





"""simple docstring"""




from __future__ import annotations

from math import ceil, floor, sqrt





def 	a_    (							_lowercase = 200_0000       ):






						"""simple docstring"""




						_UpperCamelCase      : list[int]         =   [0]
						_UpperCamelCase      : int

						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      : int         =   0
						# the area corresponding to the grid that gives the product closest to target
						_UpperCamelCase      : int         =   0
						# an estimate of b, using the quadratic formula
						_UpperCamelCase      : float
						# the largest integer less than b_estimate
						_UpperCamelCase      : int
						# the largest integer less than b_estimate
						_UpperCamelCase      : int
						# the triangle number corresponding to b_floor
						_UpperCamelCase      : int
						# the triangle number corresponding to b_ceil
						_UpperCamelCase      : int

						for idx_a, triangle_a in enumerate(triangle_numbers[1:]	,						1       ):
												_UpperCamelCase      : Union[str, Any]         =   (-1 + sqrt(1 + 8 * target / triangle_a       )) / 2
												_UpperCamelCase      : Optional[Any]         =   floor(UpperCAmelCase__       )
												_UpperCamelCase      : int         =   ceil(UpperCAmelCase__       )
												_UpperCamelCase      : List[Any]         =   triangle_numbers[b_floor]
												_UpperCamelCase      : List[Any]         =   triangle_numbers[b_ceil]

												if abs(target - triangle_b_first_guess * triangle_a       ) < abs(
												    target - best_product       ):
																		_UpperCamelCase      : int         =   triangle_b_first_guess * triangle_a
																		_UpperCamelCase      : List[str]         =   idx_a * b_floor

												if abs(target - triangle_b_second_guess * triangle_a       ) < abs(
												    target - best_product       ):
																		_UpperCamelCase      : str         =   triangle_b_second_guess * triangle_a
																		_UpperCamelCase      : Any         =   idx_a * b_ceil

						return area


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










"""simple docstring"""




from __future__ import annotations

from random import choice





def 	a_    (							_lowercase       ):
						return choice(_lowercase       )





def 	a_    (							_lowercase	,						_lowercase       ):
						_UpperCamelCase      : Optional[int]         =   random_pivot(_lowercase       )

						# partition based on pivot
						# linear time
						_UpperCamelCase      : Union[str, Any]         =   [e for e in lst if e < pivot]
						_UpperCamelCase      : int         =   [e for e in lst if e > pivot]

						# if we get lucky, pivot might be the element we want.
						# we can easily see this:
						# small (elements smaller than k)
						# + pivot (kth element)
						# + big (elements larger than k)
						if len(_lowercase       ) == k - 1:
												return pivot
						# pivot is in elements bigger than k
						elif len(_lowercase       ) < k - 1:
												return kth_number(_lowercase	,						k - len(_lowercase       ) - 1       )
						# pivot is in elements smaller than k
						else:
												return kth_number(_lowercase	,						_lowercase       )


if __name__ == "__main__":
						import doctest

						doctest.testmod()



