
"""simple docstring"""





from __future__ import annotations

from math import pi, pow







def       __a	(						__lowerCamelCase    ):
   if side_length < 0:
      raise ValueError("vol_cube() only accepts non-negative values"    )
   return pow(__lowerCamelCase,						3    )







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or radius < 0:
      raise ValueError("vol_spherical_cap() only accepts non-negative values"    )
   # Volume is 1/3 pi * height squared * (3 * radius - height)
   return 1 / 3 * pi * pow(__lowerCamelCase,						2    ) * (3 * radius - height)







def       __a	(						__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    ):
   if radius_a < 0 or radius_a < 0 or centers_distance < 0:
      raise ValueError("vol_spheres_intersect() only accepts non-negative values"    )
   if centers_distance == 0:
      return vol_sphere(min(__lowerCamelCase,						__lowerCamelCase    )    )

   UpperCAmelCase_   :   List[str]     	=       (
       (radius_a - radius_a + centers_distance)
       * (radius_a + radius_a - centers_distance)
       / (2 * centers_distance)
   )
   UpperCAmelCase_   :   Any     	=       (
       (radius_a - radius_a + centers_distance)
       * (radius_a + radius_a - centers_distance)
       / (2 * centers_distance)
   )

   return vol_spherical_cap(__lowerCamelCase,						__lowerCamelCase    ) + vol_spherical_cap(__lowerCamelCase,						__lowerCamelCase    )







def       __a	(						__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    ):

   if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
      raise ValueError(
          "vol_spheres_union() only accepts non-negative values, non-zero radius"    )

   if centers_distance == 0:
      return vol_sphere(max(__lowerCamelCase,						__lowerCamelCase    )    )

   return (
       vol_sphere(__lowerCamelCase    )
       + vol_sphere(__lowerCamelCase    )
       - vol_spheres_intersect(__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    )
   )







def       __a	(						__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    ):
   if width < 0 or height < 0 or length < 0:
      raise ValueError("vol_cuboid() only accepts non-negative values"    )
   return float(width * height * length    )







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or area_of_base < 0:
      raise ValueError("vol_cone() only accepts non-negative values"    )
   return area_of_base * height / 3.0







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or radius < 0:
      raise ValueError("vol_right_circ_cone() only accepts non-negative values"    )
   return pi * pow(__lowerCamelCase,						2    ) * height / 3.0







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or area_of_base < 0:
      raise ValueError("vol_prism() only accepts non-negative values"    )
   return float(area_of_base * height    )







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or area_of_base < 0:
      raise ValueError("vol_pyramid() only accepts non-negative values"    )
   return area_of_base * height / 3.0







def       __a	(						__lowerCamelCase    ):
   if radius < 0:
      raise ValueError("vol_sphere() only accepts non-negative values"    )
   # Volume is 4/3 * pi * radius cubed
   return 4 / 3 * pi * pow(__lowerCamelCase,						3    )







def       __a	(						__lowerCamelCase    ):
   if radius < 0:
      raise ValueError("vol_hemisphere() only accepts non-negative values"    )
   # Volume is radius cubed * pi * 2/3
   return pow(__lowerCamelCase,						3    ) * pi * 2 / 3







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if height < 0 or radius < 0:
      raise ValueError("vol_circular_cylinder() only accepts non-negative values"    )
   # Volume is radius squared * height * pi
   return pow(__lowerCamelCase,						2    ) * height * pi







def       __a	(						__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    ):
   # Volume - (outer_radius squared - inner_radius squared) * pi * height
   if inner_radius < 0 or outer_radius < 0 or height < 0:
      raise ValueError(
          "vol_hollow_circular_cylinder() only accepts non-negative values"    )
   if outer_radius <= inner_radius:
      raise ValueError("outer_radius must be greater than inner_radius"    )
   return pi * (pow(__lowerCamelCase,						2    ) - pow(__lowerCamelCase,						2    )) * height







def       __a	(						__lowerCamelCase,						__lowerCamelCase,						__lowerCamelCase    ):
   # Volume is 1/3 * pi * height *
   #           (radius_1 squared + radius_2 squared + radius_1 * radius_2)
   if radius_a < 0 or radius_a < 0 or height < 0:
      raise ValueError("vol_conical_frustum() only accepts non-negative values"    )
   return (
       1
       / 3
       * pi
       * height
       * (pow(__lowerCamelCase,						2    ) + pow(__lowerCamelCase,						2    ) + radius_a * radius_a)
   )







def       __a	(						__lowerCamelCase,						__lowerCamelCase    ):
   if torus_radius < 0 or tube_radius < 0:
      raise ValueError("vol_torus() only accepts non-negative values"    )
   return 2 * pow(__lowerCamelCase,						2    ) * torus_radius * pow(__lowerCamelCase,						2    )







def       __a	(						):
   print("Volumes:"    )
   print(f"""Cube: {vol_cube(2    ) = }"""    )  # = 8
   print(f"""Cuboid: {vol_cuboid(2,						2,						2    ) = }"""    )  # = 8
   print(f"""Cone: {vol_cone(2,						2    ) = }"""    )  # ~= 1.33
   print(f"""Right Circular Cone: {vol_right_circ_cone(2,						2    ) = }"""    )  # ~= 8.38
   print(f"""Prism: {vol_prism(2,						2    ) = }"""    )  # = 4
   print(f"""Pyramid: {vol_pyramid(2,						2    ) = }"""    )  # ~= 1.33
   print(f"""Sphere: {vol_sphere(2    ) = }"""    )  # ~= 33.5
   print(f"""Hemisphere: {vol_hemisphere(2    ) = }"""    )  # ~= 16.75
   print(f"""Circular Cylinder: {vol_circular_cylinder(2,						2    ) = }"""    )  # ~= 25.1
   print(f"""Torus: {vol_torus(2,						2    ) = }"""    )  # ~= 157.9
   print(f"""Conical Frustum: {vol_conical_frustum(2,						2,						4    ) = }"""    )  # ~= 58.6
   print(f"""Spherical cap: {vol_spherical_cap(1,						2    ) = }"""    )  # ~= 5.24
   print(f"""Spheres intersetion: {vol_spheres_intersect(2,						2,						1    ) = }"""    )  # ~= 21.21
   print(f"""Spheres union: {vol_spheres_union(2,						2,						1    ) = }"""    )  # ~= 45.81
   print(
       f"""Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1,						2,						3    ) = }"""    )  # ~= 28.3


if __name__ == "__main__":
    main()




_A       =					{
    "0": 1,
    "1": 1,
    "2": 2,
    "3": 6,
    "4": 24,
    "5": 1_20,
    "6": 7_20,
    "7": 50_40,
    "8": 4_03_20,
    "9": 36_28_80,
}

_A       =					{1_45: 1_45}

_A       =					{
    1_45: 0,
    1_69: 3,
    3_63_01: 3,
    14_54: 3,
    8_71: 2,
    4_53_61: 2,
    8_72: 2,
}



def 		lowercase_     (  A__						)				->				int:
    """simple docstring"""

    if n in CACHE_SUM_DIGIT_FACTORIALS:
        return CACHE_SUM_DIGIT_FACTORIALS[n]
    snake_case						=   sum(DIGIT_FACTORIALS[let] for let in str(A__						)						)
    snake_case						=   ret
    return ret



def 		lowercase_     (  A__					,						A__ = None						)				->				int:
    """simple docstring"""

    snake_case						=   previous or set()
    if n in CHAIN_LENGTH_CACHE:
        return CHAIN_LENGTH_CACHE[n]
    snake_case						=   sum_digit_factorials(A__						)
    if next_number in previous:
        snake_case						=   0
        return 0
    else:
        previous.add(A__						)
        snake_case						=   1 + chain_length(A__					,						A__						)
        snake_case						=   ret
        return ret



def 		lowercase_     (  A__ = 60					,						A__ = 100_0000						)				->				int:
    """simple docstring"""

    return sum(1 for i in range(1					,						A__						) if chain_length(A__						) == num_terms						)


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

