
"""simple docstring"""




from __future__ import annotations

import random
from collections.abc import Iterable
class SCREAMING_SNAKE_CASE       :






	'''simple docstring'''







	def __init__(  self :Tuple       ,    __magic_name__ :list[str]			)			->   None:

			'''simple docstring'''






			a__   		=     {literal: None for literal in literals}


	def __str__(  self :List[str]			)			->   str:

			'''simple docstring'''






			return "{" + " , ".join(self.literals			) + "}"


	def __len__(  self :Optional[int]			)			->   int:

			'''simple docstring'''






			return len(self.literals			)


	def    _UpperCamelCase							(  self :Dict       ,    __magic_name__ :dict[str, bool | None]			)			->   None:

			'''simple docstring'''






			for literal in self.literals:
					a__   		=     literal[:2]
					if symbol in model:
							a__   		=     model[symbol]
					else:
							continue
					if value is not None:
							# Complement assignment if literal is in complemented form
							if literal.endswith('''\''''			):
									a__   		=     not value
					a__   		=     value






	def    _UpperCamelCase							(  self :Union[str, Any]       ,    __magic_name__ :dict[str, bool | None]			)			->   bool | None:

			'''simple docstring'''






			for literal in self.literals:
					a__   		=     literal.rstrip('''\''''			) if literal.endswith('''\''''			) else literal + '''\''''
					if symbol in self.literals:
							return True

			self.assign(__magic_name__			)
			for value in self.literals.values():
					if value in (True, None):
							return value
			return any(self.literals.values()			)

class SCREAMING_SNAKE_CASE       :






	'''simple docstring'''







	def __init__(  self :Dict       ,    __magic_name__ :Iterable[Clause]			)			->   None:

			'''simple docstring'''






			a__   		=     list(__magic_name__			)






	def __str__(  self :List[str]			)			->   str:

			'''simple docstring'''






			return "{" + " , ".join(str(__magic_name__			) for clause in self.clauses			) + "}"

def __snake_case (      )       ->    Clause:


		"""simple docstring"""





		a__   		=     []
		a__   		=     random.randint(1						,					5    )
		a__   		=     '''A'''
		a__   		=     0
		while i < no_of_literals:
				a__   		=     random.randint(1						,					5    )
				a__   		=     base_var + str(UpperCamelCase    )
				a__   		=     random.randint(0						,					1    )
				if var_complement == 1:
						var_name += "'"
				if var_name in literals:
						i -= 1
				else:
						literals.append(UpperCamelCase    )
				i += 1
		return Clause(UpperCamelCase    )

def __snake_case (      )       ->    Formula:


		"""simple docstring"""





		a__   		=     set()
		a__   		=     random.randint(1						,					10    )
		while len(UpperCamelCase    ) < no_of_clauses:
				clauses.add(generate_clause()    )
		return Formula(UpperCamelCase    )

def __snake_case (      UpperCamelCase    )       ->    tuple[list[Clause], list[str]]:


		"""simple docstring"""





		a__   		=     formula.clauses
		a__   		=     []
		for clause in formula.clauses:
				for literal in clause.literals:
						a__   		=     literal[:2]
						if symbol not in symbols_set:
								symbols_set.append(UpperCamelCase    )
		return clauses, symbols_set

def __snake_case (      UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )       ->    tuple[list[str], dict[str, bool | None]]:


		"""simple docstring"""





		a__   		=     []
		a__   		=     {}
		a__   		=     []

		for clause in clauses:
				if clause.evaluate(UpperCamelCase    ):
						continue
				for literal in clause.literals:
						literals.append(UpperCamelCase    )

		for s in symbols:
				a__   		=     s + '''\''''
				if (s in literals and sym not in literals) or (
				    s not in literals and sym in literals
				):
						pure_symbols.append(UpperCamelCase    )
		for p in pure_symbols:
				a__   		=     None
		for s in pure_symbols:
				a__   		=     s + '''\''''
				if s in literals:
						a__   		=     True
				elif sym in literals:
						a__   		=     False
		return pure_symbols, assignment

def __snake_case (      UpperCamelCase						,					UpperCamelCase    )       ->    tuple[list[str], dict[str, bool | None]]:


		"""simple docstring"""





		a__   		=     []
		for clause in clauses:
				if len(UpperCamelCase    ) == 1:
						unit_symbols.append(list(clause.literals.keys()    )[0]    )
				else:
						a__		,	a__   		=     0, 0
						for literal, value in clause.literals.items():
								if value is False:
										f_count += 1
								elif value is None:
										a__   		=     literal
										n_count += 1
						if f_count == len(UpperCamelCase    ) - 1 and n_count == 1:
								unit_symbols.append(UpperCamelCase    )
		a__   		=     {}
		for i in unit_symbols:
				a__   		=     i[:2]
				a__   		=     len(UpperCamelCase    ) == 2
		a__   		=     [i[:2] for i in unit_symbols]

		return unit_symbols, assignment

def __snake_case (      UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )       ->    tuple[bool | None, dict[str, bool | None] | None]:


		"""simple docstring"""





		a__   		=     True
		for clause in clauses:
				a__   		=     clause.evaluate(UpperCamelCase    )
				if clause_check is False:
						return False, None
				elif clause_check is None:
						a__   		=     False
						continue

		if check_clause_all_true:
				return True, model

		try:
				a__		,	a__   		=     find_pure_symbols(UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )
		except RecursionError:
				print('''raises a RecursionError and is'''    )
				return None, {}
		a__   		=     None
		if len(UpperCamelCase    ) > 0:
				a__		,	a__   		=     pure_symbols[0], assignment[pure_symbols[0]]

		if p:
				a__   		=     model
				a__   		=     value
				a__   		=     list(UpperCamelCase    )
				if p in tmp_symbols:
						tmp_symbols.remove(UpperCamelCase    )
				return dpll_algorithm(UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )

		a__		,	a__   		=     find_unit_clauses(UpperCamelCase						,					UpperCamelCase    )
		a__   		=     None
		if len(UpperCamelCase    ) > 0:
				a__		,	a__   		=     unit_symbols[0], assignment[unit_symbols[0]]
		if p:
				a__   		=     model
				a__   		=     value
				a__   		=     list(UpperCamelCase    )
				if p in tmp_symbols:
						tmp_symbols.remove(UpperCamelCase    )
				return dpll_algorithm(UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )
		a__   		=     symbols[0]
		a__   		=     symbols[1:]
		a__		,	a__   		=     model, model
		a__		,	a__   		=     True, False

		return dpll_algorithm(UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    ) or dpll_algorithm(UpperCamelCase						,					UpperCamelCase						,					UpperCamelCase    )


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					__lowerCAmelCase							:  str 				=		generate_formula()
					print(f"The formula {formula} is", end=''' ''')

					__lowerCAmelCase				,__lowerCAmelCase							:  Optional[Any] 				=		generate_parameters(formula)
					__lowerCAmelCase				,__lowerCAmelCase							:  List[str] 				=		dpll_algorithm(clauses, symbols, {})

					if solution:
										print(f"satisfiable with the assignment {model}.")
					else:
										print('''not satisfiable.''')






"""simple docstring"""




def __snake_case (      UpperCamelCase = 10**6    )       ->    int:


		"""simple docstring"""





		if n <= 0:
				raise ValueError('''Please enter an integer greater than 0'''    )

		a__   		=     list(range(n + 1    )    )
		for number in range(2						,					n + 1    ):
				if phi[number] == number:
						phi[number] -= 1
						for multiple in range(number * 2						,					n + 1						,					UpperCamelCase    ):
								a__   		=     (phi[multiple] // number) * (number - 1)

		a__   		=     1
		for number in range(1						,					n + 1    ):
				if (answer / phi[answer]) < (number / phi[number]):
						a__   		=     number

		return answer


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




