



from __future__ import annotations

from math import ceil, floor, sqrt



def __lowerCAmelCase	(       SCREAMING_SNAKE_CASE_ = 200_0000					):
			lowercase__   					=			[0]
			lowercase__   					=			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
			lowercase__   					=			0
			# the area corresponding to the grid that gives the product closest to target
			lowercase__   					=			0
			# an estimate of b, using the quadratic formula
			lowercase__   					=			42
			# the largest integer less than b_estimate
			lowercase__   					=			42
			# the largest integer less than b_estimate
			lowercase__   					=			42
			# the triangle number corresponding to b_floor
			lowercase__   					=			42
			# the triangle number corresponding to b_ceil
			lowercase__   					=			42

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

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

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

			return area


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









from __future__ import annotations



def __lowerCAmelCase	(       SCREAMING_SNAKE_CASE_					):
			return len(set(SCREAMING_SNAKE_CASE_					)					) == len(SCREAMING_SNAKE_CASE_					)


if __name__ == "__main__":
							import doctest

							doctest.testmod()




