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from itertools import combinations, chain
from math import prod


def _powA258484(i, n):
    """Return i**n for two integers i and n."""

    # NOTE: Putting in these special cases sped up the search
    #       process enough to be worth it (measured via timeit)
    if n == 0:

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# miscellaneous descriptor-protocol classes, implemented mostly as an exercise
#

# The MIT License
#
# Copyright (c) 2022 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights

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class SetattrThenRestore:
    """Save the old value of an attribute, set it, then restore it.

    with SetattrThenRestore(obj, attrname, val):
        ...bunch of code here...

    is SOMEWHAT equivalent to:

        oldattrval = getattr(obj, attrname)
        setattr(obj, attrname, val)

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# This implements the "Knights and Pages" problem
# from The Chicken from Minsk, Chapter 1
#
# PROBLEM:
#   Many years ago three knights waited to cross the river Neva.
#   Each knight had his own page, so there were six people. The
#   boat they had could only carry two. However, the knights were
#   ferocious killers and the pages were terrified. In fact it was
#   certain that any one of the pages would die of heart failure
#   if he were not protected at every instant from the other knights

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# MIT License
#
# Copyright (c) 2022,2023 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:

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# The MIT License (MIT)
#
# Copyright (c) 2022 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:

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from fractions import Fraction
from math import gcd


# compute the sequence OEIS A092317
#    https://oeis.org/A092317
def _slowway(n):
    b = Fraction(0)
    i = 3
    while b < n:

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import functools

# Given a function such as this:
#
#    def example(a, b=17):
#        pass
#
# If code needs to conditionally call example() sometimes with b supplied
# and sometimes with b defaulted, that ends up looking like this:
#

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import math
import random
import functools
import itertools
from collections import Counter

try:
    from scipy.special import gammaincc
except ModuleNotFoundError:
    def gammaincc(x1, x2):

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# MIT License
#
# Copyright (c) 2023 Neil Webber
#
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions: