


def 			UpperCamelCase (     __magic_name__					:							list[list]					)       ->  list[list]:
		"""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 			UpperCamelCase (     __magic_name__					:							list[list]					)       ->  list:
		"""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()
	A	:		Optional[int]    =       [
	    [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]]))








from functools import lru_cache






def 			UpperCamelCase (     __magic_name__					:							int					)       ->  set:
		"""simple docstring"""




		lowercase__					      =		2
		lowercase__					      =		set()
		while i * i <= n:
				if n % i:
						i += 1
				else:
						n //= i
						factors.add(__magic_name__					)
		if n > 1:
				factors.add(__magic_name__					)
		return factors






@lru_cache
def 			UpperCamelCase (     __magic_name__					:							int					)       ->  int:
		"""simple docstring"""




		return len(unique_prime_factors(__magic_name__					)					)






def 			UpperCamelCase (     __magic_name__					:							list					)       ->  bool:
		"""simple docstring"""




		return len(set(__magic_name__					)					) in (0, 1)






def 			UpperCamelCase (     __magic_name__					:							int					)       ->  list:
		"""simple docstring"""




		lowercase__					      =		2

		while True:
				# Increment each value of a generated range
				lowercase__					      =		[base + i for i in range(__magic_name__					)]

				# Run elements through out unique_prime_factors function
				# Append our target number to the end.
				lowercase__					      =		[upf_len(__magic_name__					) for x in group]
				checker.append(__magic_name__					)

				# If all numbers in the list are equal, return the group variable.
				if equality(__magic_name__					):
						return group

				# Increment our base variable by 1
				base += 1






def 			UpperCamelCase (     __magic_name__					:							int = 4					)       ->  int:
		"""simple docstring"""




		lowercase__					      =		run(__magic_name__					)
		return results[0] if len(__magic_name__					) else None


if __name__ == "__main__":
	print(solution())

