



from __future__ import annotations

from math import ceil, floor, sqrt



def 	__a					(   lowerCAmelCase_       : int = 2_00_00_00    )				->			int:






      '''simple docstring'''



      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(lowerCAmelCase_    )
            UpperCAmelCase_=       ceil(lowerCAmelCase_    )
            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() = }')









from typing import Optional

import pyspark

from .. import Features, NamedSplit
from ..download import DownloadMode
from ..packaged_modules.spark.spark import Spark
from .abc import AbstractDatasetReader






class 		lowercase						(			snake_case__):






	"""simple docstring"""


	def __init__(     self     :						int						,  __UpperCAmelCase     :						pyspark.sql.DataFrame						,  __UpperCAmelCase     :						Optional[NamedSplit] = None						,  __UpperCAmelCase     :						Optional[Features] = None						,  __UpperCAmelCase     :						bool = True						,  __UpperCAmelCase     :						str = None						,  __UpperCAmelCase     :						bool = False						,  __UpperCAmelCase     :						str = None						,  __UpperCAmelCase     :						bool = True						,  __UpperCAmelCase     :						str = "arrow"						,  **__UpperCAmelCase     :						str						,  )      ->			Dict:
							super().__init__(
							    split=__UpperCAmelCase						,  features=__UpperCAmelCase						,  cache_dir=__UpperCAmelCase						,  keep_in_memory=__UpperCAmelCase						,  streaming=__UpperCAmelCase						,  **__UpperCAmelCase						,  )
							UpperCAmelCase_=       load_from_cache_file
							UpperCAmelCase_=       file_format
							UpperCAmelCase_=       Spark(
							    df=__UpperCAmelCase						,  features=__UpperCAmelCase						,  cache_dir=__UpperCAmelCase						,  working_dir=__UpperCAmelCase						,  **__UpperCAmelCase						,  )




	def   _SCREAMING_SNAKE_CASE		(     self     :						Any					)      ->			Dict:
							if self.streaming:
													return self.builder.as_streaming_dataset(split=self.split					)
							UpperCAmelCase_=       None if self._load_from_cache_file else DownloadMode.FORCE_REDOWNLOAD
							self.builder.download_and_prepare(
							    download_mode=__UpperCAmelCase						,  file_format=self._file_format						,  )
							return self.builder.as_dataset(split=self.split					)




