


'''simple docstring'''





import math



def 			lowerCAmelCase (		lowerCAmelCase_ )-> bool:
				lowerCAmelCase_ :      int    				=				math.loga(math.sqrt(4 * positive_integer + 1 ) / 2 + 1 / 2 )

				return exponent == int(lowerCAmelCase_ )



def 			lowerCAmelCase (		lowerCAmelCase_ = 1 / 12_345 )-> int:
				lowerCAmelCase_ :      Optional[int]    				=				0
				lowerCAmelCase_ :      List[Any]    				=				0

				lowerCAmelCase_ :      Union[str, Any]    				=				3
				while True:
								lowerCAmelCase_ :      Tuple    				=				(integer**2 - 1) / 4
								# if candidate is an integer, then there is a partition for k
								if partition_candidate == int(lowerCAmelCase_ ):
												lowerCAmelCase_ :      Dict    				=				int(lowerCAmelCase_ )
												total_partitions += 1
												if check_partition_perfect(lowerCAmelCase_ ):
																perfect_partitions += 1
								if perfect_partitions > 0:
												if perfect_partitions / total_partitions < max_proportion:
																return int(lowerCAmelCase_ )
								integer += 1


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



_UpperCAmelCase			:  int     =frozenset(
    [
        """prompt""",
        """height""",
        """width""",
        """guidance_scale""",
        """negative_prompt""",
        """prompt_embeds""",
        """negative_prompt_embeds""",
        """cross_attention_kwargs""",
    ]
)

_UpperCAmelCase			:  List[Any]     =frozenset(["""prompt""", """negative_prompt"""])

_UpperCAmelCase			:  Dict     =frozenset([])

_UpperCAmelCase			:  int     =frozenset(["""image"""])

_UpperCAmelCase			:  Tuple     =frozenset(
    [
        """image""",
        """height""",
        """width""",
        """guidance_scale""",
    ]
)

_UpperCAmelCase			:  int     =frozenset(["""image"""])

_UpperCAmelCase			:  str     =frozenset(
    [
        """prompt""",
        """image""",
        """height""",
        """width""",
        """guidance_scale""",
        """negative_prompt""",
        """prompt_embeds""",
        """negative_prompt_embeds""",
    ]
)

_UpperCAmelCase			:  int     =frozenset(["""prompt""", """image""", """negative_prompt"""])

_UpperCAmelCase			:  Optional[int]     =frozenset(
    [
        # Text guided image variation with an image mask
        """prompt""",
        """image""",
        """mask_image""",
        """height""",
        """width""",
        """guidance_scale""",
        """negative_prompt""",
        """prompt_embeds""",
        """negative_prompt_embeds""",
    ]
)

_UpperCAmelCase			:  Optional[int]     =frozenset(["""prompt""", """image""", """mask_image""", """negative_prompt"""])

_UpperCAmelCase			:  Optional[Any]     =frozenset(
    [
        # image variation with an image mask
        """image""",
        """mask_image""",
        """height""",
        """width""",
        """guidance_scale""",
    ]
)

_UpperCAmelCase			:  Optional[Any]     =frozenset(["""image""", """mask_image"""])

_UpperCAmelCase			:  Union[str, Any]     =frozenset(
    [
        """example_image""",
        """image""",
        """mask_image""",
        """height""",
        """width""",
        """guidance_scale""",
    ]
)

_UpperCAmelCase			:  Tuple     =frozenset(["""example_image""", """image""", """mask_image"""])

_UpperCAmelCase			:  Any     =frozenset(["""class_labels"""])

_UpperCAmelCase			:  List[Any]     =frozenset(["""class_labels"""])

_UpperCAmelCase			:  int     =frozenset(["""batch_size"""])

_UpperCAmelCase			:  str     =frozenset([])

_UpperCAmelCase			:  str     =frozenset(["""batch_size"""])

_UpperCAmelCase			:  Optional[Any]     =frozenset([])

_UpperCAmelCase			:  Tuple     =frozenset(
    [
        """prompt""",
        """audio_length_in_s""",
        """guidance_scale""",
        """negative_prompt""",
        """prompt_embeds""",
        """negative_prompt_embeds""",
        """cross_attention_kwargs""",
    ]
)

_UpperCAmelCase			:  Tuple     =frozenset(["""prompt""", """negative_prompt"""])
_UpperCAmelCase			:  List[str]     =frozenset(["""input_tokens"""])

_UpperCAmelCase			:  Optional[Any]     =frozenset(["""input_tokens"""])