import numpy as np
from numpy import floataa
from numpy.typing import NDArray





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase    )->	NDArray[floataa]:






      '''simple docstring'''





      UpperCAmelCase     :			str     	=np.shape(__UpperCAmelCase    )

      UpperCAmelCase     :			NDArray[floataa]     	=np.zeros((rows, 1)		,       dtype=__UpperCAmelCase    )
      for row in reversed(range(__UpperCAmelCase    )    ):
            UpperCAmelCase     :			Optional[Any]     	=0
            for col in range(row + 1		,       __UpperCAmelCase    ):
                  total += coefficients[row, col] * x[col]

            UpperCAmelCase     :			List[Any]     	=(vector[row] - total) / coefficients[row, row]

      return x





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase    )->	NDArray[floataa]:






      '''simple docstring'''





      UpperCAmelCase     :			List[str]     	=np.shape(__UpperCAmelCase    )
      if rows != columns:
            return np.array(()		,       dtype=__UpperCAmelCase    )

      # augmented matrix
      UpperCAmelCase     :			NDArray[floataa]     	=np.concatenate((coefficients, vector)		,       axis=1    )
      UpperCAmelCase     :			Optional[int]     	=augmented_mat.astype('''float64'''    )

      # scale the matrix leaving it triangular
      for row in range(rows - 1    ):
            UpperCAmelCase     :			Dict     	=augmented_mat[row, row]
            for col in range(row + 1		,       __UpperCAmelCase    ):
                  UpperCAmelCase     :			List[str]     	=augmented_mat[col, row] / pivot
                  augmented_mat[col, :] -= factor * augmented_mat[row, :]

      UpperCAmelCase     :			int     	=retroactive_resolution(
          augmented_mat[:, 0:columns]		,       augmented_mat[:, columns : columns + 1]    )

      return x


if __name__ == "__main__":
     import doctest

     doctest.testmod()

'''simple docstring'''
from __future__ import annotations

import math
import random







def 				__UpperCamelCase (					__UpperCAmelCase  :					int ,   __UpperCAmelCase  :					int ,   __UpperCAmelCase  :					int			)							->	list[int]:
  lowercase				:							Union[str, Any]									=   d * e - 1
  lowercase				:							List[str]									=   0
  lowercase				:							Tuple									=   0
  while p == 0:
    lowercase				:							Optional[Any]									=   random.randint(2 ,   n - 1			)
    lowercase				:							Dict									=   k
    while True:
      if t % 2 == 0:
        lowercase				:							List[Any]									=   t // 2
        lowercase				:							Union[str, Any]									=   (g**t) % n
        lowercase				:							Optional[Any]									=   math.gcd(x - 1 ,   __UpperCAmelCase			)
        if x > 1 and y > 1:
          lowercase				:							Dict									=   y
          lowercase				:							Any									=   n // y
          break  # find the correct factors
      else:
        break  # t is not divisible by 2, break and choose another g
  return sorted([p, q]			)


if __name__ == "__main__":
 import doctest

 doctest.testmod()