'''simple docstring'''







from math import pow


def    lowerCamelCase       (			UpperCAmelCase__				:  int				,						UpperCAmelCase__				:  int				,						UpperCAmelCase__				:  int				,						UpperCAmelCase__				:  int				,						UpperCAmelCase__				:  int				,						)			->	tuple[int, int]:


							'''simple docstring'''






							if current_sum == needed_sum:
														# If the sum of the powers is equal to needed_sum, then we have a solution.
														solutions_count += 1
														return current_sum, solutions_count

							SCREAMING_SNAKE_CASE__     :Dict						    = int(pow(UpperCAmelCase__				,						UpperCAmelCase__		)		)
							if current_sum + i_to_n <= needed_sum:
														# If the sum of the powers is less than needed_sum, then continue adding powers.
														current_sum += i_to_n
														SCREAMING_SNAKE_CASE__    ,      SCREAMING_SNAKE_CASE__     :Any						    = backtrack(
														    UpperCAmelCase__				,						UpperCAmelCase__				,						current_number + 1				,						UpperCAmelCase__				,						UpperCAmelCase__		)
														current_sum -= i_to_n
							if i_to_n < needed_sum:
														# If the power of i is less than needed_sum, then try with the next power.
														SCREAMING_SNAKE_CASE__    ,      SCREAMING_SNAKE_CASE__     :Tuple						    = backtrack(
														    UpperCAmelCase__				,						UpperCAmelCase__				,						current_number + 1				,						UpperCAmelCase__				,						UpperCAmelCase__		)
							return current_sum, solutions_count


def    lowerCamelCase       (			UpperCAmelCase__				:  int				,						UpperCAmelCase__				:  int		)			->	int:


							'''simple docstring'''






							if not (1 <= needed_sum <= 1_0_0_0 and 2 <= power <= 1_0):
														raise ValueError(
														    'Invalid input\n'
														    'needed_sum must be between 1 and 1000, power between 2 and 10.'		)

							return backtrack(UpperCAmelCase__				,						UpperCAmelCase__				,						1				,						0				,						0		)[1]  # Return the solutions_count


if __name__ == "__main__":
		import doctest

		doctest.testmod()







'''simple docstring'''







from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


UpperCamelCase_      	=	{
    '''configuration_blenderbot_small''': [
        '''BLENDERBOT_SMALL_PRETRAINED_CONFIG_ARCHIVE_MAP''',
        '''BlenderbotSmallConfig''',
        '''BlenderbotSmallOnnxConfig''',
    ],
    '''tokenization_blenderbot_small''': ['''BlenderbotSmallTokenizer'''],
}

try:
		if not is_tokenizers_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		UpperCamelCase_      	=	['''BlenderbotSmallTokenizerFast''']

try:
		if not is_torch_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		UpperCamelCase_      	=	[
		    '''BLENDERBOT_SMALL_PRETRAINED_MODEL_ARCHIVE_LIST''',
		    '''BlenderbotSmallForCausalLM''',
		    '''BlenderbotSmallForConditionalGeneration''',
		    '''BlenderbotSmallModel''',
		    '''BlenderbotSmallPreTrainedModel''',
		]

try:
		if not is_tf_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		UpperCamelCase_      	=	[
		    '''TFBlenderbotSmallForConditionalGeneration''',
		    '''TFBlenderbotSmallModel''',
		    '''TFBlenderbotSmallPreTrainedModel''',
		]

try:
		if not is_flax_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		UpperCamelCase_      	=	[
		    '''FlaxBlenderbotSmallForConditionalGeneration''',
		    '''FlaxBlenderbotSmallModel''',
		    '''FlaxBlenderbotSmallPreTrainedModel''',
		]

if TYPE_CHECKING:
		from .configuration_blenderbot_small import (
		    BLENDERBOT_SMALL_PRETRAINED_CONFIG_ARCHIVE_MAP,
		    BlenderbotSmallConfig,
		    BlenderbotSmallOnnxConfig,
		)
		from .tokenization_blenderbot_small import BlenderbotSmallTokenizer

		try:
				if not is_tokenizers_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .tokenization_blenderbot_small_fast import BlenderbotSmallTokenizerFast

		try:
				if not is_torch_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_blenderbot_small import (
				    BLENDERBOT_SMALL_PRETRAINED_MODEL_ARCHIVE_LIST,
				    BlenderbotSmallForCausalLM,
				    BlenderbotSmallForConditionalGeneration,
				    BlenderbotSmallModel,
				    BlenderbotSmallPreTrainedModel,
				)

		try:
				if not is_tf_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_tf_blenderbot_small import (
				    TFBlenderbotSmallForConditionalGeneration,
				    TFBlenderbotSmallModel,
				    TFBlenderbotSmallPreTrainedModel,
				)

		try:
				if not is_flax_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_flax_blenderbot_small import (
				    FlaxBlenderbotSmallForConditionalGeneration,
				    FlaxBlenderbotSmallModel,
				    FlaxBlenderbotSmallPreTrainedModel,
				)

else:
		import sys

		UpperCamelCase_      	=	_LazyModule(__name__, globals()['''__file__'''], _import_structure, module_spec=__spec__)






