






from typing import Any



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			list			,				__lowercase   :			list			,				__lowercase   :			dict			,				__lowercase   :			dict			,				__lowercase   :			dict			,				)						->      list:





    """simple docstring"""

    _validation(
        __lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				)
    # Creates data structures and fill initial step
    __A         =	{}
    __A         =	{}
    for state in states_space:
        __A         =	observations_space[0]
        __A         =	(
            initial_probabilities[state] * emission_probabilities[state][observation]
        )
        __A         =	None

    # Fills the data structure with the probabilities of
    # different transitions and pointers to previous states
    for o in range(1			,				len(__lowercase    )    ):
        __A         =	observations_space[o]
        __A         =	observations_space[o - 1]
        for state in states_space:
            # Calculates the argmax for probability function
            __A         =	""""""
            __A         =	-1
            for k_state in states_space:
                __A         =	(
                    probabilities[(k_state, prior_observation)]
                    * transition_probabilities[k_state][state]
                    * emission_probabilities[state][observation]
                )
                if probability > max_probability:
                    __A         =	probability
                    __A         =	k_state

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

            __A         =	arg_max

    # The final observation
    __A         =	observations_space[len(__lowercase    ) - 1]

    # argmax for given final observation
    __A         =	""""""
    __A         =	-1
    for k_state in states_space:
        __A         =	probabilities[(k_state, final_observation)]
        if probability > max_probability:
            __A         =	probability
            __A         =	k_state
    __A         =	arg_max

    # Process pointers backwards
    __A         =	last_state
    __A         =	[]
    for o in range(len(__lowercase    ) - 1			,				-1			,				-1    ):
        result.append(__lowercase    )
        __A         =	pointers[previous, observations_space[o]]
    result.reverse()

    return result



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				)						->      None:





    """simple docstring"""

    _validate_not_empty(
        __lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				)
    _validate_lists(__lowercase			,				__lowercase    )
    _validate_dicts(
        __lowercase			,				__lowercase			,				__lowercase    )



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				)						->      None:





    """simple docstring"""

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



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			Any    )						->      None:





    """simple docstring"""

    _validate_list(__lowercase			,				"""observations_space"""    )
    _validate_list(__lowercase			,				"""states_space"""    )



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			str    )						->      None:





    """simple docstring"""

    if not isinstance(_object			,				__lowercase    ):
        __A         =	f"{var_name} must be a list"
        raise ValueError(__lowercase    )
    else:
        for x in _object:
            if not isinstance(__lowercase			,				__lowercase    ):
                __A         =	f"{var_name} must be a list of strings"
                raise ValueError(__lowercase    )



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			Any			,				__lowercase   :			Any			,				)						->      None:





    """simple docstring"""

    _validate_dict(__lowercase			,				"""initial_probabilities"""			,				__lowercase    )
    _validate_nested_dict(__lowercase			,				"""transition_probabilities"""    )
    _validate_nested_dict(__lowercase			,				"""emission_probabilities"""    )



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			str    )						->      None:





    """simple docstring"""

    _validate_dict(_object			,				__lowercase			,				__lowercase    )
    for x in _object.values():
        _validate_dict(__lowercase			,				__lowercase			,				__lowercase			,				__lowercase    )



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			Any			,				__lowercase   :			str			,				__lowercase   :			type			,				__lowercase   :			bool = False    )						->      None:





    """simple docstring"""

    if not isinstance(_object			,				__lowercase    ):
        __A         =	f"{var_name} must be a dict"
        raise ValueError(__lowercase    )
    if not all(isinstance(__lowercase			,				__lowercase    ) for x in _object    ):
        __A         =	f"{var_name} all keys must be strings"
        raise ValueError(__lowercase    )
    if not all(isinstance(__lowercase			,				__lowercase    ) for x in _object.values()    ):
        __A         =	"""nested dictionary """ if nested else """"""
        __A         =	f"{var_name} {nested_text}all values must be {value_type.__name__}"
        raise ValueError(__lowercase    )


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

  testmod()












from __future__ import annotations



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			list[int]			,				__lowercase   :			int    )						->      list[list[int]]:





    """simple docstring"""

    __A         =	[]
    __A         =	[]
    __A         =	0
    __A         =	sum(__lowercase    )
    create_state_space_tree(__lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				__lowercase			,				__lowercase    )
    return result



def 			_SCREAMING_SNAKE_CASE						( __lowercase   :			list[int]			,				__lowercase   :			int			,				__lowercase   :			int			,				__lowercase   :			list[int]			,				__lowercase   :			list[list[int]]			,				__lowercase   :			int			,				)						->      None:





    """simple docstring"""

    if sum(__lowercase    ) > max_sum or (remaining_nums_sum + sum(__lowercase    )) < max_sum:
        return
    if sum(__lowercase    ) == max_sum:
        result.append(__lowercase    )
        return
    for index in range(__lowercase			,				len(__lowercase    )    ):
        create_state_space_tree(
            __lowercase			,				__lowercase			,				index + 1			,				[*path, nums[index]]			,				__lowercase			,				remaining_nums_sum - nums[index]			,				)
__a      :       str				  =   [3, 34, 4, 12, 5, 2]
__a      :       Optional[Any]				  =   9
__a      :       List[str]				  =   generate_sum_of_subsets_soln(nums, max_sum)






print(*result)




