


from typing import Any

import numpy as np





class     snake_case  :


   '''simple docstring'''







   def __init__(					self     :	Any						,    lowerCamelCase_     :	Any						,    lowerCamelCase_     :	Tuple			)					->List[Any]:
     '''simple docstring'''







     if np.any(tableau[:, -1]						,    where=tableau[:, -1] < 0			):
       raise ValueError("""RHS must be > 0"""			)

     UpperCAmelCase__    						=   tableau
     UpperCAmelCase__						,					UpperCAmelCase__    						=   tableau.shape

     # Number of decision variables x1, x2, x3...
     UpperCAmelCase__    						=   n_vars

     # Number of artificial variables to be minimised
     UpperCAmelCase__    						=   len(np.where(tableau[self.n_vars : -1] == -1			)[0]			)

     # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
     UpperCAmelCase__    						=   (self.n_art_vars > 0) + 1

     # Number of slack variables added to make inequalities into equalities
     UpperCAmelCase__    						=   self.n_rows - self.n_stages

     # Objectives for each stage
     UpperCAmelCase__    						=   ["""max"""]

     # In two stage simplex, first minimise then maximise
     if self.n_art_vars:
       self.objectives.append("""min"""			)

     UpperCAmelCase__    						=   [""""""]

     # Index of current pivot row and column
     UpperCAmelCase__    						=   None
     UpperCAmelCase__    						=   None

     # Does objective row only contain (non)-negative values?
     UpperCAmelCase__    						=   False




   @staticmethod
   def 	UpperCAmelCase       (					*lowerCamelCase_     :	Dict			)					->List[Any]:
     '''simple docstring'''







     if len(__SCREAMING_SNAKE_CASE			) != 3:
       raise ValueError("""Must provide n_vars, n_slack, and n_art_vars"""			)

     if not all(x >= 0 and isinstance(__SCREAMING_SNAKE_CASE						,    __SCREAMING_SNAKE_CASE			) for x in args			):
       raise ValueError("""All arguments must be non-negative integers"""			)

     # decision | slack | artificial
     UpperCAmelCase__    						=   ["""x""", """s""", """a"""]
     UpperCAmelCase__    						=   []
     for i in range(3			):
       for j in range(args[i]			):
         titles.append(string_starts[i] + str(j + 1			)			)
     titles.append("""RHS"""			)
     return titles




   def 	UpperCAmelCase       (					self     :	Optional[int]						,    lowerCamelCase_     :	Any			)					->Any:
     '''simple docstring'''







     UpperCAmelCase__    						=   self.objectives[-1]

     # Find entries of highest magnitude in objective rows
     UpperCAmelCase__    						=   (objective == """min""") - (objective == """max""")
     UpperCAmelCase__    						=   np.argmax(sign * tableau[0, : self.n_vars]			)

     # Choice is only valid if below 0 for maximise, and above for minimise
     if sign * self.tableau[0, col_idx] <= 0:
       UpperCAmelCase__    						=   True
       return 0, 0

     # Pivot row is chosen as having the lowest quotient when elements of
     # the pivot column divide the right-hand side

     # Slice excluding the objective rows
     UpperCAmelCase__    						=   slice(self.n_stages						,    self.n_rows			)

     # RHS
     UpperCAmelCase__    						=   tableau[s, -1]

     # Elements of pivot column within slice
     UpperCAmelCase__    						=   tableau[s, col_idx]

     # Array filled with nans
     UpperCAmelCase__    						=   np.full(self.n_rows - self.n_stages						,    np.nan			)

     # If element in pivot column is greater than zeron_stages, return
     # quotient or nan otherwise
     UpperCAmelCase__    						=   np.divide(__SCREAMING_SNAKE_CASE						,    __SCREAMING_SNAKE_CASE						,    out=__SCREAMING_SNAKE_CASE						,    where=divisor > 0			)

     # Arg of minimum quotient excluding the nan values. n_stages is added
     # to compensate for earlier exclusion of objective columns
     UpperCAmelCase__    						=   np.nanargmin(__SCREAMING_SNAKE_CASE			) + self.n_stages
     return row_idx, col_idx




   def 	UpperCAmelCase       (					self     :	str						,    lowerCamelCase_     :	int						,    lowerCamelCase_     :	Union[str, Any]						,    lowerCamelCase_     :	Optional[Any]			)					->Any:
     '''simple docstring'''







     UpperCAmelCase__    						=   tableau[row_idx].copy()

     UpperCAmelCase__    						=   piv_row[col_idx]

     # Entry becomes 1
     piv_row *= 1 / piv_val

     # Variable in pivot column becomes basic, ie the only non-zero entry
     for idx, coeff in enumerate(tableau[:, col_idx]			):
       tableau[idx] += -coeff * piv_row
     UpperCAmelCase__    						=   piv_row
     return tableau




   def 	UpperCAmelCase       (					self     :	Tuple						,    lowerCamelCase_     :	Optional[Any]			)					->Optional[int]:
     '''simple docstring'''







     self.objectives.pop()

     if not self.objectives:
       return tableau

     # Slice containing ids for artificial columns
     UpperCAmelCase__    						=   slice(-self.n_art_vars - 1						,    -1			)

     # Delete the artificial variable columns
     UpperCAmelCase__    						=   np.delete(__SCREAMING_SNAKE_CASE						,    __SCREAMING_SNAKE_CASE						,    axis=1			)

     # Delete the objective row of the first stage
     UpperCAmelCase__    						=   np.delete(__SCREAMING_SNAKE_CASE						,    0						,    axis=0			)

     UpperCAmelCase__    						=   1
     self.n_rows -= 1
     UpperCAmelCase__    						=   0
     UpperCAmelCase__    						=   False
     return tableau




   def 	UpperCAmelCase       (					self     :	Optional[int]			)					->Dict:
     '''simple docstring'''







     for _ in range(100			):
       # Completion of each stage removes an objective. If both stages
       # are complete, then no objectives are left
       if not self.objectives:
         UpperCAmelCase__    						=   self.generate_col_titles(
             self.n_vars						,    self.n_slack						,    self.n_art_vars			)

         # Find the values of each variable at optimal solution
         return self.interpret_tableau(self.tableau						,    self.col_titles			)

       UpperCAmelCase__						,					UpperCAmelCase__    						=   self.find_pivot(self.tableau			)

       # If there are no more negative values in objective row
       if self.stop_iter:
         # Delete artificial variable columns and rows. Update attributes
         UpperCAmelCase__    						=   self.change_stage(self.tableau			)
       else:
         UpperCAmelCase__    						=   self.pivot(self.tableau						,    __SCREAMING_SNAKE_CASE						,    __SCREAMING_SNAKE_CASE			)
     return {}







   def 	UpperCAmelCase       (					self     :	Any						,    lowerCamelCase_     :	Tuple						,    lowerCamelCase_     :	Tuple			)					->Union[str, Any]:
     '''simple docstring'''







     UpperCAmelCase__    						=   {"""P""": abs(tableau[0, -1]			)}

     for i in range(self.n_vars			):
       # Gives ids of nonzero entries in the ith column
       UpperCAmelCase__    						=   np.nonzero(tableau[:, i]			)
       UpperCAmelCase__    						=   len(nonzero[0]			)

       # First entry in the nonzero ids
       UpperCAmelCase__    						=   nonzero[0][0]
       UpperCAmelCase__    						=   tableau[nonzero_rowidx, i]

       # If there is only one nonzero value in column, which is one
       if n_nonzero == nonzero_val == 1:
         UpperCAmelCase__    						=   tableau[nonzero_rowidx, -1]
         UpperCAmelCase__    						=   rhs_val

        # Check for basic variables
     for title in col_titles:
       # Don't add RHS or slack variables to output dict
       if title[0] not in "R-s-a":
         output_dict.setdefault(__SCREAMING_SNAKE_CASE						,    0			)
     return output_dict


if __name__ == "__main__":
   import doctest

   doctest.testmod()










from __future__ import annotations

from collections import Counter


def 						lowercase		(     _lowerCAmelCase      ):
  UpperCAmelCase__    						=   [True] * n
  UpperCAmelCase__    						=   False
  UpperCAmelCase__    						=   False
  UpperCAmelCase__    						=   True

  for i in range(3   ,     int(n**0.5 + 1      )   ,     2      ):
    UpperCAmelCase__    						=   i * 2
    while index < n:
      UpperCAmelCase__    						=   False
      UpperCAmelCase__    						=   index + i

  UpperCAmelCase__    						=   [2]

  for i in range(3   ,     _lowerCAmelCase   ,     2      ):
    if is_prime[i]:
      primes.append(_lowerCAmelCase      )

  return primes


def 						lowercase		(     _lowerCAmelCase      ):
  UpperCAmelCase__    						=   str(_lowerCAmelCase      )
  UpperCAmelCase__    						=   []
  UpperCAmelCase__    						=   ["""0""", """1""", """2""", """3""", """4""", """5""", """6""", """7""", """8""", """9"""]

  for duplicate in Counter(_lowerCAmelCase      ) - Counter(set(_lowerCAmelCase      )      ):
    UpperCAmelCase__    						=   [int(number_str.replace(_lowerCAmelCase   ,     _lowerCAmelCase      )      ) for digit in digits]
    replacements.append(_lowerCAmelCase      )

  return replacements


def 						lowercase		(     _lowerCAmelCase = 8      ):
  UpperCAmelCase__    						=   set()

  # Filter primes with less than 3 replaceable digits
  UpperCAmelCase__    						=   {
      x for x in set(prime_sieve(100_0000      )      ) if len(str(_lowerCAmelCase      )      ) - len(set(str(_lowerCAmelCase      )      )      ) >= 3
  }

  for prime in primes:
    if prime in numbers_checked:
      continue

    UpperCAmelCase__    						=   digit_replacements(_lowerCAmelCase      )

    for family in replacements:
      numbers_checked.update(_lowerCAmelCase      )
      UpperCAmelCase__    						=   primes.intersection(_lowerCAmelCase      )

      if len(_lowerCAmelCase      ) != family_length:
        continue

      return min(_lowerCAmelCase      )

  return -1


if __name__ == "__main__":
   print(solution())




