
from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt




def 	A_ (							snake_case							:   Union[str, Any] ):



						'''simple docstring'''

						__UpperCamelCase       =			int(number**0.5 )
						return number == sq * sq




def 	A_ (							snake_case							:   Dict ,      snake_case							:   List[Any] ,      snake_case							:   str ,      snake_case							:   Optional[int] ,      snake_case							:   str ,      snake_case							:   Any ):



						'''simple docstring'''

						__UpperCamelCase       =			x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
						__UpperCamelCase       =			x_den * y_den * z_den
						__UpperCamelCase       =			gcd(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
						top //= hcf
						bottom //= hcf
						return top, bottom




def 	A_ (							snake_case							:   Any = 35 ):



						'''simple docstring'''

						__UpperCamelCase       =			set()
						__UpperCamelCase       =			42
						__UpperCamelCase       =			Fraction(0 )
						__UpperCamelCase       =			42

						for x_num in range(1 ,      order + 1 ):
												for x_den in range(x_num + 1 ,      order + 1 ):
																		for y_num in range(1 ,      order + 1 ):
																								for y_den in range(y_num + 1 ,      order + 1 ):
																														# n=1
																														__UpperCamelCase       =			x_num * y_den + x_den * y_num
																														__UpperCamelCase       =			x_den * y_den
																														__UpperCamelCase       =			gcd(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																														z_num //= hcf
																														z_den //= hcf
																														if 0 < z_num < z_den <= order:
																																				__UpperCamelCase       =			add_three(
																																				    SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																				unique_s.add(SCREAMING_SNAKE_CASE_ )

																														# n=2
																														__UpperCamelCase       =			(
																														    x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
																														)
																														__UpperCamelCase       =			x_den * x_den * y_den * y_den
																														if is_sq(SCREAMING_SNAKE_CASE_ ) and is_sq(SCREAMING_SNAKE_CASE_ ):
																																				__UpperCamelCase       =			int(sqrt(SCREAMING_SNAKE_CASE_ ) )
																																				__UpperCamelCase       =			int(sqrt(SCREAMING_SNAKE_CASE_ ) )
																																				__UpperCamelCase       =			gcd(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																				z_num //= hcf
																																				z_den //= hcf
																																				if 0 < z_num < z_den <= order:
																																										__UpperCamelCase       =			add_three(
																																										    SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																										unique_s.add(SCREAMING_SNAKE_CASE_ )

																																				  # n=-1
																														__UpperCamelCase       =			x_num * y_num
																														__UpperCamelCase       =			x_den * y_num + x_num * y_den
																														__UpperCamelCase       =			gcd(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																														z_num //= hcf
																														z_den //= hcf
																														if 0 < z_num < z_den <= order:
																																				__UpperCamelCase       =			add_three(
																																				    SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																				unique_s.add(SCREAMING_SNAKE_CASE_ )

																														# n=2
																														__UpperCamelCase       =			x_num * x_num * y_num * y_num
																														__UpperCamelCase       =			(
																														    x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
																														)
																														if is_sq(SCREAMING_SNAKE_CASE_ ) and is_sq(SCREAMING_SNAKE_CASE_ ):
																																				__UpperCamelCase       =			int(sqrt(SCREAMING_SNAKE_CASE_ ) )
																																				__UpperCamelCase       =			int(sqrt(SCREAMING_SNAKE_CASE_ ) )
																																				__UpperCamelCase       =			gcd(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																				z_num //= hcf
																																				z_den //= hcf
																																				if 0 < z_num < z_den <= order:
																																										__UpperCamelCase       =			add_three(
																																										    SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )
																																										unique_s.add(SCREAMING_SNAKE_CASE_ )

						for num, den in unique_s:
												total += Fraction(SCREAMING_SNAKE_CASE_ ,      SCREAMING_SNAKE_CASE_ )

						return total.denominator + total.numerator


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






from typing import List, Optional, Tuple, Union

import torch

from ...schedulers import DDIMScheduler
from ...utils import randn_tensor
from ..pipeline_utils import DiffusionPipeline, ImagePipelineOutput






class        SCREAMING_SNAKE_CASE__    (							SCREAMING_SNAKE_CASE_    ):

							"""simple docstring"""






							def __init__(  self     ,	SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_			)->   str:





													'''simple docstring'''






													super().__init__()

													# make sure scheduler can always be converted to DDIM
													__UpperCamelCase       =			DDIMScheduler.from_config(scheduler.config			)

													self.register_modules(unet=SCREAMING_SNAKE_CASE_     ,	scheduler=SCREAMING_SNAKE_CASE_			)



							@torch.no_grad()
							def __call__(  self     ,	SCREAMING_SNAKE_CASE_ = 1     ,	SCREAMING_SNAKE_CASE_ = None     ,	SCREAMING_SNAKE_CASE_ = 0.0     ,	SCREAMING_SNAKE_CASE_ = 50     ,	SCREAMING_SNAKE_CASE_ = None     ,	SCREAMING_SNAKE_CASE_ = "pil"     ,	SCREAMING_SNAKE_CASE_ = True     ,	)->   Union[ImagePipelineOutput, Tuple]:





													'''simple docstring'''






													if isinstance(self.unet.config.sample_size     ,	SCREAMING_SNAKE_CASE_			):
																			__UpperCamelCase       =			(
																			    batch_size,
																			    self.unet.config.in_channels,
																			    self.unet.config.sample_size,
																			    self.unet.config.sample_size,
																			)
													else:
																			__UpperCamelCase       =			(batch_size, self.unet.config.in_channels, *self.unet.config.sample_size)

													if isinstance(SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_			) and len(SCREAMING_SNAKE_CASE_			) != batch_size:
																			raise ValueError(
																			    F"You have passed a list of generators of length {len(SCREAMING_SNAKE_CASE_			)}, but requested an effective batch"
																			    F" size of {batch_size}. Make sure the batch size matches the length of the generators."			)

													__UpperCamelCase       =			randn_tensor(SCREAMING_SNAKE_CASE_     ,	generator=SCREAMING_SNAKE_CASE_     ,	device=self.device     ,	dtype=self.unet.dtype			)

													# set step values
													self.scheduler.set_timesteps(SCREAMING_SNAKE_CASE_			)

													for t in self.progress_bar(self.scheduler.timesteps			):
																			# 1. predict noise model_output
																			__UpperCamelCase       =			self.unet(SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_			).sample

																			# 2. predict previous mean of image x_t-1 and add variance depending on eta
																			# eta corresponds to η in paper and should be between [0, 1]
																			# do x_t -> x_t-1
																			__UpperCamelCase       =			self.scheduler.step(
																			    SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_     ,	eta=SCREAMING_SNAKE_CASE_     ,	use_clipped_model_output=SCREAMING_SNAKE_CASE_     ,	generator=SCREAMING_SNAKE_CASE_			).prev_sample

													__UpperCamelCase       =			(image / 2 + 0.5).clamp(0     ,	1			)
													__UpperCamelCase       =			image.cpu().permute(0     ,	2     ,	3     ,	1			).numpy()
													if output_type == "pil":
																			__UpperCamelCase       =			self.numpy_to_pil(SCREAMING_SNAKE_CASE_			)

													if not return_dict:
																			return (image,)

													return ImagePipelineOutput(images=SCREAMING_SNAKE_CASE_			)




