code: 
  
  
  
  
  
  
  from itertools import count
  from math import asin, pi, sqrt
  
  
  
  
  
  
  
  def       lowerCAmelCase__(	lowercase       :    float				) ->    float:
    return (
        (1 - 2 * point) * sqrt(point - point**2				) + 2 * point + asin(sqrt(1 - point				)				)
    ) / 4
  
  
  
  
  
  
  
  def       lowerCAmelCase__(	lowercase       :    int				) ->    float:
    __snake_case   :				int        =       (circles_number + 1 - sqrt(2 * circles_number				)) / (
        2 * (circles_number**2 + 1)
    )
    __snake_case   :				Union[str, Any]        =       circles_number * intersection_y
  
    __snake_case   :				List[str]        =       intersection_x * intersection_y / 2
    __snake_case   :				Dict        =       circle_bottom_arc_integral(
        1 / 2				) - circle_bottom_arc_integral(lowercase				)
  
    return triangle_area + concave_region_area
  
  
  
  
  
  
  
  def       lowerCAmelCase__(	lowercase       :    float = 1 / 1000				) ->    int:
    __snake_case   :				Tuple        =       (1 - pi / 4) / 4
  
    for n in count(1				):
      if concave_triangle_area(lowercase				) / l_section_area < fraction:
        return n
  
    return -1
  
  
  if __name__ == "__main__":
       print(F'''{solution() = }''')
  
  
style_context: 
  
  
  
  
  
  from typing import TYPE_CHECKING
  
  from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available, is_vision_available
  
  
  _UpperCamelCase       = {
      '''configuration_deta''': ['''DETA_PRETRAINED_CONFIG_ARCHIVE_MAP''', '''DetaConfig'''],
  }
  
  try:
       if not is_vision_available():
            raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
       pass
  else:
       _UpperCamelCase       = ['''DetaImageProcessor''']
  
  try:
       if not is_torch_available():
            raise OptionalDependencyNotAvailable()
  except OptionalDependencyNotAvailable:
       pass
  else:
       _UpperCamelCase       = [
           '''DETA_PRETRAINED_MODEL_ARCHIVE_LIST''',
           '''DetaForObjectDetection''',
           '''DetaModel''',
           '''DetaPreTrainedModel''',
       ]
  
  
  if TYPE_CHECKING:
       from .configuration_deta import DETA_PRETRAINED_CONFIG_ARCHIVE_MAP, DetaConfig
  
       try:
            if not is_vision_available():
                 raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
            pass
       else:
            from .image_processing_deta import DetaImageProcessor
  
       try:
            if not is_torch_available():
                 raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
            pass
       else:
            from .modeling_deta import (
                DETA_PRETRAINED_MODEL_ARCHIVE_LIST,
                DetaForObjectDetection,
                DetaModel,
                DetaPreTrainedModel,
            )
  
  else:
       import sys
  
       _UpperCamelCase       = _LazyModule(__name__, globals()['''__file__'''], _import_structure, module_spec=__spec__)
  
  