
def  __snake_case							(	__magic_name__						):







				'''simple docstring'''


				lowercase         =      current_set.copy()
				for row_index, row in enumerate(__magic_name__						):
								lowercase         =      row[0]
								for column_index, column in enumerate(__magic_name__						):
												if magnitude == 0:
																lowercase         =      column
																continue
												lowercase         =      column / magnitude
    # Subtract to cancel term
				lowercase         =      current_set[0]
				lowercase         =      [first_row]
				lowercase         =      current_set[1::]
				for row in current_set:
								lowercase         =      []
								# If first term is 0, it is already in form we want, so we preserve it
								if row[0] == 0:
												final_set.append(__magic_name__						)
												continue
								for column_index in range(len(__magic_name__						)						):
												temp_row.append(first_row[column_index] - row[column_index]						)
								final_set.append(__magic_name__						)
				# Create next recursion iteration set
				if len(final_set[0]						) != 3:
								lowercase         =      final_set[0]
								lowercase         =      []
								lowercase         =      []
								for row in final_set[1::]:
												current_first_column.append(row[0]						)
												next_iteration.append(row[1::]						)
								lowercase         =      simplify(__magic_name__						)
								for i in range(len(__magic_name__						)						):
												resultant[i].insert(0		,    current_first_column[i]						)
								resultant.insert(0		,    __magic_name__						)
								lowercase         =      resultant
				return final_set

def  __snake_case							(	__magic_name__						):







				'''simple docstring'''


				if len(__magic_name__						) == 0:
								raise IndexError("solve_simultaneous() requires n lists of length n+1"						)
				lowercase         =      len(__magic_name__						) + 1
				if any(len(__magic_name__						) != _length for item in equations						):
								raise IndexError("solve_simultaneous() requires n lists of length n+1"						)
				for row in equations:
								if any(not isinstance(__magic_name__		,    (int, float)						) for column in row						):
												raise ValueError("solve_simultaneous() requires lists of integers"						)
				if len(__magic_name__						) == 1:
								return [equations[0][-1] / equations[0][0]]
				lowercase         =      equations.copy()
				if any(0 in row for row in data_set						):
								lowercase         =      data_set.copy()
								lowercase         =      []
								for row_index, row in enumerate(__magic_name__						):
												if 0 not in row:
																lowercase         =      data_set.pop(__magic_name__						)
																break
								if not full_row:
												raise ValueError("solve_simultaneous() requires at least 1 full equation"						)
								data_set.insert(0		,    __magic_name__						)
				lowercase         =      data_set.copy()
				lowercase         =      simplify(__magic_name__						)
				lowercase         =      simplified[::-1]
				lowercase         =      []
				for row in simplified:
								lowercase         =      row[-1]
								if not solutions:
												if row[-2] == 0:
																solutions.append(0						)
																continue
												solutions.append(current_solution / row[-2]						)
												continue
								lowercase         =      row.copy()[: len(__magic_name__						) - 1 :]
								while temp_row[0] == 0:
												temp_row.pop(0						)
								if len(__magic_name__						) == 0:
												solutions.append(0						)
												continue
								lowercase         =      temp_row[1::]
								lowercase         =      temp_row[::-1]
								for column_index, column in enumerate(__magic_name__						):
												current_solution -= column * solutions[column_index]
								solutions.append(__magic_name__						)
				lowercase         =      []
				for item in solutions:
								final.append(float(round(__magic_name__		,    5						)						)						)
				return final[::-1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						_snake_case      :							Union[str, Any]     	=    [
						    [2, 1, 1, 1, 1, 4],
						    [1, 2, 1, 1, 1, 5],
						    [1, 1, 2, 1, 1, 6],
						    [1, 1, 1, 2, 1, 7],
						    [1, 1, 1, 1, 2, 8],
						]
						print(solve_simultaneous(eq))
						print(solve_simultaneous([[4, 2]]))


def  __snake_case							(	__magic_name__		,    __magic_name__ = False						):







				'''simple docstring'''


				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 > 3317044064679887385961981 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,
				    1373653,
				    25326001,
				    3215031751,
				    2152302898747,
				    3474749660383,
				    341550071728321,
				    1,
				    3825123056546413051,
				    1,
				    1,
				    318665857834031151167461,
				    3317044064679887385961981,
				]

				lowercase         =      [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
				for idx, _p in enumerate(__magic_name__		,    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(__magic_name__						):
												lowercase         =      pow(__magic_name__		,    d * 2**r		,    __magic_name__						)
												# 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  __snake_case							(	):







				'''simple docstring'''


				assert not miller_rabin(561						)
				assert miller_rabin(563						)
				# 2047

				assert not miller_rabin(838201						)
				assert miller_rabin(838207						)
				# 1_373_653

				assert not miller_rabin(17316001						)
				assert miller_rabin(17316017						)
				# 25_326_001

				assert not miller_rabin(3078386641						)
				assert miller_rabin(3078386653						)
				# 3_215_031_751

				assert not miller_rabin(1713045574801						)
				assert miller_rabin(1713045574819						)
				# 2_152_302_898_747

				assert not miller_rabin(2779799728307						)
				assert miller_rabin(2779799728327						)
				# 3_474_749_660_383

				assert not miller_rabin(113850023909441						)
				assert miller_rabin(113850023909527						)
				# 341_550_071_728_321

				assert not miller_rabin(1275041018848804351						)
				assert miller_rabin(1275041018848804391						)
				# 3_825_123_056_546_413_051

				assert not miller_rabin(79666464458507787791867						)
				assert miller_rabin(79666464458507787791951						)
				# 318_665_857_834_031_151_167_461

				assert not miller_rabin(552840677446647897660333						)
				assert miller_rabin(552840677446647897660359						)
				# 3_317_044_064_679_887_385_961_981
				# upper limit for probabilistic test


if __name__ == "__main__":
						test_miller_rabin()
