
import copy

from ...configuration_utils import PretrainedConfig
from ...utils import add_start_docstrings


_A   =			R"\n    [`RagConfig`] stores the configuration of a *RagModel*. Configuration objects inherit from [`PretrainedConfig`] and\n    can be used to control the model outputs. Read the documentation from [`PretrainedConfig`] for more information.\n\n    Args:\n        title_sep (`str`, *optional*, defaults to  `\" / \"`):\n            Separator inserted between the title and the text of the retrieved document when calling [`RagRetriever`].\n        doc_sep (`str`, *optional*, defaults to  `\" // \"`):\n            Separator inserted between the text of the retrieved document and the original input when calling\n            [`RagRetriever`].\n        n_docs (`int`, *optional*, defaults to 5):\n            Number of documents to retrieve.\n        max_combined_length (`int`, *optional*, defaults to 300):\n            Max length of contextualized input returned by [`~RagRetriever.__call__`].\n        retrieval_vector_size (`int`, *optional*, defaults to 768):\n            Dimensionality of the document embeddings indexed by [`RagRetriever`].\n        retrieval_batch_size (`int`, *optional*, defaults to 8):\n            Retrieval batch size, defined as the number of queries issues concurrently to the faiss index encapsulated\n            [`RagRetriever`].\n        dataset (`str`, *optional*, defaults to `\"wiki_dpr\"`):\n            A dataset identifier of the indexed dataset in HuggingFace Datasets (list all available datasets and ids\n            using `datasets.list_datasets()`).\n        dataset_split (`str`, *optional*, defaults to `\"train\"`)\n            Which split of the `dataset` to load.\n        index_name (`str`, *optional*, defaults to `\"compressed\"`)\n            The index name of the index associated with the `dataset`. One can choose between `\"legacy\"`, `\"exact\"` and\n            `\"compressed\"`.\n        index_path (`str`, *optional*)\n            The path to the serialized faiss index on disk.\n        passages_path (`str`, *optional*):\n            A path to text passages compatible with the faiss index. Required if using\n            [`~models.rag.retrieval_rag.LegacyIndex`]\n        use_dummy_dataset (`bool`, *optional*, defaults to `False`)\n            Whether to load a \"dummy\" variant of the dataset specified by `dataset`.\n        label_smoothing (`float`, *optional*, defaults to 0.0):\n            Only relevant if `return_loss` is set to `True`. Controls the `epsilon` parameter value for label smoothing\n            in the loss calculation. If set to 0, no label smoothing is performed.\n        do_marginalize (`bool`, *optional*, defaults to `False`):\n            If `True`, the logits are marginalized over all documents by making use of\n            `torch.nn.functional.log_softmax`.\n        reduce_loss (`bool`, *optional*, defaults to `False`):\n            Whether or not to reduce the NLL loss using the `torch.Tensor.sum` operation.\n        do_deduplication (`bool`, *optional*, defaults to `True`):\n            Whether or not to deduplicate the generations from different context documents for a given input. Has to be\n            set to `False` if used while training with distributed backend.\n        exclude_bos_score (`bool`, *optional*, defaults to `False`):\n            Whether or not to disregard the BOS token when computing the loss.\n        output_retrieved(`bool`, *optional*, defaults to `False`):\n            If set to `True`, `retrieved_doc_embeds`, `retrieved_doc_ids`, `context_input_ids` and\n            `context_attention_mask` are returned. See returned tensors for more detail.\n        use_cache (`bool`, *optional*, defaults to `True`):\n            Whether or not the model should return the last key/values attentions (not used by all models).\n        forced_eos_token_id (`int`, *optional*):\n            The id of the token to force as the last generated token when `max_length` is reached. Usually set to\n            `eos_token_id`.\n"







@add_start_docstrings(A_  )
class 					lowerCamelCase		(  A_  ):
     UpperCAmelCase__    :							Union[str, Any]									=		"rag"
     UpperCAmelCase__    :							int									=		True





     def __init__(self				:   Optional[int]    ,   _A				:   List[Any]=None    ,   _A				:   Optional[int]=True    ,   _A				:   Dict=None    ,   _A				:   Any=None    ,   _A				:   Optional[int]=None    ,   _A				:   Any=None    ,   _A				:   Union[str, Any]=None    ,   _A				:   Union[str, Any]=" / "    ,   _A				:   Tuple=" // "    ,   _A				:   Union[str, Any]=5    ,   _A				:   Dict=3_0_0    ,   _A				:   Optional[int]=7_6_8    ,   _A				:   Tuple=8    ,   _A				:   str="wiki_dpr"    ,   _A				:   List[Any]="train"    ,   _A				:   List[str]="compressed"    ,   _A				:   int=None    ,   _A				:   Any=None    ,   _A				:   Tuple=False    ,   _A				:   Optional[Any]=False    ,   _A				:   str=0.0    ,   _A				:   Tuple=True    ,   _A				:   Union[str, Any]=False    ,   _A				:   Tuple=False    ,   _A				:   Optional[int]=False    ,   _A				:   List[str]=True    ,   _A				:   Optional[Any]=None    ,   **_A				:   Union[str, Any]    ,   )						->						int:
         super().__init__(
             bos_token_id=_A    ,   pad_token_id=_A    ,   eos_token_id=_A    ,   decoder_start_token_id=_A    ,   forced_eos_token_id=_A    ,   is_encoder_decoder=_A    ,   prefix=_A    ,   vocab_size=_A    ,   **_A    ,   )
         assert (
             "question_encoder" in kwargs and "generator" in kwargs
         ), "Config has to be initialized with question_encoder and generator config"
         snake_case								= kwargs.pop("question_encoder"       )
         snake_case								= question_encoder_config.pop("model_type"       )
         snake_case								= kwargs.pop("generator"       )
         snake_case								= decoder_config.pop("model_type"       )

         from ..auto.configuration_auto import AutoConfig

         snake_case								= AutoConfig.for_model(_A    ,   **_A       )
         snake_case								= AutoConfig.for_model(_A    ,   **_A       )

         snake_case								= reduce_loss
         snake_case								= label_smoothing
         snake_case								= exclude_bos_score
         snake_case								= do_marginalize

         snake_case								= title_sep
         snake_case								= doc_sep
         snake_case								= n_docs
         snake_case								= max_combined_length

         snake_case								= dataset
         snake_case								= dataset_split
         snake_case								= index_name

         snake_case								= retrieval_vector_size
         snake_case								= retrieval_batch_size
         snake_case								= passages_path
         snake_case								= index_path
         snake_case								= use_dummy_dataset

         snake_case								= output_retrieved

         snake_case								= do_deduplication

         snake_case								= use_cache

         if self.forced_eos_token_id is None:
             snake_case								= getattr(self.generator    ,   "forced_eos_token_id"    ,   _A       )





     @classmethod
     def 		UpperCAmelCase(cls				:   List[Any]    ,   _A				:   PretrainedConfig    ,   _A				:   PretrainedConfig    ,   **_A				:   str       )						->						PretrainedConfig:
         return cls(question_encoder=question_encoder_config.to_dict()    ,   generator=generator_config.to_dict()    ,   **_A       )





     def 		UpperCAmelCase(self				:   List[str]       )						->						Union[str, Any]:
         snake_case								= copy.deepcopy(self.__dict__       )
         snake_case								= self.question_encoder.to_dict()
         snake_case								= self.generator.to_dict()
         snake_case								= self.__class__.model_type
         return output



from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt



def 						lowercase_      (       A__		)							->							bool:
    """simple docstring"""

    snake_case								= int(number**0.5		)
    return number == sq * sq



def 						lowercase_      (       A__			,		A__			,		A__			,		A__			,		A__			,		A__		)							->							tuple[int, int]:
    """simple docstring"""

    snake_case								= x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
    snake_case								= x_den * y_den * z_den
    snake_case								= gcd(A__			,		A__		)
    top //= hcf
    bottom //= hcf
    return top, bottom



def 						lowercase_      (       A__ = 35		)							->							int:
    """simple docstring"""

    snake_case								= set()
    snake_case								= 42
    snake_case								= Fraction(0		)
    snake_case								= 42

    for x_num in range(1			,		order + 1		):
        for x_den in range(x_num + 1			,		order + 1		):
            for y_num in range(1			,		order + 1		):
                for y_den in range(y_num + 1			,		order + 1		):
                    # n=1
                    snake_case								= x_num * y_den + x_den * y_num
                    snake_case								= x_den * y_den
                    snake_case								= gcd(A__			,		A__		)
                    z_num //= hcf
                    z_den //= hcf
                    if 0 < z_num < z_den <= order:
                        snake_case								= add_three(
                            A__			,		A__			,		A__			,		A__			,		A__			,		A__		)
                        unique_s.add(A__		)

                    # n=2
                    snake_case								= (
                        x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
                    )
                    snake_case								= x_den * x_den * y_den * y_den
                    if is_sq(A__		) and is_sq(A__		):
                        snake_case								= int(sqrt(A__		)		)
                        snake_case								= int(sqrt(A__		)		)
                        snake_case								= gcd(A__			,		A__		)
                        z_num //= hcf
                        z_den //= hcf
                        if 0 < z_num < z_den <= order:
                            snake_case								= add_three(
                                A__			,		A__			,		A__			,		A__			,		A__			,		A__		)
                            unique_s.add(A__		)

                    # n=-1
                    snake_case								= x_num * y_num
                    snake_case								= x_den * y_num + x_num * y_den
                    snake_case								= gcd(A__			,		A__		)
                    z_num //= hcf
                    z_den //= hcf
                    if 0 < z_num < z_den <= order:
                        snake_case								= add_three(
                            A__			,		A__			,		A__			,		A__			,		A__			,		A__		)
                        unique_s.add(A__		)

                    # n=2
                    snake_case								= x_num * x_num * y_num * y_num
                    snake_case								= (
                        x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
                    )
                    if is_sq(A__		) and is_sq(A__		):
                        snake_case								= int(sqrt(A__		)		)
                        snake_case								= int(sqrt(A__		)		)
                        snake_case								= gcd(A__			,		A__		)
                        z_num //= hcf
                        z_den //= hcf
                        if 0 < z_num < z_den <= order:
                            snake_case								= add_three(
                                A__			,		A__			,		A__			,		A__			,		A__			,		A__		)
                            unique_s.add(A__		)

    for num, den in unique_s:
        total += Fraction(A__			,		A__		)

    return total.denominator + total.numerator


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

