			,_snake_case						:  Tuple						)					->		int:






				"""simple docstring"""



				lowercase__   :    List[Any]				=			set(self.languages						)
				if self.languages and set(_snake_case						) - lang_set:
						raise ValueError(
						    f"""Some languages in example ({", ".join(sorted(set(_snake_case						) - lang_set						)						)}) are not in valid set ({", ".join(_snake_case						)})."""						)

				# Convert dictionary into tuples, splitting out cases where there are
				# multiple translations for a single language.
				lowercase__   :    str				=			[]
				for lang, text in translation_dict.items():
						if isinstance(_snake_case			,_snake_case						):
								translation_tuples.append((lang, text)						)
						else:
								translation_tuples.extend([(lang, el) for el in text]						)

        # Ensure translations are in ascending order by language code.
				lowercase__		,			lowercase__   :    Optional[Any]				=			zip(*sorted(_snake_case						)						)

				return {"language": languages, "translation": translations}







		def 					UpperCAmelCase    (	self						:  List[Any]						)					->		Union["FeatureType", Dict[str, "FeatureType"]]:






				"""simple docstring"""



				from .features import Sequence, Value

				return {
				    "language": Sequence(Value('''string'''						)						),
				    "translation": Sequence(Value('''string'''						)						),
				}
