"""simple docstring"""






from __future__ import annotations

from math import ceil, floor, sqrt



def   a_						(					lowerCamelCase = 2_0_0_0_0_0_0       ):
				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 __future__ import annotations

import time
from collections.abc import Sequence
from random import randint

from matplotlib import pyplot as plt


def     __lowerCAmelCase			(			a__						,    a__						,    a__  )       ->  tuple[int | None, int | None, float]:
				if not arr:
								return None, None, 0
				if low == high:
								return low, high, arr[low]

				__a										=    (low + high) // 2
				__a							,     __a							,     __a										=    max_subarray(a__						,    a__						,    a__  )
				__a							,     __a							,     __a										=    max_subarray(a__						,    mid + 1						,    a__  )
				__a							,     __a							,     __a										=    max_cross_sum(a__						,    a__						,    a__						,    a__  )
				if left_sum >= right_sum and left_sum >= cross_sum:
								return left_low, left_high, left_sum
				elif right_sum >= left_sum and right_sum >= cross_sum:
								return right_low, right_high, right_sum
				return cross_left, cross_right, cross_sum


def     __lowerCAmelCase			(			a__						,    a__						,    a__						,    a__  )       ->  tuple[int, int, float]:
				__a							,     __a										=    float('''-inf'''  ), -1
				__a							,     __a										=    float('''-inf'''  ), -1

				__a										=    0
				for i in range(a__						,    low - 1						,    -1  ):
								summ += arr[i]
								if summ > left_sum:
												__a										=    summ
												__a										=    i

				__a										=    0
				for i in range(mid + 1						,    high + 1  ):
								summ += arr[i]
								if summ > right_sum:
												__a										=    summ
												__a										=    i

				return max_left, max_right, (left_sum + right_sum)


def     __lowerCAmelCase			(			a__  )       ->  float:
				__a										=    [randint(1						,    a__  ) for _ in range(a__  )]
				__a										=    time.time()
				max_subarray(a__						,    0						,    input_size - 1  )
				__a										=    time.time()
				return end - start


def     __lowerCAmelCase			(			)       ->  None:
				__a										=    [10, 100, 1000, 1_0000, 5_0000, 10_0000, 20_0000, 30_0000, 40_0000, 50_0000]
				__a										=    [time_max_subarray(a__  ) for input_size in input_sizes]
				print('''No of Inputs\t\tTime Taken'''  )
				for input_size, runtime in zip(a__						,    a__  ):
								print(a__						,    '''\t\t'''						,    a__  )
				plt.plot(a__						,    a__  )
				plt.xlabel('''Number of Inputs'''  )
				plt.ylabel('''Time taken in seconds'''  )
				plt.show()


if __name__ == "__main__":
					from doctest import testmod

					testmod()