



"""simple docstring"""
A:   int								=		range(2, 2_0 + 1)
A:   Any								=		[1_0**k for k in range(ks[-1] + 1)]
A:   dict[int, dict[int, list[list[int]]]]								=		{}

def       _snake_case						(						UpperCamelCase						:		Dict			,		UpperCamelCase						:		Any			,		UpperCamelCase						:		Any			,		UpperCamelCase						:		int						):
 UpperCAmelCase	:					List[str]	=	sum(a_i[j] for j in range(UpperCamelCase			,		len(UpperCamelCase						)						)						)
 UpperCAmelCase	:					str	=	sum(a_i[j] * base[j] for j in range(min(len(UpperCamelCase						)			,		UpperCamelCase						)						)						)

 UpperCAmelCase	,		UpperCAmelCase	:					str	=	0, 0
 UpperCAmelCase	:					Optional[Any]	=	n - i

 UpperCAmelCase	:					Optional[int]	=	memo.get(UpperCamelCase						)

 if sub_memo is not None:
  UpperCAmelCase	:					str	=	sub_memo.get(UpperCamelCase						)

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

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

  else:
   UpperCAmelCase	:					int	=	[]
 else:
  UpperCAmelCase	:					List[str]	=	{c: []}
  UpperCAmelCase	:					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
   UpperCAmelCase	,		UpperCAmelCase	:					List[str]	=	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
  UpperCAmelCase	,		UpperCAmelCase	:					int	=	compute(UpperCamelCase			,		UpperCamelCase			,		i + dn			,		UpperCamelCase						)
  diff += _diff
  dn += terms_jumped

 UpperCAmelCase	:					Dict	=	sub_memo[c]

 # keep jumps sorted by # of terms skipped
 UpperCAmelCase	:					str	=	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       _snake_case						(						UpperCamelCase						:		int			,		UpperCamelCase						:		Tuple			,		UpperCamelCase						:		Union[str, Any]			,		UpperCamelCase						:		Any						):
 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)
 UpperCAmelCase	:					List[str]	=	i
 UpperCAmelCase	,		UpperCAmelCase	,		UpperCAmelCase	:					Optional[int]	=	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
  UpperCAmelCase	:					Optional[int]	=	ds_c + ds_b
  diff += addend
  UpperCAmelCase	:					str	=	0
  for j in range(UpperCamelCase						):
   UpperCAmelCase	:					Any	=	a_i[j] + addend
   UpperCAmelCase	,		UpperCAmelCase	:					Any	=	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       _snake_case						(						UpperCamelCase						:		Union[str, Any]			,		UpperCamelCase						:		str			,		UpperCamelCase						:		Optional[int]						):
 for j in range(UpperCamelCase			,		len(UpperCamelCase						)						):
  UpperCAmelCase	:					Optional[int]	=	digits[j] + addend
  if s >= 10:
   UpperCAmelCase	,		UpperCAmelCase	:					int	=	divmod(UpperCamelCase			,		10						)
   UpperCAmelCase	:					str	=	addend // 10 + quotient
  else:
   UpperCAmelCase	:					Any	=	s
   UpperCAmelCase	:					Union[str, Any]	=	addend // 10

  if addend == 0:
   break

 while addend > 0:
  UpperCAmelCase	,		UpperCAmelCase	:					Any	=	divmod(UpperCamelCase			,		10						)
  digits.append(UpperCamelCase						)

def       _snake_case						(						UpperCamelCase						:		int = 10**15						):
 UpperCAmelCase	:					Dict	=	[1]
 UpperCAmelCase	:					int	=	1
 UpperCAmelCase	:					Tuple	=	0
 while True:
  UpperCAmelCase	,		UpperCAmelCase	:					Tuple	=	next_term(UpperCamelCase			,		20			,		i + dn			,		UpperCamelCase						)
  dn += terms_jumped
  if dn == n - i:
   break

 UpperCAmelCase	:					Any	=	0
 for j in range(len(UpperCamelCase						)						):
  a_n += digits[j] * 10**j
 return a_n


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




"""simple docstring"""





import gc
import unittest

from diffusers import FlaxDPMSolverMultistepScheduler, FlaxStableDiffusionPipeline
from diffusers.utils import is_flax_available, slow
from diffusers.utils.testing_utils import require_flax


if is_flax_available():
 import jax
 import jax.numpy as jnp
 from flax.jax_utils import replicate
 from flax.training.common_utils import shard



@slow
@require_flax
class 						snake_case_(				unittest.TestCase	):






     def    lowerCamelCase__   (		self     :    int ):
           # clean up the VRAM after each test
           super().tearDown()
           gc.collect()






     def    lowerCamelCase__   (		self     :    Optional[Any] ):
           lowerCAmelCase,					lowerCAmelCase    :		Optional[int]				=      FlaxStableDiffusionPipeline.from_pretrained(
               '''stabilityai/stable-diffusion-2'''    , revision='''bf16'''    , dtype=jnp.bfloataa    , )

           lowerCAmelCase    :		Optional[int]				=      '''A painting of a squirrel eating a burger'''
           lowerCAmelCase    :		List[str]				=      jax.device_count()
           lowerCAmelCase    :		Optional[int]				=      num_samples * [prompt]
           lowerCAmelCase    :		Any				=      sd_pipe.prepare_inputs(UpperCamelCase_ )

           lowerCAmelCase    :		Optional[int]				=      replicate(UpperCamelCase_ )
           lowerCAmelCase    :		Union[str, Any]				=      shard(UpperCamelCase_ )

           lowerCAmelCase    :		Optional[int]				=      jax.random.PRNGKey(0 )
           lowerCAmelCase    :		Optional[Any]				=      jax.random.split(UpperCamelCase_    , jax.device_count() )

           lowerCAmelCase    :		str				=      sd_pipe(UpperCamelCase_    , UpperCamelCase_    , UpperCamelCase_    , num_inference_steps=2_5    , jit=UpperCamelCase_ )[0]
           assert images.shape == (jax.device_count(), 1, 7_6_8, 7_6_8, 3)

           lowerCAmelCase    :		str				=      images.reshape((images.shape[0] * images.shape[1],) + images.shape[-3:] )
           lowerCAmelCase    :		List[str]				=      images[0, 2_5_3:2_5_6, 2_5_3:2_5_6, -1]

           lowerCAmelCase    :		Dict				=      jnp.asarray(jax.device_get(image_slice.flatten() ) )
           lowerCAmelCase    :		List[str]				=      jnp.array([0.4_238, 0.4_414, 0.4_395, 0.4_453, 0.4_629, 0.4_590, 0.4_531, 0.45_508, 0.4_512] )
           print(F'''output_slice: {output_slice}''' )
           assert jnp.abs(output_slice - expected_slice ).max() < 1E-2






     def    lowerCamelCase__   (		self     :    Union[str, Any] ):
           lowerCAmelCase    :		Union[str, Any]				=      '''stabilityai/stable-diffusion-2'''
           lowerCAmelCase,					lowerCAmelCase    :		Dict				=      FlaxDPMSolverMultistepScheduler.from_pretrained(UpperCamelCase_    , subfolder='''scheduler''' )
           lowerCAmelCase,					lowerCAmelCase    :		int				=      FlaxStableDiffusionPipeline.from_pretrained(
               UpperCamelCase_    , scheduler=UpperCamelCase_    , revision='''bf16'''    , dtype=jnp.bfloataa    , )
           lowerCAmelCase    :		List[Any]				=      scheduler_params

           lowerCAmelCase    :		List[Any]				=      '''A painting of a squirrel eating a burger'''
           lowerCAmelCase    :		Any				=      jax.device_count()
           lowerCAmelCase    :		int				=      num_samples * [prompt]
           lowerCAmelCase    :		int				=      sd_pipe.prepare_inputs(UpperCamelCase_ )

           lowerCAmelCase    :		Dict				=      replicate(UpperCamelCase_ )
           lowerCAmelCase    :		Tuple				=      shard(UpperCamelCase_ )

           lowerCAmelCase    :		int				=      jax.random.PRNGKey(0 )
           lowerCAmelCase    :		Optional[int]				=      jax.random.split(UpperCamelCase_    , jax.device_count() )

           lowerCAmelCase    :		Tuple				=      sd_pipe(UpperCamelCase_    , UpperCamelCase_    , UpperCamelCase_    , num_inference_steps=2_5    , jit=UpperCamelCase_ )[0]
           assert images.shape == (jax.device_count(), 1, 7_6_8, 7_6_8, 3)

           lowerCAmelCase    :		Any				=      images.reshape((images.shape[0] * images.shape[1],) + images.shape[-3:] )
           lowerCAmelCase    :		str				=      images[0, 2_5_3:2_5_6, 2_5_3:2_5_6, -1]

           lowerCAmelCase    :		Optional[int]				=      jnp.asarray(jax.device_get(image_slice.flatten() ) )
           lowerCAmelCase    :		Tuple				=      jnp.array([0.4_336, 0.42_969, 0.4_453, 0.4_199, 0.4_297, 0.4_531, 0.4_434, 0.4_434, 0.4_297] )
           print(F'''output_slice: {output_slice}''' )
           assert jnp.abs(output_slice - expected_slice ).max() < 1E-2


