





from __future__ import annotations

from math import ceil, floor, sqrt


def 	UpperCamelCase__(       UpperCamelCase__			:       int = 2_00_00_00  )->int:
       A__											=						[0]
       A__											=						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
       A__											=						0
       # the area corresponding to the grid that gives the product closest to target
       A__											=						0
       # an estimate of b, using the quadratic formula
       A__											=						42
       # the largest integer less than b_estimate
       A__											=						42
       # the largest integer less than b_estimate
       A__											=						42
       # the triangle number corresponding to b_floor
       A__											=						42
       # the triangle number corresponding to b_ceil
       A__											=						42

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

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

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

       return area


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












import random


def 	UpperCamelCase__(       UpperCamelCase__			:       list ,     UpperCamelCase__			:       List[Any]  )->tuple:
       A__					,	A__					,	A__											=						[], [], []
       for element in data:
              if element < pivot:
                     less.append(UpperCamelCase__  )
              elif element > pivot:
                     greater.append(UpperCamelCase__  )
              else:
                     equal.append(UpperCamelCase__  )
       return less, equal, greater


def 	UpperCamelCase__(       UpperCamelCase__			:       list ,     UpperCamelCase__			:       int  )->Optional[int]:
       # index = len(items) // 2 when trying to find the median
       #   (value of index when items is sorted)

       # invalid input
       if index >= len(UpperCamelCase__  ) or index < 0:
              return None

       A__											=						items[random.randint(0 ,     len(UpperCamelCase__  ) - 1  )]
       A__											=						0
       A__					,	A__					,	A__											=						_partition(UpperCamelCase__ ,     UpperCamelCase__  )
       A__											=						len(UpperCamelCase__  )
       A__											=						len(UpperCamelCase__  )

       # index is the pivot
       if m <= index < m + count:
              return pivot
       # must be in smaller
       elif m > index:
              return quick_select(UpperCamelCase__ ,     UpperCamelCase__  )
       # must be in larger
       else:
              return quick_select(UpperCamelCase__ ,     index - (m + count)  )





