import mpmath  # for roots of unity
import numpy as np






class 		a__						:

					def __init__(    self      :       int,_A      :       Optional[int]=None,_A      :       Any=None    ):







							"""simple docstring"""






							SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    list(poly_a or [0]    )[:]
							SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    list(poly_b or [0]    )[:]

							# Remove leading zero coefficients
							while self.polyA[-1] == 0:
									self.polyA.pop()
							SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    len(self.polyA    )

							while self.polyB[-1] == 0:
									self.polyB.pop()
							SCREAMING_SNAKE_CASE_  :		Tuple				      =    len(self.polyB    )

							# Add 0 to make lengths equal a power of 2
							SCREAMING_SNAKE_CASE_  :		List[Any]				      =    int(
							    2 ** np.ceil(np.loga(len(self.polyA    ) + len(self.polyB    ) - 1    )    )    )

							while len(self.polyA    ) < self.c_max_length:
									self.polyA.append(0    )
							while len(self.polyB    ) < self.c_max_length:
									self.polyB.append(0    )
							# A complex root used for the fourier transform
							SCREAMING_SNAKE_CASE_  :		List[str]				      =    complex(mpmath.root(x=1,n=self.c_max_length,k=1    )    )

							# The product
							SCREAMING_SNAKE_CASE_  :		List[Any]				      =    self.__multiply()

					def 				__UpperCamelCase   (    self      :       str,_A      :       str    ):







							"""simple docstring"""






							SCREAMING_SNAKE_CASE_  :		Any				      =    [[x] for x in self.polyA] if which == "A" else [[x] for x in self.polyB]
							# Corner case
							if len(_A    ) <= 1:
									return dft[0]
							#
							SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    self.c_max_length // 2
							while next_ncol > 0:
									SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    [[] for i in range(_A    )]
									SCREAMING_SNAKE_CASE_  :		str				      =    self.root**next_ncol

									# First half of next step
									SCREAMING_SNAKE_CASE_  :		int				      =    1
									for j in range(self.c_max_length // (next_ncol * 2)    ):
											for i in range(_A    ):
													new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]    )
											current_root *= root
									# Second half of next step
									SCREAMING_SNAKE_CASE_  :		Dict				      =    1
									for j in range(self.c_max_length // (next_ncol * 2)    ):
											for i in range(_A    ):
													new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]    )
											current_root *= root
									# Update
									SCREAMING_SNAKE_CASE_  :		Dict				      =    new_dft
									SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    next_ncol // 2
							return dft[0]

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







							"""simple docstring"""






							SCREAMING_SNAKE_CASE_  :		int				      =    self.__dft("A"    )
							SCREAMING_SNAKE_CASE_  :		Any				      =    self.__dft("B"    )
							SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    [[dft_a[i] * dft_b[i] for i in range(self.c_max_length    )]]
							del dft_a
							del dft_b

							# Corner Case
							if len(inverce_c[0]    ) <= 1:
									return inverce_c[0]
							# Inverse DFT
							SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    2
							while next_ncol <= self.c_max_length:
									SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    [[] for i in range(_A    )]
									SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    self.root ** (next_ncol // 2)
									SCREAMING_SNAKE_CASE_  :		Dict				      =    1
									# First half of next step
									for j in range(self.c_max_length // next_ncol    ):
											for i in range(next_ncol // 2    ):
													# Even positions
													new_inverse_c[i].append(
													    (
													        inverce_c[i][j]
													        + inverce_c[i][j + self.c_max_length // next_ncol]
													    )
													    / 2    )
													# Odd positions
													new_inverse_c[i + next_ncol // 2].append(
													    (
													        inverce_c[i][j]
													        - inverce_c[i][j + self.c_max_length // next_ncol]
													    )
													    / (2 * current_root)    )
											current_root *= root
									# Update
									SCREAMING_SNAKE_CASE_  :		Dict				      =    new_inverse_c
									next_ncol *= 2
							# Unpack
							SCREAMING_SNAKE_CASE_  :		int				      =    [round(x[0].real,8    ) + round(x[0].imag,8    ) * 1j for x in inverce_c]

							# Remove leading 0's
							while inverce_c[-1] == 0:
									inverce_c.pop()
							return inverce_c






					def __str__(    self      :       Dict    ):







							"""simple docstring"""






							SCREAMING_SNAKE_CASE_  :		List[str]				      =    "A = " + " + ".join(
							    F'{coef}*x^{i}' for coef, i in enumerate(self.polyA[: self.len_A]    )    )
							SCREAMING_SNAKE_CASE_  :		int				      =    "B = " + " + ".join(
							    F'{coef}*x^{i}' for coef, i in enumerate(self.polyB[: self.len_B]    )    )
							SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    "A*B = " + " + ".join(
							    F'{coef}*x^{i}' for coef, i in enumerate(self.product    )    )

							return F'{a}\n{b}\n{c}'


# Unit tests
if __name__ == "__main__":
		import doctest

		doctest.testmod()







'''simple docstring'''




import re

from filelock import FileLock


try:
  import nltk

  __SCREAMING_SNAKE_CASE     :Optional[int]       =				True
except (ImportError, ModuleNotFoundError):
  __SCREAMING_SNAKE_CASE     :str       =				False

if NLTK_AVAILABLE:
  with FileLock('''.lock''') as lock:
    nltk.download('''punkt''', quiet=True)
def 	UpperCAmelCase_							(					__lowercase							:      str       )		->       str:


     '''simple docstring'''



     re.sub("<n>"			,     ""			,     __lowercase       )  # remove pegasus newline char
     assert NLTK_AVAILABLE, "nltk must be installed to separate newlines between sentences. (pip install nltk)"
     return "\n".join(nltk.sent_tokenize(__lowercase       )       )




