


import enum
import inspect
from typing import Iterable, List, Optional, Tuple, Union





class       _a							(       enum.Enum):
 _a						:						Dict   					=							'''timm'''
 _a						:						str   					=							'''transformers'''






def   lowerCamelCase_					( _a			,     _a			,     _a ):




  """simple docstring"""






  if stage_names is None:
    raise ValueError('''Stage_names must be set for transformers backbones''' )

  if out_features is not None:
    if not isinstance(UpperCamelCase__			,     (list,) ):
      raise ValueError(f'out_features must be a list {type(UpperCamelCase__ )}' )
    if any(feat not in stage_names for feat in out_features ):
      raise ValueError(f'out_features must be a subset of stage_names: {stage_names} got {out_features}' )

  if out_indices is not None:
    if not isinstance(UpperCamelCase__			,     (list, tuple) ):
      raise ValueError(f'out_indices must be a list or tuple, got {type(UpperCamelCase__ )}' )
    if any(idx >= len(UpperCamelCase__ ) for idx in out_indices ):
      raise ValueError('''out_indices must be valid indices for stage_names {stage_names}, got {out_indices}''' )

  if out_features is not None and out_indices is not None:
    if len(UpperCamelCase__ ) != len(UpperCamelCase__ ):
      raise ValueError('''out_features and out_indices should have the same length if both are set''' )
    if out_features != [stage_names[idx] for idx in out_indices]:
      raise ValueError('''out_features and out_indices should correspond to the same stages if both are set''' )






def   lowerCamelCase_					( _a			,     _a			,     _a			,     ):




  """simple docstring"""






  if out_indices is None and out_features is None:
    lowerCAmelCase__     :    Optional[int]					  =      [len(UpperCamelCase__ ) - 1]
    lowerCAmelCase__     :    List[Any]					  =      [stage_names[-1]]
  elif out_indices is None and out_features is not None:
    lowerCAmelCase__     :    Union[str, Any]					  =      [stage_names.index(UpperCamelCase__ ) for layer in out_features]
  elif out_features is None and out_indices is not None:
    lowerCAmelCase__     :    str					  =      [stage_names[idx] for idx in out_indices]
  return out_features, out_indices






def   lowerCamelCase_					( _a			,     _a			,     _a			,     ):




  """simple docstring"""






  verify_out_features_out_indices(out_features=UpperCamelCase__			,     out_indices=UpperCamelCase__			,     stage_names=UpperCamelCase__ )
  lowerCAmelCase__  ,			lowerCAmelCase__     :    Union[str, Any]					  =      _align_output_features_output_indices(
      out_features=UpperCamelCase__			,     out_indices=UpperCamelCase__			,     stage_names=UpperCamelCase__ )
  # Verify that the aligned out_features and out_indices are valid
  verify_out_features_out_indices(out_features=UpperCamelCase__			,     out_indices=UpperCamelCase__			,     stage_names=UpperCamelCase__ )
  return output_features, output_indices





class       _a							:
 _a						:						Optional[BackboneType]   					=							None




 def 							UpperCAmelCase__(      self		:	Optional[int]	,      _SCREAMING_SNAKE_CASE		:	Union[str, Any]	)-> None:
   if getattr(self	,      '''_backbone'''	,      a_	) is None:
     raise ValueError('''self._backbone must be set before calling _init_timm_backbone'''	)

   # These will diagree with the defaults for the transformers models e.g. for resnet50
   # the transformer model has out_features = ['stem', 'stage1', 'stage2', 'stage3', 'stage4']
   # the timm model has out_features = ['act', 'layer1', 'layer2', 'layer3', 'layer4']
   lowerCAmelCase__     :    Tuple					  =      [stage['''module'''] for stage in self._backbone.feature_info.info]
   lowerCAmelCase__     :    Tuple					  =      [stage['''num_chs'''] for stage in self._backbone.feature_info.info]
   lowerCAmelCase__     :    List[str]					  =      self._backbone.feature_info.out_indices
   lowerCAmelCase__     :    Optional[int]					  =      self._backbone.feature_info.module_name()

   # We verify the out indices and out features are valid
   verify_out_features_out_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=self.stage_names	)
   lowerCAmelCase__  ,			lowerCAmelCase__     :    Union[str, Any]					  =      out_features, out_indices




 def 							UpperCAmelCase__(      self		:	Dict	,      _SCREAMING_SNAKE_CASE		:	Optional[int]	)-> None:
   lowerCAmelCase__     :    int					  =      getattr(a_	,      '''stage_names'''	)
   lowerCAmelCase__     :    Union[str, Any]					  =      getattr(a_	,      '''out_features'''	,      a_	)
   lowerCAmelCase__     :    Optional[Any]					  =      getattr(a_	,      '''out_indices'''	,      a_	)

   lowerCAmelCase__     :    Dict					  =      stage_names
   lowerCAmelCase__  ,			lowerCAmelCase__     :    Tuple					  =      get_aligned_output_features_output_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=a_	)
   # Number of channels for each stage. This is set in the transformer backbone model init
   lowerCAmelCase__     :    Optional[Any]					  =      None




 def 							UpperCAmelCase__(      self		:	Union[str, Any]	,      _SCREAMING_SNAKE_CASE		:	List[Any]	)-> None:
   lowerCAmelCase__     :    int					  =      config

   lowerCAmelCase__     :    Tuple					  =      getattr(a_	,      '''use_timm_backbone'''	,      a_	)
   lowerCAmelCase__     :    Tuple					  =      BackboneType.TIMM if self.use_timm_backbone else BackboneType.TRANSFORMERS

   if self.backbone_type == BackboneType.TIMM:
     self._init_timm_backbone(a_	)
   elif self.backbone_type == BackboneType.TRANSFORMERS:
     self._init_transformers_backbone(a_	)
   else:
     raise ValueError(F'backbone_type {self.backbone_type} not supported.'	)




 @property
 def 							UpperCAmelCase__(      self		:	int	)-> Tuple:
   return self._out_features




 @out_features.setter
 def 							UpperCAmelCase__(      self		:	int	,      _SCREAMING_SNAKE_CASE		:	Union[str, Any]	)-> Any:
   lowerCAmelCase__  ,			lowerCAmelCase__     :    str					  =      get_aligned_output_features_output_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=self.stage_names	)




 @property
 def 							UpperCAmelCase__(      self		:	List[Any]	)-> str:
   return self._out_indices




 @out_indices.setter
 def 							UpperCAmelCase__(      self		:	Union[str, Any]	,      _SCREAMING_SNAKE_CASE		:	str	)-> List[Any]:
   lowerCAmelCase__  ,			lowerCAmelCase__     :    Optional[int]					  =      get_aligned_output_features_output_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=self.stage_names	)




 @property
 def 							UpperCAmelCase__(      self		:	str	)-> Dict:
   # the current backbones will output the number of channels for each stage
   # even if that stage is not in the out_features list.
   return {stage: self.num_features[i] for i, stage in enumerate(self.stage_names	)}




 @property
 def 							UpperCAmelCase__(      self		:	List[Any]	)-> int:
   return [self.out_feature_channels[name] for name in self.out_features]




 def 							UpperCAmelCase__(      self		:	Optional[Any]	,      *_SCREAMING_SNAKE_CASE		:	int	,      **_SCREAMING_SNAKE_CASE		:	str	)-> Tuple:
   lowerCAmelCase__     :    List[Any]					  =      dict(inspect.signature(self.forward	).parameters	)
   lowerCAmelCase__     :    str					  =      {k: v for k, v in kwargs.items() if k in signature}
   return self(*a_	,      **a_	)




 def 							UpperCAmelCase__(      self		:	Tuple	,      _SCREAMING_SNAKE_CASE		:	Optional[Any]	,      _SCREAMING_SNAKE_CASE		:	Optional[int] = None	,      _SCREAMING_SNAKE_CASE		:	Union[str, Any] = None	,      _SCREAMING_SNAKE_CASE		:	List[str] = None	,      )-> Any:
   raise NotImplementedError('''This method should be implemented by the derived class.'''	)




 def 							UpperCAmelCase__(      self		:	Any	)-> int:
   lowerCAmelCase__     :    Optional[Any]					  =      super().to_dict()
   lowerCAmelCase__     :    List[Any]					  =      output.pop('''_out_features'''	)
   lowerCAmelCase__     :    Dict					  =      output.pop('''_out_indices'''	)
   return output





class       _a							:




 @property
 def 							UpperCAmelCase__(      self		:	str	)-> Optional[Any]:
   return self._out_features




 @out_features.setter
 def 							UpperCAmelCase__(      self		:	Any	,      _SCREAMING_SNAKE_CASE		:	List[str]	)-> Dict:
   lowerCAmelCase__  ,			lowerCAmelCase__     :    Optional[int]					  =      get_aligned_output_features_output_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=self.stage_names	)




 @property
 def 							UpperCAmelCase__(      self		:	Tuple	)-> Tuple:
   return self._out_indices




 @out_indices.setter
 def 							UpperCAmelCase__(      self		:	Any	,      _SCREAMING_SNAKE_CASE		:	int	)-> str:
   lowerCAmelCase__  ,			lowerCAmelCase__     :    List[str]					  =      get_aligned_output_features_output_indices(
       out_features=a_	,      out_indices=a_	,      stage_names=self.stage_names	)




 def 							UpperCAmelCase__(      self		:	List[str]	)-> Union[str, Any]:
   lowerCAmelCase__     :    List[str]					  =      super().to_dict()
   lowerCAmelCase__     :    str					  =      output.pop('''_out_features'''	)
   lowerCAmelCase__     :    str					  =      output.pop('''_out_indices'''	)
   return output








from itertools import count
from math import asin, pi, sqrt






def   lowerCamelCase_					( _a ):




		"""simple docstring"""






		return (
		    (1 - 2 * point) * sqrt(point - point**2 ) + 2 * point + asin(sqrt(1 - point ) )
		) / 4






def   lowerCamelCase_					( _a ):




		"""simple docstring"""






		lowerCAmelCase__     :    Union[str, Any]					  =      (circles_number + 1 - sqrt(2 * circles_number )) / (
		    2 * (circles_number**2 + 1)
		)
		lowerCAmelCase__     :    Tuple					  =      circles_number * intersection_y

		lowerCAmelCase__     :    Any					  =      intersection_x * intersection_y / 2
		lowerCAmelCase__     :    Optional[Any]					  =      circle_bottom_arc_integral(
		    1 / 2 ) - circle_bottom_arc_integral(_a )

		return triangle_area + concave_region_area






def   lowerCamelCase_					( _a = 1 / 1_000 ):




		"""simple docstring"""






		lowerCAmelCase__     :    Any					  =      (1 - pi / 4) / 4

		for n in count(1 ):
				if concave_triangle_area(_a ) / l_section_area < fraction:
						return n

		return -1


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




