


from typing import Any


def        _UpperCamelCase							(      lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       ):
						_validation(
						    lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       )
						# Creates data structures and fill initial step
						__SCREAMING_SNAKE_CASE						:     dict					=		{}
						__SCREAMING_SNAKE_CASE						:     dict					=		{}
						for state in states_space:
												__SCREAMING_SNAKE_CASE						:     List[Any]					=		observations_space[0]
												__SCREAMING_SNAKE_CASE						:     str					=		(
												    initial_probabilities[state] * emission_probabilities[state][observation]
												)
												__SCREAMING_SNAKE_CASE						:     List[Any]					=		None

						# Fills the data structure with the probabilities of
						# different transitions and pointers to previous states
						for o in range(1    ,       len(lowercase__      )      ):
												__SCREAMING_SNAKE_CASE						:     List[Any]					=		observations_space[o]
												__SCREAMING_SNAKE_CASE						:     Optional[Any]					=		observations_space[o - 1]
												for state in states_space:
																		# Calculates the argmax for probability function
																		__SCREAMING_SNAKE_CASE						:     int					=		''''''
																		__SCREAMING_SNAKE_CASE						:     Any					=		-1
																		for k_state in states_space:
																								__SCREAMING_SNAKE_CASE						:     int					=		(
																								    probabilities[(k_state, prior_observation)]
																								    * transition_probabilities[k_state][state]
																								    * emission_probabilities[state][observation]
																								)
																								if probability > max_probability:
																														__SCREAMING_SNAKE_CASE						:     Tuple					=		probability
																														__SCREAMING_SNAKE_CASE						:     Union[str, Any]					=		k_state

            # Update probabilities and pointers dicts
																		__SCREAMING_SNAKE_CASE						:     str					=		(
																		    probabilities[(arg_max, prior_observation)]
																		    * transition_probabilities[arg_max][state]
																		    * emission_probabilities[state][observation]
																		)

																		__SCREAMING_SNAKE_CASE						:     Union[str, Any]					=		arg_max

    # The final observation
						__SCREAMING_SNAKE_CASE						:     Union[str, Any]					=		observations_space[len(lowercase__      ) - 1]

						# argmax for given final observation
						__SCREAMING_SNAKE_CASE						:     Tuple					=		''''''
						__SCREAMING_SNAKE_CASE						:     Optional[int]					=		-1
						for k_state in states_space:
												__SCREAMING_SNAKE_CASE						:     Dict					=		probabilities[(k_state, final_observation)]
												if probability > max_probability:
																		__SCREAMING_SNAKE_CASE						:     int					=		probability
																		__SCREAMING_SNAKE_CASE						:     Optional[int]					=		k_state
						__SCREAMING_SNAKE_CASE						:     Optional[int]					=		arg_max

						# Process pointers backwards
						__SCREAMING_SNAKE_CASE						:     List[Any]					=		last_state
						__SCREAMING_SNAKE_CASE						:     Optional[Any]					=		[]
						for o in range(len(lowercase__      ) - 1    ,       -1    ,       -1      ):
												result.append(lowercase__      )
												__SCREAMING_SNAKE_CASE						:     Tuple					=		pointers[previous, observations_space[o]]
						result.reverse()

						return result


def        _UpperCamelCase							(      lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       ):
						_validate_not_empty(
						    lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       )
						_validate_lists(lowercase__    ,       lowercase__      )
						_validate_dicts(
						    lowercase__    ,       lowercase__    ,       lowercase__      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__    ,       ):
						if not all(
						    [
						        observations_space,
						        states_space,
						        initial_probabilities,
						        transition_probabilities,
						        emission_probabilities,
						    ]      ):
												raise ValueError('''There\'s an empty parameter'''      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__      ):
						_validate_list(lowercase__    ,       '''observations_space'''      )
						_validate_list(lowercase__    ,       '''states_space'''      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__      ):
						if not isinstance(_object    ,       lowercase__      ):
												__SCREAMING_SNAKE_CASE						:     str					=		F'''{var_name} must be a list'''
												raise ValueError(lowercase__      )
						else:
												for x in _object:
																		if not isinstance(lowercase__    ,       lowercase__      ):
																								__SCREAMING_SNAKE_CASE						:     str					=		F'''{var_name} must be a list of strings'''
																								raise ValueError(lowercase__      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__    ,       lowercase__    ,       ):
						_validate_dict(lowercase__    ,       '''initial_probabilities'''    ,       lowercase__      )
						_validate_nested_dict(lowercase__    ,       '''transition_probabilities'''      )
						_validate_nested_dict(lowercase__    ,       '''emission_probabilities'''      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__      ):
						_validate_dict(_object    ,       lowercase__    ,       lowercase__      )
						for x in _object.values():
												_validate_dict(lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__      )


def        _UpperCamelCase							(      lowercase__    ,       lowercase__    ,       lowercase__    ,       lowercase__ = False      ):
						if not isinstance(_object    ,       lowercase__      ):
												__SCREAMING_SNAKE_CASE						:     str					=		F'''{var_name} must be a dict'''
												raise ValueError(lowercase__      )
						if not all(isinstance(lowercase__    ,       lowercase__      ) for x in _object      ):
												__SCREAMING_SNAKE_CASE						:     Dict					=		F'''{var_name} all keys must be strings'''
												raise ValueError(lowercase__      )
						if not all(isinstance(lowercase__    ,       lowercase__      ) for x in _object.values()      ):
												__SCREAMING_SNAKE_CASE						:     Tuple					=		'''nested dictionary ''' if nested else ''''''
												__SCREAMING_SNAKE_CASE						:     Optional[Any]					=		F'''{var_name} {nested_text}all values must be {value_type.__name__}'''
												raise ValueError(lowercase__      )


if __name__ == "__main__":
						from doctest import testmod

						testmod()







def 		lowerCAmelCase_    (  __UpperCAmelCase:			int      )      ->	bool:
    return str(__UpperCAmelCase      ) == str(__UpperCAmelCase      )[::-1]




def 		lowerCAmelCase_    (  __UpperCAmelCase:			int      )      ->	int:
    return int(__UpperCAmelCase      ) + int(str(__UpperCAmelCase      )[::-1]      )




def 		lowerCAmelCase_    (  __UpperCAmelCase:			int = 1_0000      )      ->	int:
    UpperCamelCase__							:				Optional[Any]          =      []
    for num in range(1		,   __UpperCAmelCase      ):
        UpperCamelCase__							:				str          =      0
        UpperCamelCase__							:				Any          =      num
        while iterations < 50:
            UpperCamelCase__							:				List[Any]          =      sum_reverse(__UpperCAmelCase      )
            iterations += 1
            if is_palindrome(__UpperCAmelCase      ):
                break
        else:
            lychrel_nums.append(__UpperCAmelCase      )
    return len(__UpperCAmelCase      )


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

