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  } e|«      ‘Œ c}Z d„ Z!dZ"de"fd„Z#dde"fd„Z$dde"fd„Z%edk(  re%Z&ne$Z&ddz  Z'dd z  Z(dd!z  Z)dd"z  Z*d#Z+d$Z,d%„ Z-d&„ Z.d'„ Z/d(„ Z0d)„ Z1e1Z2edk(  rN ej(                  «       dk\  rejf                  xZ4xZ5Z3ejl                  Z7n=ejp                  xZ4xZ5Z3ejn                  Z7n edk(  r e9ed*d+„ «      xZ4xZ5Z3d,„ Z7ne-Z4e.Z5e0Z3e/Z7i fd-„Z:d.Z;ddd/œfd0„Z<ddiddigfd1„Z=edk(  rej|                  Z<nedk(  rd2„ Z<ej~                  Z:d3„ Z@edk(  rd4„ Z@d5ZA eBeA«      ZCd6„ ZDd7„ ZEd8„ ZFd9ZGde
ifd:„ZHd;„ ZId<„ ZJyc c} w c c}w c c}w )>zw
Utility functions for integer math.

TODO: rename, cleanup, perhaps move the gmpy wrapper code
here from settings.py

é    N)Úbisecté   )Úxrange)ÚBACKENDÚgmpyÚsageÚ
sage_utilsÚMPZÚMPZ_ONEÚMPZ_ZEROé   é   é   c                 ód   — |g}|d   | |z  kD  r||d   |z  dz   gz   }|d   | |z  kD  rŒ|ddd…   S )a  
    Return a list of integers ~=

    [start, n*start, ..., target/n^2, target/n, target]

    but conservatively rounded so that the quotient between two
    successive elements is actually slightly less than n.

    With n = 2, this describes suitable precision steps for a
    quadratically convergent algorithm such as Newton's method;
    with n = 3 steps for cubic convergence (Halley's method), etc.

        >>> giant_steps(50,1000)
        [66, 128, 253, 502, 1000]
        >>> giant_steps(50,1000,4)
        [65, 252, 1000]

    éÿÿÿÿé   N© )ÚstartÚtargetÚnÚLs       úU/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/mpmath/libmp/libintmath.pyÚgiant_stepsr      sP   € ð& 
ˆ€AØ
ˆB‰%�%˜‘'Š/Ø��2‘˜‘˜A‘�Ñˆð ˆB‰%�%˜‘'‹/à‰TˆrˆT‰7€Nó    c                 ó"   — |dk\  r| |z	  S | | z  S )zÀFor an integer x, calculate x >> n with the fastest (floor)
    rounding. Unlike the plain Python expression (x >> n), n is
    allowed to be negative, in which case a left shift is performed.r   r   ©Úxr   s     r   Úrshiftr   +   ó   € ð 	ˆA‚v�a˜1‘fˆ}Ø˜Q˜B‘iÐr   c                 ó"   — |dk\  r| |z  S | | z	  S )z½For an integer x, calculate x << n. Unlike the plain Python
    expression (x << n), n is allowed to be negative, in which case a
    right shift with default (floor) rounding is performed.r   r   r   s     r   Úlshiftr!   2   r   r   r   c                 ó~   — | sy| dz  }|r	t         |   S d}| dz  } | dz  s| dz  } |dz  }| dz  sŒ|t         | dz     z   S )z1Count the number of trailing zero bits in abs(n).r   éÿ   r   )Úsmall_trailing)r   Úlow_byteÚts      r   Úpython_trailingr'   >   se   € áØØ�4‰x€HÙÜ˜hÑ'Ð'Ø	€AØˆ!�G€AØ�$ŠhØ	ˆa‰ˆØ	ˆQ‰ˆð �$‹hð Œ~˜a $™hÑ'Ñ'Ð'r   r   Ú2c                 ó:   — | rt        | «      j                  «       S y©z<Count the number of trailing zero bits in abs(n) using gmpy.r   )r
   Ú	bit_scan1©r   s    r   Úgmpy_trailingr-   N   s   € áœ˜Q›×)Ñ)Ó+Ð+Ør   c                 ó:   — | rt        | «      j                  «       S yr*   )r
   Úscan1r,   s    r   r-   r-   S   s   € áœ˜Q›Ÿ™›Ð'Ør   é,  c                 ó’   — t        t        | «      }|dk7  r|S t        t        j                  | d«      «      dz
  }|t
        | |z	     z   S )ú0Calculate bit size of the nonnegative integer n.r0   r   é   )r   ÚpowersÚintÚmathÚlogÚbctable)r   Úbcs     r   Úpython_bitcountr:   [   sF   € ä	”˜Ó	€BØ	ˆS‚yØˆ	Ü	ŒT�X‰X�a˜‹^Ó	˜qÑ	 €BØ”˜˜2™‘ÑÐr   c                 ó<   — | rt        | «      j                  d«      S y)r2   r   r   )r
   Ú	numdigitsr,   s    r   Úgmpy_bitcountr=   c   s   € á”�Q“×!Ñ! !Ó$Ð
$Ør   c                 ó4   — t        | «      j                  «       S ©N)r
   Útrailing_zero_bitsr,   s    r   Úsage_trailingrA   l   s   € Üˆq‹6×$Ñ$Ó&Ð&r   Ú
bit_lengthi   c                 ó*   — | t        |«      |z  z  |z	  S )zaChanges radix of a fixed-point number; i.e., converts
    x * 2**xbits to floor(x * 10**bdigits).)r
   )r   ÚxbitsÚbaseÚbdigitss       r   Úbin_to_radixrG   ƒ   s   € ð ”�D“	˜7Ñ"Ñ# uÑ,Ð,r   Ú$0123456789abcdefghijklmnopqrstuvwxyzé
   c                 ó¤   — |dk(  rt        | «      S g }| r&t        | |«      \  } }|j                  ||   «       | rŒ&dj                  |ddd…   «      S )ziReturn the string numeral of a positive integer in an arbitrary
    base. Most efficient for small input.rI   Ú Nr   )ÚstrÚdivmodÚappendÚjoin)r   rE   ÚdigitsÚdigsÚdigits        r   Úsmall_numeralrS   Š   sY   € ð ˆr‚zÜ�1‹vˆØ€DÙ
Ü˜!˜T“?‰ˆˆ5Ø�‰�F˜5‘MÔ"ò ð �7‰7�4™˜"˜‘:ÓÐr   c                 óö   — | dk  r| sydt        |  |||«      z   S |dk  rt        | ||«      S |dz  |dz  z   }t        | ||z  «      \  }}t        ||||«      }t        ||||«      j                  |d«      }||z   S )á_  Represent the integer n as a string of digits in the given base.
    Recursive division is used to make this function about 3x faster
    than Python's str() for converting integers to decimal strings.

    The 'size' parameters specifies the number of digits in n; this
    number is only used to determine splitting points and need not be
    exact.r   Ú0ú-éú   r   r   )ÚnumeralrS   rM   Úrjust©	r   rE   ÚsizerP   ÚhalfÚAÚBÚadÚbds	            r   Únumeral_pythonrb   •   s›   € ð 	ˆA‚vÙØØ”W˜a˜R  t¨VÓ4Ñ4Ð4àˆc‚zÜ˜Q  fÓ-Ð-à�A‰I˜$ ™(Ñ#€DÜ�!�T˜4‘ZÓ �D€A€qÜ	��D˜$ Ó	'€BÜ	��D˜$ Ó	'×	-Ñ	-¨d°CÓ	8€BØ�‰7€Nr   c                 ó  — | dk  rdt        |  |||«      z   S |dk  rt        j                  | |«      S |dz  |dz  z   }t        | t	        |«      |z  «      \  }}t        ||||«      }t        ||||«      j                  |d«      }||z   S )rU   r   rW   i`ã r   r   rV   )rY   r   rP   rM   r
   rZ   r[   s	            r   Únumeral_gmpyrd   «   s�   € ð 	ˆ1‚uØ”W˜a˜R  t¨VÓ4Ñ4Ð4ð ˆg‚~Ü�{‰{˜1˜dÓ#Ð#à�A‰I˜$ ™(Ñ#€DÜ�!”S˜“Y ‘_Ó%�D€A€qÜ	��D˜$ Ó	'€BÜ	��D˜$ Ó	'×	-Ñ	-¨d°CÓ	8€BØ�‰7€Nr   i   iX  i�  éÈ   l                l           c                 ó   — | s| S | t         k  r,| t        k  rt        | dz  «      S t        | dz  dz  «      dz   }n0t        | «      }|dz  }t        | d|z  dz
  z	  dz  dz   «      |dz
  z  }	 || |z  z   dz	  }||k\  r|S |}Œ)zd
    Correctly (floor) rounded integer square root, using
    division. Fast up to ~200 digits.
    ç      à?g-     ð?r   r   éd   é2   )Ú_1_800Ú_1_50r5   Úbitcount)r   Úrr9   r   Úys        r   Úisqrt_small_pythonro   Í   s¨   € ñ
 ØˆØŒ6‚zàŒuŠ9Ü�q˜#‘v“;Ðä��3‘Ð)Ñ)Ó*¨QÑ.‰ä�a‹[ˆØ�‰EˆÜ��Q�q‘S˜‘W‘ Ñ# AÑ%Ó&¨¨2©Ñ.ˆð Øˆq�!‰t‰V�a‰KˆØ�Š6ØˆHØˆð	 r   c                 óø  — | t         k  rLt        | dz  «      }| t        k\  r3|| |z  z   dz	  }| t        k\  r|| |z  z   dz	  }| t        k\  r|| |z  z   dz	  }|S t        | «      }d}| d|z  z  } |d|z  z  }||dz  z  }|dz  }t        d|«      }t        dd|z  z  | |d|z  z
  z	  dz  z  «      }|}t        ||«      D ]1  }||z  d|z  |z
  z	  }	| ||z
  z	  |	z  |z	  }
|d|z  |
z
  z  |dz   z	  }|}Œ3 || |z	  z  |z   z	  S )	a  
    Fast approximate integer square root, computed using division-free
    Newton iteration for large x. For random integers the result is almost
    always correct (floor(sqrt(x))), but is 1 ulp too small with a roughly
    0.1% probability. If x is very close to an exact square, the answer is
    1 ulp wrong with high probability.

    With 0 guard bits, the largest error over a set of 10^5 random
    inputs of size 1-10^5 bits was 3 ulp. The use of 10 guard bits
    almost certainly guarantees a max 1 ulp error.
    rg   r   rI   r   ri   g       @g      à¿é   )rj   r5   Ú_1_100Ú_1_200Ú_1_400rl   Úminr   )r   rn   r9   Ú
guard_bitsÚhbcÚ	startprecrm   ÚppÚpÚr2Úxr2s              r   Úisqrt_fast_pythonr}   ç   s[  € ð$ 	Œ6‚zÜ��3‘‹KˆØ”Š;Ø�Q˜‘T‘˜a‘ˆAØ”FŠ{Ø˜˜A™‘X !‘O�Øœ’;Ø˜Q ™T™ a™�AØˆÜ	�!‹€BØ€JØˆ!ˆJ‰,Ñ€AØˆ!ˆJ‰,Ñ€BØˆ2ˆa‰4�L€BØ
ˆa‰%€CÜ�B˜“€IäˆC�!�I‘+Ñ !¨¨1¨Y©;©Ñ"7¸DÑ!@Ñ@ÓA€AØ	€BÜ˜ CÓ(ò ˆà�‰c�q˜‘t˜a‘xÑ ˆà�b˜‘d‘˜rÑ! aÑ'ˆà�1�a‘4˜3‘,Ñ R¨¡TÑ*ˆØ‰ðð ˆq�#‰v‰J˜A˜j™LÑ)Ð)r   c                 óú   — | t         k  rt        | «      }|| ||z  z
  fS t        | «      dz   }| ||z  z
  }|dk  r|dz  }|dd|z  z   z  }|dk  rŒ|r'|dd|z   z  kD  r|dz  }|dd|z  z   z  }|dd|z   z  kD  rŒ||fS )z=Correctly rounded integer (floor) square root with remainder.r   r   r   )Ú_1_600ro   r}   )r   rn   Úrems      r   Úsqrtrem_pythonr�     s·   € ð 	Œ6‚zÜ˜qÓ!ˆØ�!�a˜‘c‘'ˆzÐÜ˜!Ó˜qÑ €AØ
ˆa�‰c‰'€Cà
�Š'Ø	ˆQ‰ˆØ��!�A‘#‘‰ˆð �‹'ñ Ø˜˜1˜Q™3™’-Ø�Q‘�Ø˜˜!˜A™#™‘�ð ˜˜1˜Q™3™“-ð ˆcˆ6€Mr   c                 ó   — t        | «      d   S )z2Integer square root with correct (floor) rounding.r   )r�   )r   s    r   Úisqrt_pythonrƒ   +  s   € ä˜!Ó˜QÑÐr   c                 ó   — t        | |z  «      S r?   )Ú
isqrt_fast)r   Úprecs     r   Ú
sqrt_fixedr‡   /  s   € Ü�a˜‘gÓÐr   Úisqrtc                 ó4   — t        | «      j                  «       S r?   )r
   rˆ   r,   s    r   ú<lambda>rŠ   =  s   € ¬s°1«v¯|©|«~€ r   c                 ó4   — t        | «      j                  «       S r?   )r
   Úsqrtremr,   s    r   rŠ   rŠ   >  s   € œ˜A›Ÿ™Ó(€ r   c                 ó,  — | dk  rd|  dz   z  t        |  «      z  S | |v r||    S | }t        t        t        t        f\  }}}}| rF| dz  r!||z  }||z  |z   ||z  z   ||z  |z   }}| dz  } n||z  }||z  |z   |d|z  |z  z   }}| dz  } | rŒF|dk  r|||<   |S )zCComputes the nth Fibonacci number as an integer, for
    integer n.r   r   r   r   rX   )Úifibr   r   )	r   Ú_cacheÚmÚaÚbrz   ÚqÚaqÚqqs	            r   rŽ   rŽ   F  sÜ   € ð 	ˆ1‚uØ�q�b˜‘d‰|œd A 2›hÑ&Ð&ØˆF�{Ø�a‰yÐØ	€Aô œ(¤H¬gÐ5�J€A€qˆ!ˆQÙ
ØˆqŠ5Ø�1‘ˆBØ�Q‘3�r‘6˜!˜A™#‘:˜q ™s 2™vˆqˆAØ�‰F‰Aà�1‘ˆBØ�Q‘3�r‘6˜2˜a ™c !™e™8ˆqˆAØ�!‰GˆAò ð 	ˆ3‚wØˆˆq‰	Ø€Hr   iè  )r   r   c                 ó    — |j                  | «      }|r|S t        |«      }||dz
     }t        }|| k  r||z  }||k  r|||<   |dz  }|| k  rŒ|S )z.Return n factorial (for integers n >= 0 only).r   )ÚgetÚlenÚMAX_FACTORIAL_CACHE)r   ÚmemoÚfÚkrz   ÚMAXs         r   Úifacrž   a  sk   € à�‰�‹€AÙØˆÜˆD‹	€AØˆQˆq‰S‰	€AÜ
€CØ
ˆqŠ&Ø	ˆQ‰ˆØ�Š8ØˆD�‰GØ	ˆQ‰ˆð	 ˆq‹&ð
 €Hr   c                 óª   — || dz     }|j                  | «      }|r|S t        |«      }||   }t        }|| k  r|dz  }||z  }||k  r|||<   || k  rŒ|S )z4Return n!! (double factorial), integers n >= 0 only.r   r   )r—   Úmaxr™   )r   Ú	memo_pairrš   r›   rœ   rz   r�   s          r   Úifac2r¢   p  st   € à�Q�q‘S‰>€DØ�‰�‹€AÙØˆÜˆD‹	€AØˆQ‰€AÜ
€CØ
ˆaŠ%Ø	ˆQ‰ˆØ	ˆQ‰ˆØ�Š8ØˆD�‰Gð	 ˆa‹%ð
 €Hr   c                 ó>   — t        t        j                  | «      «      S r?   )r5   r   Ú	factorialr,   s    r   rŠ   rŠ   ƒ  s   € ”SœŸ™¨Ó*Ó+€ r   c                 óò   — | dz   } t        t        | «      «      }ddg|d d t        dt        | dz  «      dz   «      D ]"  }||   sŒ	t        |dz  | |«      D ]  }d||<   Œ	 Œ$ |D �cg c]  }|sŒ|‘Œ	 c}S c c}w )Nr   r   r   rg   )Úlistr   r5   )r   ÚsieveÚiÚjrz   s        r   Úlist_primesrª   †  s�   € Ø	ˆA‰€AÜ”˜“‹O€EØ�A�€Eˆ"ˆ1€IÜ�A”s˜1˜c™6“{ 1‘}Ó%ò ˆØ�‹8Ü˜A˜q™D ! QÓ'ò �Ø��a’ñðð Ö"�!¢ŠAÒ"Ð"ùÒ"s   Á%A4Á-A4c                 ój   — t        j                  | dz   «      D �cg c]  }t        |«      ‘Œ c}S c c}w )Nr   )r   Úprimesr5   )r   Ú_s     r   rª   rª   “  s'   € Ü $§¡¨A¨a©CÓ 0Ö1˜1”�A•Ò1Ð1ùÒ1s   ›0)rq   é   r   é   é   é   é   é   é   é   é%   é)   é+   é/   c                 ó  ‡ ‡‡‡— t        ‰ «      Š ‰ dz  s‰ dk(  S ‰ dk  r‰ t        v S t        D ]	  }‰ |z  rŒ	 y ‰ dz
  Št        ‰«      Š‰‰z	  Šˆˆˆ ˆfd„}‰ dk  rddg}n‰ dk  rg d	¢}nt        }|D ]  } ||«      rŒ y y
)a&  
    Determines whether n is a prime number. A probabilistic test is
    performed if n is very large. No special trick is used for detecting
    perfect powers.

        >>> sum(list_primes(100000))
        454396537
        >>> sum(n*isprime(n) for n in range(100000))
        454396537

    r   r   ri   Fc                 óv   •— t        | ‰‰«      }|dk(  s|‰k(  ryt        d‰«      D ]  }|dz  ‰z  }|‰k(  sŒ y y)Nr   Tr   F)Úpowr   )r‘   r   rm   Údr�   r   Úss      €€€€r   Útestzisprime.<locals>.test°  sQ   ø€ Ü��!�A‹JˆØ�Š6�Q˜!’VØÜ˜˜!“ò 	ˆAØ�1‘�q‘ˆAØ�A‹vÙð	ð r   iÕõ rq   l   ÁHe%�Z	 )r   rq   r®   r   r¯   r°   r±   T)r5   Úsmall_odd_primes_setÚsmall_odd_primesÚtrailing)r   rz   r¿   Ú	witnessesr‘   r½   r�   r¾   s   `    @@@r   ÚisprimerÄ   ™  s®   û€ ô 	ˆA‹€AØˆqŠ5Ø�A‰vˆØˆ2‚vØÔ(Ð(Ð(Üò ˆØ�1‹uÙðð 	
ˆ!‰€AÜ�‹€AØ	ˆQ‰€A÷ð 	ˆ7‚{Ø�q�E‰	Ø	
ˆ_Ò	Ú&‰	ä$ˆ	Øò ˆÙ�A�wÙðð r   c                 óî   ‡— t        t        | «      «      } | dk  r| S g }t        d| dz   «      D ]8  Š| ‰z  rŒ	| ‰dz  z  s yt        ˆfd„|D «       «      rŒ(|j	                  ‰«       Œ: dt        |«      z  S )z´
    Evaluates the Moebius function which is `mu(n) = (-1)^k` if `n`
    is a product of `k` distinct primes and `mu(n) = 0` otherwise.

    TODO: speed up using factorization
    r   r   r   c              3   ó(   •K  — | ]	  }‰|z  –— Œ y ­wr?   r   )Ú.0r›   rz   s     €r   ú	<genexpr>zmoebius.<locals>.<genexpr>Ô  s   øè ø€ Ò. �q˜1•uÑ.ùs   ƒr   )Úabsr5   r   ÚsumrN   r˜   )r   Úfactorsrz   s     @r   ÚmoebiusrÌ   Å  s|   ø€ ô 	ŒC�‹F‹€AØˆ1‚uØˆØ€GÜ�A�q˜‘s‹^ò "ˆØ�A“Ø˜˜1™’HÙÜÓ. gÔ.Õ.Ø—‘˜qÕ!ð"ð ”�W“ÑÐr   c                  ó<   — d}| D ]  }|r|sŒ|||z  }}|rŒ
Œ|}Œ |S )Nr   r   )Úargsr‘   r’   s      r   ÚgcdrÏ   Ø  s<   € Ø	€AØò ˆÙÚØ˜!˜a™%�1�ó ð ‰Aðð €Hr   iô  c                 óê  — | dz  rt         S |j                  | «      }|r|S t        }| }dD �cg c]  }t        |«      ‘Œ }}t	        d| dz   «      D ]œ  }t	        |dz   dd«      D ]"  }|dz
  ||   z  |dz   ||dz      z  z   ||dz   <   Œ$ |j                  d«       d}t	        |dz   dd«      D ]'  }	|||	dz      z  }||k  sŒd|dz  z  |d|z  z  z  ||<   Œ) || k(  sŒ‹d|dz  z  |z  d|z  z  c S  yc c}w )a¼  
    Computes the Euler numbers `E(n)`, which can be defined as
    coefficients of the Taylor expansion of `1/cosh x`:

    .. math ::

        \frac{1}{\cosh x} = \sum_{n=0}^\infty \frac{E_n}{n!} x^n

    Example::

        >>> [int(eulernum(n)) for n in range(11)]
        [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
        >>> [int(eulernum(n)) for n in range(11)]   # test cache
        [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]

    r   )r   r   r   r   r   r   r   éþÿÿÿr   r   N)r   r—   ÚMAX_EULER_CACHEr
   ÚrangerN   )
r�   r�   r›   r�   r   r­   r‘   r©   Úsumarœ   s
             r   ÚeulernumrÕ   ÿ  s5  € ð$ 	ˆ1‚uÜˆØ�
‰
�1‹€AÙØˆÜ
€CØ	€AØ&Ö'�AŒˆQ�Ð'€AÐ'Ü�A�q˜‘s‹mò 
/ˆÜ�q˜‘s˜B Ó#ò 	/ˆAØ˜‘c˜1˜Q™4‘Z 1 Q¡3¨¨!¨A©#©¡,Ñ.ˆAˆa�‰cŠFð	/à	�‰�ŒØˆÜ�q˜‘s˜B Ó#ò 	:ˆAØ�A�a˜‘c‘F‰NˆDØ�C‹xØ  A q¡D™\¨D°A°q±D©LÑ9��q’	ð	:ð �‹6Ø˜1˜a™4‘L $Ñ&¨!¨Q©$Ñ.Ò.ñ
/ùò 	(s   ­C0c                 ó4  — | dk  s|dk  rt         ‚|| k\  rt        | |k(  «      S |dk  rt        S t        g|dz   z  }t        |d<   t	        d| dz   «      D ]5  }t	        t        ||«      dd«      D ]  }|dz
  ||   z  ||dz
     z   ||<   Œ Œ7 d| |z   z  ||   z  S )z,
    Stirling number of the first kind.
    r   r   r   r   )Ú
ValueErrorr
   r   r   r   ru   )r   rœ   r   r�   r©   s        r   Ú	stirling1rØ   %  sÃ   € ð 	ˆ1‚u��A’ÜÐØˆA‚vÜ�1˜‘6‹{ÐØˆ1‚uÜˆÜ	ˆ
�a˜‘cÑ€AÜ€A€a�DÜ�A�q˜‘s‹^ò )ˆÜœ˜A˜q›	 1 bÓ)ò 	)ˆAØ�a‘C˜1˜Q™4‘< ! A a¡C¡&Ñ(ˆAˆaŠDñ	)ð)ð �!�A‘#‰;˜˜1™ÑÐr   c                 óF  — | dk  s|dk  rt         ‚|| k\  rt        | |k(  «      S |dk  rt        |dk(  «      S t        }t        }t	        |dz   «      D ]A  }||z   dz  r||t        |«      | z  z  z  }n||t        |«      | z  z  z  }|||z
  z  |dz   z  }ŒC |t        |«      z  S )z-
    Stirling number of the second kind.
    r   r   )r×   r
   r   r   r   rž   )r   rœ   r¾   r&   r©   s        r   Ú	stirling2rÚ   6  s¿   € ð 	ˆ1‚u��A’ÜÐØˆA‚vÜ�1˜‘6‹{ÐØˆA‚vÜ�1˜‘6‹{ÐÜ€AÜ€AÜ�A�a‘C‹[ò #ˆØ�‰E�QŠ;Ø�”S˜“V˜Q‘Y‘Ñ‰Aà�”S˜“V˜Q‘Y‘ÑˆAØ��Q‘‰K˜A ™EÑ"‰ð#ð ”�Q“‰<Ðr   )r   )KÚ__doc__r6   r   Úbackendr   r   r   r   r	   r
   r   r   r$   rÓ   r©   r   r   r!   Úoperatorr'   Úversionr-   r4   r:   r=   rA   rl   rÂ   Úsage_bitcountÚdirrB   Ú
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