



'''simple docstring'''




a   :			Union[str, Any]					  =    range(2, 2_0 + 1)
a   :			Tuple					  =    [1_0**k for k in range(ks[-1] + 1)]
a   :			dict[int, dict[int, list[list[int]]]]					  =    {}



def       __lowerCamelCase  (					_lowercase	,				_lowercase	,				_lowercase	,				_lowercase		)      ->			Optional[int]:
							UpperCAmelCase     :		Tuple							     =		sum(a_i[j] for j in range(_lowercase	,				len(_lowercase		)		)		)
							UpperCAmelCase     :		Dict							     =		sum(a_i[j] * base[j] for j in range(min(len(_lowercase		)	,				_lowercase		)		)		)

							UpperCAmelCase     ,     UpperCAmelCase     :		Any							     =		0, 0
							UpperCAmelCase     :		List[Any]							     =		n - i

							UpperCAmelCase     :		int							     =		memo.get(_lowercase		)

							if sub_memo is not None:
														UpperCAmelCase     :		Optional[Any]							     =		sub_memo.get(_lowercase		)

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

																					if max_jump >= 0:
																												UpperCAmelCase     ,     UpperCAmelCase     ,     UpperCAmelCase     :		Union[str, Any]							     =		jumps[max_jump]
																												# since the difference between jumps is cached, add c
																												UpperCAmelCase     :		Optional[Any]							     =		diff + c
																												for j in range(min(_lowercase	,				len(_lowercase		)		)		):
																																			UpperCAmelCase     ,     UpperCAmelCase     :		List[Any]							     =		divmod(_lowercase	,				1_0		)
																												if new_c > 0:
																																			add(_lowercase	,				_lowercase	,				_lowercase		)

														else:
																					UpperCAmelCase     :		Optional[int]							     =		[]
							else:
														UpperCAmelCase     :		Dict							     =		{c: []}
														UpperCAmelCase     :		Optional[int]							     =		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(_lowercase	,				k - 1	,				i + dn	,				_lowercase		)
																					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     :		List[Any]							     =		compute(_lowercase	,				_lowercase	,				i + dn	,				_lowercase		)
														diff += _diff
														dn += terms_jumped

							UpperCAmelCase     :		List[Any]							     =		sub_memo[c]

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

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



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

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

							while i < n:
														i += 1
														UpperCAmelCase     :		Optional[Any]							     =		ds_c + ds_b
														diff += addend
														UpperCAmelCase     :		int							     =		0
														for j in range(_lowercase		):
																					UpperCAmelCase     :		str							     =		a_i[j] + addend
																					UpperCAmelCase     ,     UpperCAmelCase     :		List[Any]							     =		divmod(_lowercase	,				1_0		)

																					ds_c += a_i[j]

														if addend > 0:
																					break

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



def       __lowerCamelCase  (					_lowercase	,				_lowercase	,				_lowercase		)      ->			int:
							for j in range(_lowercase	,				len(_lowercase		)		):
														UpperCAmelCase     :		Optional[Any]							     =		digits[j] + addend
														if s >= 1_0:
																					UpperCAmelCase     ,     UpperCAmelCase     :		List[str]							     =		divmod(_lowercase	,				1_0		)
																					UpperCAmelCase     :		str							     =		addend // 1_0 + quotient
														else:
																					UpperCAmelCase     :		str							     =		s
																					UpperCAmelCase     :		str							     =		addend // 1_0

														if addend == 0:
																					break

							while addend > 0:
														UpperCAmelCase     ,     UpperCAmelCase     :		Dict							     =		divmod(_lowercase	,				1_0		)
														digits.append(_lowercase		)



def       __lowerCamelCase  (					_lowercase = 1_0**1_5		)      ->			int:
							UpperCAmelCase     :		List[str]							     =		[1]
							UpperCAmelCase     :		int							     =		1
							UpperCAmelCase     :		Dict							     =		0
							while True:
														UpperCAmelCase     ,     UpperCAmelCase     :		int							     =		next_term(_lowercase	,				2_0	,				i + dn	,				_lowercase		)
														dn += terms_jumped
														if dn == n - i:
																					break

							UpperCAmelCase     :		Optional[Any]							     =		0
							for j in range(len(_lowercase		)		):
														a_n += digits[j] * 1_0**j
							return a_n


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






'''simple docstring'''




import contextlib
import importlib
import io
import unittest

import transformers

# Try to import everything from transformers to ensure every object can be loaded.
from transformers import *  # noqa F406
from transformers.testing_utils import DUMMY_UNKNOWN_IDENTIFIER, require_flax, require_tf, require_torch
from transformers.utils import ContextManagers, find_labels, is_flax_available, is_tf_available, is_torch_available


if is_torch_available():
							from transformers import BertForPreTraining, BertForQuestionAnswering, BertForSequenceClassification

if is_tf_available():
							from transformers import TFBertForPreTraining, TFBertForQuestionAnswering, TFBertForSequenceClassification

if is_flax_available():
							from transformers import FlaxBertForPreTraining, FlaxBertForQuestionAnswering, FlaxBertForSequenceClassification


a   :			str					  =    DUMMY_UNKNOWN_IDENTIFIER
# An actual model hosted on huggingface.co

a   :			int					  =    """main"""
# Default branch name
a   :			Any					  =    """f2c752cfc5c0ab6f4bdec59acea69eefbee381c2"""
# One particular commit (not the top of `main`)
a   :			str					  =    """aaaaaaa"""
# This commit does not exist, so we should 404.

a   :			int					  =    """d9e9f15bc825e4b2c9249e9578f884bbcb5e3684"""
# Sha-1 of config.json on the top of `main`, for checking purposes
a   :			Any					  =    """4b243c475af8d0a7754e87d7d096c92e5199ec2fe168a2ee7998e3b8e9bcb1d3"""



@contextlib.contextmanager
def       __lowerCamelCase  (					)      ->			List[str]:
							print("""Welcome!"""		)
							yield
							print("""Bye!"""		)



@contextlib.contextmanager
def       __lowerCamelCase  (					)      ->			Optional[int]:
							print("""Bonjour!"""		)
							yield
							print("""Au revoir!"""		)

class 			UpperCamelCase_ (							unittest.TestCase			):





	def  _lowercase(  self    )			->			List[Any]:
								# If the spec is missing, importlib would not be able to import the module dynamically.
								assert transformers.__spec__ is not None
								assert importlib.util.find_spec("""transformers"""    ) is not None




class 			UpperCamelCase_ (							unittest.TestCase			):





	@unittest.mock.patch("""sys.stdout"""		,		new_callable=io.StringIO    )
	def  _lowercase(  self		,		A    )			->			Tuple:
								with ContextManagers([]    ):
															print("""Transformers are awesome!"""    )
								# The print statement adds a new line at the end of the output
								self.assertEqual(mock_stdout.getvalue()		,		"""Transformers are awesome!\n"""    )





	@unittest.mock.patch("""sys.stdout"""		,		new_callable=io.StringIO    )
	def  _lowercase(  self		,		A    )			->			Dict:
								with ContextManagers([context_en()]    ):
															print("""Transformers are awesome!"""    )
								# The output should be wrapped with an English welcome and goodbye
								self.assertEqual(mock_stdout.getvalue()		,		"""Welcome!\nTransformers are awesome!\nBye!\n"""    )





	@unittest.mock.patch("""sys.stdout"""		,		new_callable=io.StringIO    )
	def  _lowercase(  self		,		A    )			->			Union[str, Any]:
								with ContextManagers([context_fr(), context_en()]    ):
															print("""Transformers are awesome!"""    )
								# The output should be wrapped with an English and French welcome and goodbye
								self.assertEqual(mock_stdout.getvalue()		,		"""Bonjour!\nWelcome!\nTransformers are awesome!\nBye!\nAu revoir!\n"""    )





	@require_torch
	def  _lowercase(  self    )			->			Optional[int]:
								self.assertEqual(find_labels(A    )		,		["""labels"""]    )
								self.assertEqual(find_labels(A    )		,		["""labels""", """next_sentence_label"""]    )
								self.assertEqual(find_labels(A    )		,		["""start_positions""", """end_positions"""]    )

								class 			UpperCamelCase_ (							__magic_name__			):
									pass




								self.assertEqual(find_labels(A    )		,		["""labels"""]    )





	@require_tf
	def  _lowercase(  self    )			->			int:
								self.assertEqual(find_labels(A    )		,		["""labels"""]    )
								self.assertEqual(find_labels(A    )		,		["""labels""", """next_sentence_label"""]    )
								self.assertEqual(find_labels(A    )		,		["""start_positions""", """end_positions"""]    )

								class 			UpperCamelCase_ (							__magic_name__			):
									pass




								self.assertEqual(find_labels(A    )		,		["""labels"""]    )






	@require_flax
	def  _lowercase(  self    )			->			Any:
								# Flax models don't have labels
								self.assertEqual(find_labels(A    )		,		[]    )
								self.assertEqual(find_labels(A    )		,		[]    )
								self.assertEqual(find_labels(A    )		,		[]    )

								class 			UpperCamelCase_ (							__magic_name__			):
									pass




								self.assertEqual(find_labels(A    )		,		[]    )

