
'''simple docstring'''






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







def 		A_							(						_lowerCAmelCase :     Dict					,						_lowerCAmelCase :     Dict					,						_lowerCAmelCase :     Optional[Any]					,						_lowerCAmelCase :     Union[str, Any]				):



      """simple docstring"""




      _lowerCamelCase					: List[Any]		    = sum(a_i[j] for j in range(UpperCamelCase__					,						len(UpperCamelCase__				)				)				)
      _lowerCamelCase					: str		    = sum(a_i[j] * base[j] for j in range(min(len(UpperCamelCase__				)					,						UpperCamelCase__				)				)				)

      _lowerCamelCase					: Optional[int]		    = 0, 0
      _lowerCamelCase					: int		    = n - i

      _lowerCamelCase					: List[Any]		    = memo.get(UpperCamelCase__				)

      if sub_memo is not None:
            _lowerCamelCase					: Optional[Any]		    = sub_memo.get(UpperCamelCase__				)

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

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

            else:
                  _lowerCamelCase					: str		    = []
      else:
            _lowerCamelCase					: Dict		    = {c: []}
            _lowerCamelCase					: 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
                  _lowerCamelCase					: Any		    = 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
            _lowerCamelCase					: Optional[Any]		    = compute(UpperCamelCase__					,						UpperCamelCase__					,						i + dn					,						UpperCamelCase__				)
            diff += _diff
            dn += terms_jumped

      _lowerCamelCase					: Optional[Any]		    = sub_memo[c]

      # keep jumps sorted by # of terms skipped
      _lowerCamelCase					: List[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 		A_							(						_lowerCAmelCase :     List[Any]					,						_lowerCAmelCase :     Dict					,						_lowerCAmelCase :     Optional[int]					,						_lowerCAmelCase :     Any				):



      """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)
      _lowerCamelCase					: int		    = i
      _lowerCamelCase					: Dict		    = 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
            _lowerCamelCase					: List[str]		    = ds_c + ds_b
            diff += addend
            _lowerCamelCase					: str		    = 0
            for j in range(UpperCamelCase__				):
                  _lowerCamelCase					: str		    = a_i[j] + addend
                  _lowerCamelCase					: Dict		    = 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 		A_							(						_lowerCAmelCase :     Union[str, Any]					,						_lowerCAmelCase :     str					,						_lowerCAmelCase :     List[str]				):



      """simple docstring"""




      for j in range(UpperCamelCase__					,						len(UpperCamelCase__				)				):
            _lowerCamelCase					: Any		    = digits[j] + addend
            if s >= 10:
                  _lowerCamelCase					: List[str]		    = divmod(UpperCamelCase__					,						10				)
                  _lowerCamelCase					: str		    = addend // 10 + quotient
            else:
                  _lowerCamelCase					: Dict		    = s
                  _lowerCamelCase					: Optional[Any]		    = addend // 10

            if addend == 0:
                  break

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







def 		A_							(						_lowerCAmelCase :     int = 10**15				):



      """simple docstring"""




      _lowerCamelCase					: List[str]		    = [1]
      _lowerCamelCase					: Optional[int]		    = 1
      _lowerCamelCase					: Union[str, Any]		    = 0
      while True:
            _lowerCamelCase					: List[str]		    = next_term(UpperCamelCase__					,						20					,						i + dn					,						UpperCamelCase__				)
            dn += terms_jumped
            if dn == n - i:
                  break

      _lowerCamelCase					: str		    = 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 os
from shutil import copyfile
from typing import Any, Dict, List, Optional, Tuple

import sentencepiece as spm

from ...tokenization_utils import AddedToken, BatchEncoding, PreTrainedTokenizer
from ...utils import logging


lowerCAmelCase_      :  Optional[int]				     =  logging.get_logger(__name__)

lowerCAmelCase_      :  str				     =  '▁'

lowerCAmelCase_      :  List[str]				     =  {'vocab_file': 'sentencepiece.bpe.model'}

lowerCAmelCase_      :  Optional[Any]				     =  {
    'vocab_file': {
        'facebook/mbart-large-50-one-to-many-mmt': (
            'https://huggingface.co/facebook/mbart-large-50-one-to-many-mmt/resolve/main/sentencepiece.bpe.model'
        ),
    }
}

lowerCAmelCase_      :  List[str]				     =  {
    'facebook/mbart-large-50-one-to-many-mmt': 1024,
}

# fmt: off
lowerCAmelCase_      :  Optional[Any]				     =  ['ar_AR', 'cs_CZ', 'de_DE', 'en_XX', 'es_XX', 'et_EE', 'fi_FI', 'fr_XX', 'gu_IN', 'hi_IN', 'it_IT', 'ja_XX', 'kk_KZ', 'ko_KR', 'lt_LT', 'lv_LV', 'my_MM', 'ne_NP', 'nl_XX', 'ro_RO', 'ru_RU', 'si_LK', 'tr_TR', 'vi_VN', 'zh_CN', 'af_ZA', 'az_AZ', 'bn_IN', 'fa_IR', 'he_IL', 'hr_HR', 'id_ID', 'ka_GE', 'km_KH', 'mk_MK', 'ml_IN', 'mn_MN', 'mr_IN', 'pl_PL', 'ps_AF', 'pt_XX', 'sv_SE', 'sw_KE', 'ta_IN', 'te_IN', 'th_TH', 'tl_XX', 'uk_UA', 'ur_PK', 'xh_ZA', 'gl_ES', 'sl_SI']







class      SCREAMING_SNAKE_CASE    (						snake_case_				):
      __magic_name__  :       Any   							=					VOCAB_FILES_NAMES
      __magic_name__  :       List[Any]   							=					PRETRAINED_POSITIONAL_EMBEDDINGS_SIZES
      __magic_name__  :       Optional[Any]   							=					PRETRAINED_VOCAB_FILES_MAP
      __magic_name__  :       Union[str, Any]   							=					['''input_ids''', '''attention_mask''']

      __magic_name__  :       List[int]   							=					[]
      __magic_name__  :       List[int]   							=					[]






      def __init__( self   :  int						,						lowercase__   :  Tuple						,						lowercase__   :  Optional[int]=None						,						lowercase__   :  str=None						,						lowercase__   :  Dict="</s>"						,						lowercase__   :  Tuple="</s>"						,						lowercase__   :  Any="<s>"						,						lowercase__   :  Tuple="<unk>"						,						lowercase__   :  List[Any]="<pad>"						,						lowercase__   :  Dict="<mask>"						,						lowercase__   :  Optional[Dict[str, Any]] = None						,						**lowercase__   :  Tuple						,						):


           '''simple docstring'''

           a_     :							Any     						=  AddedToken(lowercase__						,						lstrip=lowercase__						,						rstrip=lowercase__       ) if isinstance(lowercase__						,						lowercase__       ) else mask_token

           a_     :							Union[str, Any]     						=  {} if sp_model_kwargs is None else sp_model_kwargs

           a_     :							int     						=  kwargs.get("""additional_special_tokens"""						,						[]       )
           kwargs["additional_special_tokens"] += [
               code for code in FAIRSEQ_LANGUAGE_CODES if code not in kwargs["additional_special_tokens"]
           ]

           super().__init__(
               src_lang=lowercase__						,						tgt_lang=lowercase__						,						eos_token=lowercase__						,						unk_token=lowercase__						,						sep_token=lowercase__						,						cls_token=lowercase__						,						pad_token=lowercase__						,						mask_token=lowercase__						,						sp_model_kwargs=self.sp_model_kwargs						,						**lowercase__						,						)

           a_     :							Union[str, Any]     						=  spm.SentencePieceProcessor(**self.sp_model_kwargs       )
           self.sp_model.Load(str(lowercase__       )       )
           a_     :							Union[str, Any]     						=  vocab_file

           # Original fairseq vocab and spm vocab must be "aligned":
           # Vocab    |    0    |    1    |   2    |    3    |  4  |  5  |  6  |   7   |   8   |  9
           # -------- | ------- | ------- | ------ | ------- | --- | --- | --- | ----- | ----- | ----
           # fairseq  | '<s>'   | '<pad>' | '</s>' | '<unk>' | ',' | '.' | '▁' | 's'   | '▁de' | '-'
           # spm      | '<unk>' | '<s>'   | '</s>' | ','     | '.' | '▁' | 's' | '▁de' | '-'   | '▁a'

           # Mimic fairseq token-to-id alignment for the first 4 token
           a_     :							Tuple     						=  {"""<s>""": 0, """<pad>""": 1, """</s>""": 2, """<unk>""": 3}

           # The first "real" token "," has position 4 in the original fairseq vocab and position 3 in the spm vocab
           a_     :							List[str]     						=  1

           a_     :							List[Any]     						=  len(self.sp_model       )
           a_     :							List[str]     						=  {
               code: self.sp_model_size + i + self.fairseq_offset for i, code in enumerate(lowercase__       )
           }
           a_     :							str     						=  {v: k for k, v in self.lang_code_to_id.items()}
           a_     :							Any     						=  len(self.sp_model       ) + len(self.lang_code_to_id       ) + self.fairseq_offset

           self.fairseq_tokens_to_ids.update(self.lang_code_to_id       )
           a_     :							int     						=  {v: k for k, v in self.fairseq_tokens_to_ids.items()}

           a_     :							List[Any]     						=  src_lang if src_lang is not None else """en_XX"""
           a_     :							Any     						=  self.lang_code_to_id[self._src_lang]
           a_     :							Union[str, Any]     						=  tgt_lang
           self.set_src_lang_special_tokens(self._src_lang       )






      @property
      def 							lowercase_						( self   :  Union[str, Any]       ):


           '''simple docstring'''

           return len(self.sp_model       ) + len(self.lang_code_to_id       ) + self.fairseq_offset + 1  # Plus 1 for the mask token






      @property
      def 							lowercase_						( self   :  Tuple       ):


           '''simple docstring'''

           return self._src_lang






      @src_lang.setter
      def 							lowercase_						( self   :  Optional[Any]						,						lowercase__   :  str       ):


           '''simple docstring'''

           a_     :							Tuple     						=  new_src_lang
           self.set_src_lang_special_tokens(self._src_lang       )






      def __getstate__( self   :  Any       ):


           '''simple docstring'''

           a_     :							str     						=  self.__dict__.copy()
           a_     :							Dict     						=  None
           return state






      def __setstate__( self   :  Optional[Any]						,						lowercase__   :  Dict       ):


           '''simple docstring'''

           a_     :							Any     						=  d

           # for backward compatibility
           if not hasattr(self						,						"""sp_model_kwargs"""       ):
                a_     :							Tuple     						=  {}

           a_     :							int     						=  spm.SentencePieceProcessor(**self.sp_model_kwargs       )
           self.sp_model.Load(self.vocab_file       )






      def 							lowercase_						( self   :  int       ):


           '''simple docstring'''

           a_     :							Dict     						=  {self.convert_ids_to_tokens(lowercase__       ): i for i in range(self.vocab_size       )}
           vocab.update(self.added_tokens_encoder       )
           return vocab






      def 							lowercase_						( self   :  Optional[int]						,						lowercase__   :  str       ):


           '''simple docstring'''

           return self.sp_model.encode(lowercase__						,						out_type=lowercase__       )






      def 							lowercase_						( self   :  Tuple						,						lowercase__   :  str       ):


           '''simple docstring'''

           if token in self.fairseq_tokens_to_ids:
                return self.fairseq_tokens_to_ids[token]
           a_     :							List[str]     						=  self.sp_model.PieceToId(lowercase__       )

           # Need to return unknown token if the SP model returned 0
           return spm_id + self.fairseq_offset if spm_id else self.unk_token_id






      def 							lowercase_						( self   :  Optional[int]						,						lowercase__   :  int       ):


           '''simple docstring'''

           if index in self.fairseq_ids_to_tokens:
                return self.fairseq_ids_to_tokens[index]
           return self.sp_model.IdToPiece(index - self.fairseq_offset       )






      def 							lowercase_						( self   :  Tuple						,						lowercase__   :  str       ):


           '''simple docstring'''

           a_     :							Union[str, Any]     						=  []
           a_     :							List[str]     						=  """"""
           a_     :							Dict     						=  False
           for token in tokens:
                # make sure that special tokens are not decoded using sentencepiece model
                if token in self.all_special_tokens:
                     if not prev_is_special:
                          out_string += " "
                     out_string += self.sp_model.decode(lowercase__       ) + token
                     a_     :							str     						=  True
                     a_     :							Optional[Any]     						=  []
                else:
                     current_sub_tokens.append(lowercase__       )
                     a_     :							Union[str, Any]     						=  False
           out_string += self.sp_model.decode(lowercase__       )
           return out_string.strip()






      def 							lowercase_						( self   :  Tuple						,						lowercase__   :  str						,						lowercase__   :  Optional[str] = None       ):


           '''simple docstring'''

           if not os.path.isdir(lowercase__       ):
                logger.error(F"Vocabulary path ({save_directory}) should be a directory"       )
                return
           a_     :							Dict     						=  os.path.join(
               lowercase__						,						(filename_prefix + """-""" if filename_prefix else """""") + VOCAB_FILES_NAMES["""vocab_file"""]       )

           if os.path.abspath(self.vocab_file       ) != os.path.abspath(lowercase__       ) and os.path.isfile(self.vocab_file       ):
                copyfile(self.vocab_file						,						lowercase__       )
           elif not os.path.isfile(self.vocab_file       ):
                with open(lowercase__						,						"""wb"""       ) as fi:
                     a_     :							Dict     						=  self.sp_model.serialized_model_proto()
                     fi.write(lowercase__       )

           return (out_vocab_file,)






      def 							lowercase_						( self   :  List[str]						,						lowercase__   :  List[int]						,						lowercase__   :  Optional[List[int]] = None						,						lowercase__   :  bool = False       ):


           '''simple docstring'''

           if already_has_special_tokens:
                return super().get_special_tokens_mask(
                    token_ids_a=lowercase__						,						token_ids_a=lowercase__						,						already_has_special_tokens=lowercase__       )

           a_     :							Union[str, Any]     						=  [1] * len(self.prefix_tokens       )
           a_     :							List[Any]     						=  [1] * len(self.suffix_tokens       )
           if token_ids_a is None:
                return prefix_ones + ([0] * len(lowercase__       )) + suffix_ones
           return prefix_ones + ([0] * len(lowercase__       )) + ([0] * len(lowercase__       )) + suffix_ones






      def 							lowercase_						( self   :  List[str]						,						lowercase__   :  List[int]						,						lowercase__   :  Optional[List[int]] = None       ):


           '''simple docstring'''

           if token_ids_a is None:
                return self.prefix_tokens + token_ids_a + self.suffix_tokens
           # We don't expect to process pairs, but leave the pair logic for API consistency
           return self.prefix_tokens + token_ids_a + token_ids_a + self.suffix_tokens






      def 							lowercase_						( self   :  Any						,						lowercase__   :  Optional[int]						,						lowercase__   :  str						,						lowercase__   :  Optional[str]						,						lowercase__   :  Optional[str]						,						**lowercase__   :  Optional[int]       ):


           '''simple docstring'''

           if src_lang is None or tgt_lang is None:
                raise ValueError("""Translation requires a `src_lang` and a `tgt_lang` for this model"""       )
           a_     :							Union[str, Any]     						=  src_lang
           a_     :							Dict     						=  self(lowercase__						,						add_special_tokens=lowercase__						,						return_tensors=lowercase__						,						**lowercase__       )
           a_     :							Tuple     						=  self.convert_tokens_to_ids(lowercase__       )
           a_     :							Optional[Any]     						=  tgt_lang_id
           return inputs






      def 							lowercase_						( self   :  Optional[Any]						,						lowercase__   :  List[str]						,						lowercase__   :  str = "en_XX"						,						lowercase__   :  Optional[List[str]] = None						,						lowercase__   :  str = "ro_RO"						,						**lowercase__   :  Union[str, Any]						,						):


           '''simple docstring'''

           a_     :							str     						=  src_lang
           a_     :							Dict     						=  tgt_lang
           return super().prepare_seqaseq_batch(lowercase__						,						lowercase__						,						**lowercase__       )






      def 							lowercase_						( self   :  List[Any]       ):


           '''simple docstring'''

           return self.set_src_lang_special_tokens(self.src_lang       )






      def 							lowercase_						( self   :  Union[str, Any]       ):


           '''simple docstring'''

           return self.set_tgt_lang_special_tokens(self.tgt_lang       )






      def 							lowercase_						( self   :  Optional[int]						,						lowercase__   :  str       ):


           '''simple docstring'''

           a_     :							Any     						=  self.lang_code_to_id[src_lang]
           a_     :							Dict     						=  [self.cur_lang_code_id]
           a_     :							Any     						=  [self.eos_token_id]






      def 							lowercase_						( self   :  Any						,						lowercase__   :  str       ):


           '''simple docstring'''

           a_     :							Union[str, Any]     						=  self.lang_code_to_id[tgt_lang]
           a_     :							str     						=  [self.cur_lang_code_id]
           a_     :							Union[str, Any]     						=  [self.eos_token_id]
