

import torch

from diffusers import DiffusionPipeline







class 					lowercase							(			lowercase_     ):




				def __init__(   self ,  snake_case ,  snake_case			):
											super().__init__()

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




				def __call__(   self			):
											snake_case_     							=       torch.randn(
											    (1, self.unet.config.in_channels, self.unet.config.sample_size, self.unet.config.sample_size) ,  )
											snake_case_     							=       1

											snake_case_     							=       self.unet(snake_case ,  snake_case			).sample
											snake_case_     							=       self.scheduler.step(snake_case ,  snake_case ,  snake_case			).prev_sample

											snake_case_     							=       scheduler_output - scheduler_output + torch.ones_like(snake_case			)

											return result







_UpperCAmelCase						:  int									=			range(2, 20 + 1)
_UpperCAmelCase						:  Dict									=			[10**k for k in range(ks[-1] + 1)]
_UpperCAmelCase						:  dict[int, dict[int, list[list[int]]]]									=			{}





def    __lowerCamelCase      (					UpperCamelCase__  ,				UpperCamelCase__  ,				UpperCamelCase__  ,				UpperCamelCase__ ):







							'''simple docstring'''






							snake_case_     							=       sum(a_i[j] for j in range(UpperCamelCase__  ,				len(UpperCamelCase__ ) ) )
							snake_case_     							=       sum(a_i[j] * base[j] for j in range(min(len(UpperCamelCase__ )  ,				UpperCamelCase__ ) ) )

							snake_case_							,       snake_case_     							=       0, 0
							snake_case_     							=       n - i

							snake_case_     							=       memo.get(UpperCamelCase__ )

							if sub_memo is not None:
														snake_case_     							=       sub_memo.get(UpperCamelCase__ )

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

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

														else:
																					snake_case_     							=       []
							else:
														snake_case_     							=       {c: []}
														snake_case_     							=       sub_memo

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

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

							snake_case_     							=       sub_memo[c]

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

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





def    __lowerCamelCase      (					UpperCamelCase__  ,				UpperCamelCase__  ,				UpperCamelCase__  ,				UpperCamelCase__ ):







							'''simple docstring'''






							if i >= n:
														return 0, i
							if k > len(UpperCamelCase__ ):
														a_i.extend([0 for _ in range(k - len(UpperCamelCase__ ) )] )

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

							while i < n:
														i += 1
														snake_case_     							=       ds_c + ds_b
														diff += addend
														snake_case_     							=       0
														for j in range(UpperCamelCase__ ):
																					snake_case_     							=       a_i[j] + addend
																					snake_case_							,       snake_case_     							=       divmod(UpperCamelCase__  ,				10 )

																					ds_c += a_i[j]

														if addend > 0:
																					break

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





def    __lowerCamelCase      (					UpperCamelCase__  ,				UpperCamelCase__  ,				UpperCamelCase__ ):







							'''simple docstring'''






							for j in range(UpperCamelCase__  ,				len(UpperCamelCase__ ) ):
														snake_case_     							=       digits[j] + addend
														if s >= 10:
																					snake_case_							,       snake_case_     							=       divmod(UpperCamelCase__  ,				10 )
																					snake_case_     							=       addend // 10 + quotient
														else:
																					snake_case_     							=       s
																					snake_case_     							=       addend // 10

														if addend == 0:
																					break

							while addend > 0:
														snake_case_							,       snake_case_     							=       divmod(UpperCamelCase__  ,				10 )
														digits.append(UpperCamelCase__ )





def    __lowerCamelCase      (					UpperCamelCase__ = 10**15 ):







							'''simple docstring'''






							snake_case_     							=       [1]
							snake_case_     							=       1
							snake_case_     							=       0
							while True:
														snake_case_							,       snake_case_     							=       next_term(UpperCamelCase__  ,				20  ,				i + dn  ,				UpperCamelCase__ )
														dn += terms_jumped
														if dn == n - i:
																					break

							snake_case_     							=       0
							for j in range(len(UpperCamelCase__ ) ):
														a_n += digits[j] * 10**j
							return a_n


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




