






'''simple docstring'''







__lowerCAmelCase       		=      range(2, 20 + 1)
__lowerCAmelCase       		=      [10**k for k in range(ks[-1] + 1)]
__lowerCAmelCase       		=      {}





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   int						,		__A							:   Optional[int]						,		__A							:   int      ):
				a_		:  Union[str, Any]											=      sum(a_i[j] for j in range(lowerCamelCase__						,		len(lowerCamelCase__      )      )      )
				a_		:  Any											=      sum(a_i[j] * base[j] for j in range(min(len(lowerCamelCase__      )						,		lowerCamelCase__      )      )      )

				a_				,			a_		:  str											=      0, 0
				a_		:  Dict											=      n - i

				a_		:  Union[str, Any]											=      memo.get(lowerCamelCase__      )

				if sub_memo is not None:
								a_		:  Union[str, Any]											=      sub_memo.get(lowerCamelCase__      )

								if jumps is not None and len(lowerCamelCase__      ) > 0:
												# find and make the largest jump without going over
												a_		:  List[Any]											=      -1
												for _k in range(len(lowerCamelCase__      ) - 1						,		-1						,		-1      ):
																if jumps[_k][2] <= k and jumps[_k][1] <= max_dn:
																				a_		:  Optional[int]											=      _k
																				break

												if max_jump >= 0:
																a_				,			a_				,			a_		:  Any											=      jumps[max_jump]
																# since the difference between jumps is cached, add c
																a_		:  str											=      diff + c
																for j in range(min(lowerCamelCase__						,		len(lowerCamelCase__      )      )      ):
																				a_				,			a_		:  Optional[Any]											=      divmod(lowerCamelCase__						,		10      )
																if new_c > 0:
																				add(lowerCamelCase__						,		lowerCamelCase__						,		lowerCamelCase__      )

								else:
												a_		:  Dict											=      []
				else:
								a_		:  Union[str, Any]											=      {c: []}
								a_		:  str											=      sub_memo

				if dn >= max_dn or c + diff >= base[k]:
								return diff, dn

				if k > ks[0]:
								while True:
												# keep doing smaller jumps
												a_				,			a_		:  Optional[Any]											=      next_term(lowerCamelCase__						,		k - 1						,		i + dn						,		lowerCamelCase__      )
												diff += _diff
												dn += terms_jumped

												if dn >= max_dn or c + diff >= base[k]:
																break
				else:
								# would be too small a jump, just compute sequential terms instead
								a_				,			a_		:  str											=      compute(lowerCamelCase__						,		lowerCamelCase__						,		i + dn						,		lowerCamelCase__      )
								diff += _diff
								dn += terms_jumped

				a_		:  Optional[int]											=      sub_memo[c]

				# keep jumps sorted by # of terms skipped
				a_		:  int											=      0
				while j < len(lowerCamelCase__      ):
								if jumps[j][1] > dn:
												break
								j += 1

				# cache the jump for this value digitsum(b) and c
				sub_memo[c].insert(lowerCamelCase__						,		(diff, dn, k)      )
				return (diff, dn)





def 		_UpperCAmelCase   (							__A							:   Optional[Any]						,		__A							:   str						,		__A							:   Dict						,		__A							:   Optional[Any]      ):
				if i >= n:
								return 0, i
				if k > len(lowerCamelCase__      ):
								a_i.extend([0 for _ in range(k - len(lowerCamelCase__      )      )]      )

				# note: a_i -> b * 10^k + c
				# ds_b -> digitsum(b)
				# ds_c -> digitsum(c)
				a_		:  str											=      i
				a_				,			a_				,			a_		:  Optional[Any]											=      0, 0, 0
				for j in range(len(lowerCamelCase__      )      ):
								if j >= k:
												ds_b += a_i[j]
								else:
												ds_c += a_i[j]

				while i < n:
								i += 1
								a_		:  Optional[int]											=      ds_c + ds_b
								diff += addend
								a_		:  int											=      0
								for j in range(lowerCamelCase__      ):
												a_		:  Tuple											=      a_i[j] + addend
												a_				,			a_		:  Tuple											=      divmod(lowerCamelCase__						,		10      )

												ds_c += a_i[j]

								if addend > 0:
												break

				if addend > 0:
								add(lowerCamelCase__						,		lowerCamelCase__						,		lowerCamelCase__      )
				return diff, i - start_i





def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   List[str]						,		__A							:   str      ):
				for j in range(lowerCamelCase__						,		len(lowerCamelCase__      )      ):
								a_		:  Any											=      digits[j] + addend
								if s >= 10:
												a_				,			a_		:  Any											=      divmod(lowerCamelCase__						,		10      )
												a_		:  Optional[int]											=      addend // 10 + quotient
								else:
												a_		:  Union[str, Any]											=      s
												a_		:  int											=      addend // 10

								if addend == 0:
												break

				while addend > 0:
								a_				,			a_		:  Any											=      divmod(lowerCamelCase__						,		10      )
								digits.append(lowerCamelCase__      )





def 		_UpperCAmelCase   (							__A							:   Any = 10**15      ):
				a_		:  List[str]											=      [1]
				a_		:  List[Any]											=      1
				a_		:  Optional[int]											=      0
				while True:
								a_				,			a_		:  Dict											=      next_term(lowerCamelCase__						,		20						,		i + dn						,		lowerCamelCase__      )
								dn += terms_jumped
								if dn == n - i:
												break

				a_		:  Optional[Any]											=      0
				for j in range(len(lowerCamelCase__      )      ):
								a_n += digits[j] * 10**j
				return a_n


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











'''simple docstring'''







from math import pi, sqrt, tan





def 		_UpperCAmelCase   (							__A							:   float      ):
				if side_length < 0:
								raise ValueError('''surface_area_cube() only accepts non-negative values'''      )
				return 6 * side_length**2





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float						,		__A							:   float      ):
				if length < 0 or breadth < 0 or height < 0:
								raise ValueError('''surface_area_cuboid() only accepts non-negative values'''      )
				return 2 * ((length * breadth) + (breadth * height) + (length * height))





def 		_UpperCAmelCase   (							__A							:   float      ):
				if radius < 0:
								raise ValueError('''surface_area_sphere() only accepts non-negative values'''      )
				return 4 * pi * radius**2





def 		_UpperCAmelCase   (							__A							:   float      ):
				if radius < 0:
								raise ValueError('''surface_area_hemisphere() only accepts non-negative values'''      )
				return 3 * pi * radius**2





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if radius < 0 or height < 0:
								raise ValueError('''surface_area_cone() only accepts non-negative values'''      )
				return pi * radius * (radius + (height**2 + radius**2) ** 0.5)





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float						,		__A							:   float      ):
				if radius_a < 0 or radius_a < 0 or height < 0:
								raise ValueError(
								    '''surface_area_conical_frustum() only accepts non-negative values'''      )
				a_		:  Any											=      (height**2 + (radius_a - radius_a) ** 2) ** 0.5
				return pi * ((slant_height * (radius_a + radius_a)) + radius_a**2 + radius_a**2)





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if radius < 0 or height < 0:
								raise ValueError('''surface_area_cylinder() only accepts non-negative values'''      )
				return 2 * pi * radius * (height + radius)





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if torus_radius < 0 or tube_radius < 0:
								raise ValueError('''surface_area_torus() only accepts non-negative values'''      )
				if torus_radius < tube_radius:
								raise ValueError(
								    '''surface_area_torus() does not support spindle or self intersecting tori'''      )
				return 4 * pow(__A						,		2      ) * torus_radius * tube_radius





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if length < 0 or width < 0:
								raise ValueError('''area_rectangle() only accepts non-negative values'''      )
				return length * width





def 		_UpperCAmelCase   (							__A							:   float      ):
				if side_length < 0:
								raise ValueError('''area_square() only accepts non-negative values'''      )
				return side_length**2





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if base < 0 or height < 0:
								raise ValueError('''area_triangle() only accepts non-negative values'''      )
				return (base * height) / 2





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float						,		__A							:   float      ):
				if sidea < 0 or sidea < 0 or sidea < 0:
								raise ValueError('''area_triangle_three_sides() only accepts non-negative values'''      )
				elif sidea + sidea < sidea or sidea + sidea < sidea or sidea + sidea < sidea:
								raise ValueError('''Given three sides do not form a triangle'''      )
				a_		:  int											=      (sidea + sidea + sidea) / 2
				a_		:  Optional[Any]											=      sqrt(
				    semi_perimeter
				    * (semi_perimeter - sidea)
				    * (semi_perimeter - sidea)
				    * (semi_perimeter - sidea)      )
				return area





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if base < 0 or height < 0:
								raise ValueError('''area_parallelogram() only accepts non-negative values'''      )
				return base * height





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float						,		__A							:   float      ):
				if basea < 0 or basea < 0 or height < 0:
								raise ValueError('''area_trapezium() only accepts non-negative values'''      )
				return 1 / 2 * (basea + basea) * height





def 		_UpperCAmelCase   (							__A							:   float      ):
				if radius < 0:
								raise ValueError('''area_circle() only accepts non-negative values'''      )
				return pi * radius**2





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if radius_x < 0 or radius_y < 0:
								raise ValueError('''area_ellipse() only accepts non-negative values'''      )
				return pi * radius_x * radius_y





def 		_UpperCAmelCase   (							__A							:   float						,		__A							:   float      ):
				if diagonal_a < 0 or diagonal_a < 0:
								raise ValueError('''area_rhombus() only accepts non-negative values'''      )
				return 1 / 2 * diagonal_a * diagonal_a





def 		_UpperCAmelCase   (							__A							:   int						,		__A							:   float      ):
				if not isinstance(__A						,		__A      ) or sides < 3:
								raise ValueError(
								    '''area_reg_polygon() only accepts integers greater than or \
equal to three as number of sides'''      )
				elif length < 0:
								raise ValueError(
								    '''area_reg_polygon() only accepts non-negative values as \
length of a side'''      )
				return (sides * length**2) / (4 * tan(pi / sides      ))
				return (sides * length**2) / (4 * tan(pi / sides      ))


if __name__ == "__main__":
		import doctest

		doctest.testmod(verbose=True)  # verbose so we can see methods missing tests

		print('[DEMO] Areas of various geometric shapes: \n')
		print(F"""Rectangle: {area_rectangle(10, 20) = }""")
		print(F"""Square: {area_square(10) = }""")
		print(F"""Triangle: {area_triangle(10, 10) = }""")
		print(F"""Triangle: {area_triangle_three_sides(5, 12, 13) = }""")
		print(F"""Parallelogram: {area_parallelogram(10, 20) = }""")
		print(F"""Rhombus: {area_rhombus(10, 20) = }""")
		print(F"""Trapezium: {area_trapezium(10, 20, 30) = }""")
		print(F"""Circle: {area_circle(20) = }""")
		print(F"""Ellipse: {area_ellipse(10, 20) = }""")
		print('\nSurface Areas of various geometric shapes: \n')
		print(F"""Cube: {surface_area_cube(20) = }""")
		print(F"""Cuboid: {surface_area_cuboid(10, 20, 30) = }""")
		print(F"""Sphere: {surface_area_sphere(20) = }""")
		print(F"""Hemisphere: {surface_area_hemisphere(20) = }""")
		print(F"""Cone: {surface_area_cone(10, 20) = }""")
		print(F"""Conical Frustum: {surface_area_conical_frustum(10, 20, 30) = }""")
		print(F"""Cylinder: {surface_area_cylinder(10, 20) = }""")
		print(F"""Torus: {surface_area_torus(20, 10) = }""")
		print(F"""Equilateral Triangle: {area_reg_polygon(3, 10) = }""")
		print(F"""Square: {area_reg_polygon(4, 10) = }""")
		print(F"""Reqular Pentagon: {area_reg_polygon(5, 10) = }""")




