





from __future__ import annotations

from math import ceil, floor, sqrt


def 							__snake_case							(    lowerCAmelCase_ = 2_0_0_0_0_0_0	)				->						int:
    SCREAMING_SNAKE_CASE__											=			[0]
    SCREAMING_SNAKE_CASE__											=			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
    SCREAMING_SNAKE_CASE__											=			0
    # the area corresponding to the grid that gives the product closest to target
    SCREAMING_SNAKE_CASE__											=			0
    # an estimate of b, using the quadratic formula
    SCREAMING_SNAKE_CASE__											=			42
    # the largest integer less than b_estimate
    SCREAMING_SNAKE_CASE__											=			42
    # the largest integer less than b_estimate
    SCREAMING_SNAKE_CASE__											=			42
    # the triangle number corresponding to b_floor
    SCREAMING_SNAKE_CASE__											=			42
    # the triangle number corresponding to b_ceil
    SCREAMING_SNAKE_CASE__											=			42

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

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

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

    return area


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







import pytest
import requests

from datasets.utils.file_utils import http_head

from .utils import OfflineSimulationMode, RequestWouldHangIndefinitelyError, offline


@pytest.mark.integration
def 							__snake_case							(    )				->						Any:
    with offline(OfflineSimulationMode.CONNECTION_TIMES_OUT	):
        with pytest.raises(lowerCAmelCase_	):
            requests.request('''GET''' ,							'''https://huggingface.co'''	)
        with pytest.raises(requests.exceptions.ConnectTimeout	):
            requests.request('''GET''' ,							'''https://huggingface.co''' ,							timeout=1.0	)


@pytest.mark.integration
def 							__snake_case							(    )				->						int:
    with offline(OfflineSimulationMode.CONNECTION_FAILS	):
        with pytest.raises(requests.exceptions.ConnectionError	):
            requests.request('''GET''' ,							'''https://huggingface.co'''	)



def 							__snake_case							(    )				->						List[Any]:
    with offline(OfflineSimulationMode.HF_DATASETS_OFFLINE_SET_TO_1	):
        with pytest.raises(lowerCAmelCase_	):
            http_head('''https://huggingface.co'''	)
