



from __future__ import annotations

from math import ceil, floor, sqrt




def    A_(			A = 2000000   ):
	UpperCAmelCase_								=     [0]
	UpperCAmelCase_								=     42

	for idx in range(1				,      ceil(sqrt(target * 2   ) * 1.1   )   ):
		triangle_numbers.append(triangle_numbers[-1] + idx   )

	# we want this to be as close as possible to target
	UpperCAmelCase_								=     0
	# the area corresponding to the grid that gives the product closest to target
	UpperCAmelCase_								=     0
	# an estimate of b, using the quadratic formula
	UpperCAmelCase_								=     42
	# the largest integer less than b_estimate
	UpperCAmelCase_								=     42
	# the largest integer less than b_estimate
	UpperCAmelCase_								=     42
	# the triangle number corresponding to b_floor
	UpperCAmelCase_								=     42
	# the triangle number corresponding to b_ceil
	UpperCAmelCase_								=     42

	for idx_a, triangle_a in enumerate(triangle_numbers[1:]				,      1   ):
		UpperCAmelCase_								=     (-1 + sqrt(1 + 8 * target / triangle_a   )) / 2
		UpperCAmelCase_								=     floor(A   )
		UpperCAmelCase_								=     ceil(A   )
		UpperCAmelCase_								=     triangle_numbers[b_floor]
		UpperCAmelCase_								=     triangle_numbers[b_ceil]

		if abs(target - triangle_b_first_guess * triangle_a   ) < abs(
		    target - best_product   ):
			UpperCAmelCase_								=     triangle_b_first_guess * triangle_a
			UpperCAmelCase_								=     idx_a * b_floor

		if abs(target - triangle_b_second_guess * triangle_a   ) < abs(
		    target - best_product   ):
			UpperCAmelCase_								=     triangle_b_second_guess * triangle_a
			UpperCAmelCase_								=     idx_a * b_ceil

	return area


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









import json
import os
import shutil
import tempfile
from unittest import TestCase

from transformers import BartTokenizer, BartTokenizerFast, DPRQuestionEncoderTokenizer, DPRQuestionEncoderTokenizerFast
from transformers.models.bart.configuration_bart import BartConfig
from transformers.models.bert.tokenization_bert import VOCAB_FILES_NAMES as DPR_VOCAB_FILES_NAMES
from transformers.models.dpr.configuration_dpr import DPRConfig
from transformers.models.roberta.tokenization_roberta import VOCAB_FILES_NAMES as BART_VOCAB_FILES_NAMES
from transformers.testing_utils import require_faiss, require_tokenizers, require_torch, slow
from transformers.utils import is_datasets_available, is_faiss_available, is_torch_available


if is_torch_available() and is_datasets_available() and is_faiss_available():
				from transformers.models.rag.configuration_rag import RagConfig
				from transformers.models.rag.tokenization_rag import RagTokenizer







@require_faiss
@require_torch
class   _UpperCamelCase  (       A_  ):







					'''simple docstring'''

					def        SCREAMING_SNAKE_CASE							(						self    :  Optional[Any]		):






						'''simple docstring'''







						UpperCAmelCase_								=     tempfile.mkdtemp()
						UpperCAmelCase_								=     8

						# DPR tok
						UpperCAmelCase_								=     [
						    """[UNK]""",
						    """[CLS]""",
						    """[SEP]""",
						    """[PAD]""",
						    """[MASK]""",
						    """want""",
						    """##want""",
						    """##ed""",
						    """wa""",
						    """un""",
						    """runn""",
						    """##ing""",
						    """,""",
						    """low""",
						    """lowest""",
						]
						UpperCAmelCase_								=     os.path.join(self.tmpdirname ,					"""dpr_tokenizer"""		)
						os.makedirs(__lowercase ,					exist_ok=__lowercase		)
						UpperCAmelCase_								=     os.path.join(__lowercase ,					DPR_VOCAB_FILES_NAMES["""vocab_file"""]		)
						with open(self.vocab_file ,					"""w""" ,					encoding="""utf-8"""		) as vocab_writer:
							vocab_writer.write("""""".join([x + """\n""" for x in vocab_tokens]		)		)

						# BART tok
						UpperCAmelCase_								=     [
						    """l""",
						    """o""",
						    """w""",
						    """e""",
						    """r""",
						    """s""",
						    """t""",
						    """i""",
						    """d""",
						    """n""",
						    """\u0120""",
						    """\u0120l""",
						    """\u0120n""",
						    """\u0120lo""",
						    """\u0120low""",
						    """er""",
						    """\u0120lowest""",
						    """\u0120newer""",
						    """\u0120wider""",
						    """<unk>""",
						]
						UpperCAmelCase_								=     dict(zip(__lowercase ,					range(len(__lowercase		)		)		)		)
						UpperCAmelCase_								=     ["""#version: 0.2""", """\u0120 l""", """\u0120l o""", """\u0120lo w""", """e r""", """"""]
						UpperCAmelCase_								=     {"""unk_token""": """<unk>"""}

						UpperCAmelCase_								=     os.path.join(self.tmpdirname ,					"""bart_tokenizer"""		)
						os.makedirs(__lowercase ,					exist_ok=__lowercase		)
						UpperCAmelCase_								=     os.path.join(__lowercase ,					BART_VOCAB_FILES_NAMES["""vocab_file"""]		)
						UpperCAmelCase_								=     os.path.join(__lowercase ,					BART_VOCAB_FILES_NAMES["""merges_file"""]		)
						with open(self.vocab_file ,					"""w""" ,					encoding="""utf-8"""		) as fp:
							fp.write(json.dumps(__lowercase		) + """\n"""		)
						with open(self.merges_file ,					"""w""" ,					encoding="""utf-8"""		) as fp:
							fp.write("""\n""".join(__lowercase		)		)





					def        SCREAMING_SNAKE_CASE							(						self    :  Union[str, Any]		):






						'''simple docstring'''







						return DPRQuestionEncoderTokenizer.from_pretrained(os.path.join(self.tmpdirname ,					"""dpr_tokenizer"""		)		)





					def        SCREAMING_SNAKE_CASE							(						self    :  List[Any]		):






						'''simple docstring'''







						return BartTokenizer.from_pretrained(os.path.join(self.tmpdirname ,					"""bart_tokenizer"""		)		)





					def        SCREAMING_SNAKE_CASE							(						self    :  str		):






						'''simple docstring'''







						shutil.rmtree(self.tmpdirname		)





					@require_tokenizers
					def        SCREAMING_SNAKE_CASE							(						self    :  Union[str, Any]		):






						'''simple docstring'''







						UpperCAmelCase_								=     os.path.join(self.tmpdirname ,					"""rag_tokenizer"""		)
						UpperCAmelCase_								=     RagConfig(question_encoder=DPRConfig().to_dict() ,					generator=BartConfig().to_dict()		)
						UpperCAmelCase_								=     RagTokenizer(question_encoder=self.get_dpr_tokenizer() ,					generator=self.get_bart_tokenizer()		)
						rag_config.save_pretrained(__lowercase		)
						rag_tokenizer.save_pretrained(__lowercase		)
						UpperCAmelCase_								=     RagTokenizer.from_pretrained(__lowercase ,					config=__lowercase		)
						self.assertIsInstance(new_rag_tokenizer.question_encoder ,					__lowercase		)
						self.assertEqual(new_rag_tokenizer.question_encoder.get_vocab() ,					rag_tokenizer.question_encoder.get_vocab()		)
						self.assertIsInstance(new_rag_tokenizer.generator ,					__lowercase		)
						self.assertEqual(new_rag_tokenizer.generator.get_vocab() ,					rag_tokenizer.generator.get_vocab()		)





					@slow
					def        SCREAMING_SNAKE_CASE							(						self    :  List[Any]		):






						'''simple docstring'''







						UpperCAmelCase_								=     RagTokenizer.from_pretrained("""facebook/rag-token-nq"""		)
						UpperCAmelCase_								=     [
						    """who got the first nobel prize in physics""",
						    """when is the next deadpool movie being released""",
						    """which mode is used for short wave broadcast service""",
						    """who is the owner of reading football club""",
						    """when is the next scandal episode coming out""",
						    """when is the last time the philadelphia won the superbowl""",
						    """what is the most current adobe flash player version""",
						    """how many episodes are there in dragon ball z""",
						    """what is the first step in the evolution of the eye""",
						    """where is gall bladder situated in human body""",
						    """what is the main mineral in lithium batteries""",
						    """who is the president of usa right now""",
						    """where do the greasers live in the outsiders""",
						    """panda is a national animal of which country""",
						    """what is the name of manchester united stadium""",
						]
						UpperCAmelCase_								=     tokenizer(__lowercase		)
						self.assertIsNotNone(__lowercase		)





					@slow
					def        SCREAMING_SNAKE_CASE							(						self    :  List[str]		):






						'''simple docstring'''







						UpperCAmelCase_								=     RagTokenizer.from_pretrained("""facebook/rag-sequence-nq"""		)
						UpperCAmelCase_								=     [
						    """who got the first nobel prize in physics""",
						    """when is the next deadpool movie being released""",
						    """which mode is used for short wave broadcast service""",
						    """who is the owner of reading football club""",
						    """when is the next scandal episode coming out""",
						    """when is the last time the philadelphia won the superbowl""",
						    """what is the most current adobe flash player version""",
						    """how many episodes are there in dragon ball z""",
						    """what is the first step in the evolution of the eye""",
						    """where is gall bladder situated in human body""",
						    """what is the main mineral in lithium batteries""",
						    """who is the president of usa right now""",
						    """where do the greasers live in the outsiders""",
						    """panda is a national animal of which country""",
						    """what is the name of manchester united stadium""",
						]
						UpperCAmelCase_								=     tokenizer(__lowercase		)
						self.assertIsNotNone(__lowercase		)


