



def __lowerCAmelCase (     SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_ = False  ):
			if n == 2:
						return True
			if not n % 2 or n < 2:
						return False
			if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
						return False
			if n > 3_3170_4406_4679_8873_8596_1981 and not allow_probable:
						raise ValueError(
						    "Warning: upper bound of deterministic test is exceeded. "
						    "Pass allow_probable=True to allow probabilistic test. "
						    "A return value of True indicates a probable prime."  )
			# array bounds provided by analysis
			lowercase__					  =				[
			    2047,
			    137_3653,
			    2532_6001,
			    32_1503_1751,
			    2_1523_0289_8747,
			    3_4747_4966_0383,
			    341_5500_7172_8321,
			    1,
			    382_5123_0565_4641_3051,
			    1,
			    1,
			    3186_6585_7834_0311_5116_7461,
			    3_3170_4406_4679_8873_8596_1981,
			]

			lowercase__					  =				[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
			for idx, _p in enumerate(SCREAMING_SNAKE_CASE_						,  1  ):
						if n < _p:
									# then we have our last prime to check
									lowercase__					  =				primes[:idx]
									break
			lowercase__  ,       lowercase__					  =				n - 1, 0
			# break up n -1 into a power of 2 (s) and
			# remaining odd component
			# essentially, solve for d * 2 ** s == n - 1
			while d % 2 == 0:
						d //= 2
						s += 1
			for prime in plist:
						lowercase__					  =				False
						for r in range(SCREAMING_SNAKE_CASE_  ):
									lowercase__					  =				pow(SCREAMING_SNAKE_CASE_						,  d * 2**r						,  SCREAMING_SNAKE_CASE_  )
									# see article for analysis explanation for m
									if (r == 0 and m == 1) or ((m + 1) % n == 0):
												lowercase__					  =				True
												# this loop will not determine compositeness
												break
						if pr:
									continue
						# if pr is False, then the above loop never evaluated to true,
						# and the n MUST be composite
						return False
			return True



def __lowerCAmelCase (     ):
			assert not miller_rabin(561  )
			assert miller_rabin(563  )
			# 2047

			assert not miller_rabin(83_8201  )
			assert miller_rabin(83_8207  )
			# 1_373_653

			assert not miller_rabin(1731_6001  )
			assert miller_rabin(1731_6017  )
			# 25_326_001

			assert not miller_rabin(30_7838_6641  )
			assert miller_rabin(30_7838_6653  )
			# 3_215_031_751

			assert not miller_rabin(1_7130_4557_4801  )
			assert miller_rabin(1_7130_4557_4819  )
			# 2_152_302_898_747

			assert not miller_rabin(2_7797_9972_8307  )
			assert miller_rabin(2_7797_9972_8327  )
			# 3_474_749_660_383

			assert not miller_rabin(113_8500_2390_9441  )
			assert miller_rabin(113_8500_2390_9527  )
			# 341_550_071_728_321

			assert not miller_rabin(127_5041_0188_4880_4351  )
			assert miller_rabin(127_5041_0188_4880_4391  )
			# 3_825_123_056_546_413_051

			assert not miller_rabin(796_6646_4458_5077_8779_1867  )
			assert miller_rabin(796_6646_4458_5077_8779_1951  )
			# 318_665_857_834_031_151_167_461

			assert not miller_rabin(5528_4067_7446_6478_9766_0333  )
			assert miller_rabin(5528_4067_7446_6478_9766_0359  )
			# 3_317_044_064_679_887_385_961_981
			# upper limit for probabilistic test


if __name__ == "__main__":
							test_miller_rabin()









def __lowerCAmelCase (     SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_  ):
			lowercase__  ,       lowercase__					  =				len(SCREAMING_SNAKE_CASE_  ), len(grid[0]  )
			if (
			    min(SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_  ) < 0
			    or row == row_length
			    or col == col_length
			    or (row, col) in visit
			    or grid[row][col] == 1
			):
						return 0
			if row == row_length - 1 and col == col_length - 1:
						return 1

			visit.add((row, col)  )

			lowercase__					  =				0
			count += depth_first_search(SCREAMING_SNAKE_CASE_						,  row + 1						,  SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_  )
			count += depth_first_search(SCREAMING_SNAKE_CASE_						,  row - 1						,  SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_  )
			count += depth_first_search(SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_						,  col + 1						,  SCREAMING_SNAKE_CASE_  )
			count += depth_first_search(SCREAMING_SNAKE_CASE_						,  SCREAMING_SNAKE_CASE_						,  col - 1						,  SCREAMING_SNAKE_CASE_  )

			visit.remove((row, col)  )
			return count


if __name__ == "__main__":
							import doctest

							doctest.testmod()




