





'''simple docstring'''

__lowerCAmelCase							:     List[Any]		     =range(2, 20 + 1)
__lowerCAmelCase							:     Optional[Any]		     =[10**k for k in range(ks[-1] + 1)]
__lowerCAmelCase							:     dict[int, dict[int, list[list[int]]]]		     ={}





def 						UpperCamelCase    (							_lowerCamelCase : Tuple						,	_lowerCamelCase : List[Any]						,	_lowerCamelCase : Tuple						,	_lowerCamelCase : List[str]	):
    A__    =      sum(a_i[j] for j in range(_lowerCamelCase						,	len(_lowerCamelCase	)	)	)
    A__    =      sum(a_i[j] * base[j] for j in range(min(len(_lowerCamelCase	)						,	_lowerCamelCase	)	)	)

    A__,     A__    =      0, 0
    A__    =      n - i

    A__    =      memo.get(_lowerCamelCase	)

    if sub_memo is not None:
        A__    =      sub_memo.get(_lowerCamelCase	)

        if jumps is not None and len(_lowerCamelCase	) > 0:
            # find and make the largest jump without going over
            A__    =      -1
            for _k in range(len(_lowerCamelCase	) - 1						,	-1						,	-1	):
                if jumps[_k][2] <= k and jumps[_k][1] <= max_dn:
                    A__    =      _k
                    break

            if max_jump >= 0:
                A__,     A__,     A__    =      jumps[max_jump]
                # since the difference between jumps is cached, add c
                A__    =      diff + c
                for j in range(min(_lowerCamelCase						,	len(_lowerCamelCase	)	)	):
                    A__,     A__    =      divmod(_lowerCamelCase						,	10	)
                if new_c > 0:
                    add(_lowerCamelCase						,	_lowerCamelCase						,	_lowerCamelCase	)

        else:
            A__    =      []
    else:
        A__    =      {c: []}
        A__    =      sub_memo

    if dn >= max_dn or c + diff >= base[k]:
        return diff, dn

    if k > ks[0]:
        while True:
            # keep doing smaller jumps
            A__,     A__    =      next_term(_lowerCamelCase						,	k - 1						,	i + dn						,	_lowerCamelCase	)
            diff += _diff
            dn += terms_jumped

            if dn >= max_dn or c + diff >= base[k]:
                break
    else:
        # would be too small a jump, just compute sequential terms instead
        A__,     A__    =      compute(_lowerCamelCase						,	_lowerCamelCase						,	i + dn						,	_lowerCamelCase	)
        diff += _diff
        dn += terms_jumped

    A__    =      sub_memo[c]

    # keep jumps sorted by # of terms skipped
    A__    =      0
    while j < len(_lowerCamelCase	):
        if jumps[j][1] > dn:
            break
        j += 1

    # cache the jump for this value digitsum(b) and c
    sub_memo[c].insert(_lowerCamelCase						,	(diff, dn, k)	)
    return (diff, dn)





def 						UpperCamelCase    (							_lowerCamelCase : Tuple						,	_lowerCamelCase : Tuple						,	_lowerCamelCase : str						,	_lowerCamelCase : Union[str, Any]	):
    if i >= n:
        return 0, i
    if k > len(_lowerCamelCase	):
        a_i.extend([0 for _ in range(k - len(_lowerCamelCase	)	)]	)

    # note: a_i -> b * 10^k + c
    # ds_b -> digitsum(b)
    # ds_c -> digitsum(c)
    A__    =      i
    A__,     A__,     A__    =      0, 0, 0
    for j in range(len(_lowerCamelCase	)	):
        if j >= k:
            ds_b += a_i[j]
        else:
            ds_c += a_i[j]

    while i < n:
        i += 1
        A__    =      ds_c + ds_b
        diff += addend
        A__    =      0
        for j in range(_lowerCamelCase	):
            A__    =      a_i[j] + addend
            A__,     A__    =      divmod(_lowerCamelCase						,	10	)

            ds_c += a_i[j]

        if addend > 0:
            break

    if addend > 0:
        add(_lowerCamelCase						,	_lowerCamelCase						,	_lowerCamelCase	)
    return diff, i - start_i





def 						UpperCamelCase    (							_lowerCamelCase : Optional[int]						,	_lowerCamelCase : Optional[Any]						,	_lowerCamelCase : Dict	):
    for j in range(_lowerCamelCase						,	len(_lowerCamelCase	)	):
        A__    =      digits[j] + addend
        if s >= 10:
            A__,     A__    =      divmod(_lowerCamelCase						,	10	)
            A__    =      addend // 10 + quotient
        else:
            A__    =      s
            A__    =      addend // 10

        if addend == 0:
            break

    while addend > 0:
        A__,     A__    =      divmod(_lowerCamelCase						,	10	)
        digits.append(_lowerCamelCase	)





def 						UpperCamelCase    (							_lowerCamelCase : int = 10**15	):
    A__    =      [1]
    A__    =      1
    A__    =      0
    while True:
        A__,     A__    =      next_term(_lowerCamelCase						,	20						,	i + dn						,	_lowerCamelCase	)
        dn += terms_jumped
        if dn == n - i:
            break

    A__    =      0
    for j in range(len(_lowerCamelCase	)	):
        a_n += digits[j] * 10**j
    return a_n


if __name__ == "__main__":
  print(f"""{solution() = }""")










'''simple docstring'''

from __future__ import annotations

from math import gcd





def 						UpperCamelCase    (							_lowerCamelCase : int						,	_lowerCamelCase : int = 2						,	_lowerCamelCase : int = 1						,	_lowerCamelCase : int = 3						,	):
    # A value less than 2 can cause an infinite loop in the algorithm.
    if num < 2:
        raise ValueError("The input value cannot be less than 2"	)

    # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
    # algorithm struggles to find factors that are divisible by two.
    # As a workaround, we specifically check for two and even inputs.
    #   See: https://math.stackexchange.com/a/2856214/165820
    if num > 2 and num % 2 == 0:
        return 2

    # Pollard's Rho algorithm requires a function that returns pseudorandom
    # values between 0 <= X < ``num``.  It doesn't need to be random in the
    # sense that the output value is cryptographically secure or difficult
    # to calculate, it only needs to be random in the sense that all output
    # values should be equally likely to appear.
    # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
    # However, the success of Pollard's algorithm isn't guaranteed and is
    # determined in part by the initial seed and the chosen random function.
    # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
    # where ``C`` is a value that we can modify between each attempt.
    def rand_fn(_lowerCamelCase : int						,	_lowerCamelCase : int						,	_lowerCamelCase : int	) -> int:
        return (pow(_lowerCamelCase						,	2	) + step) % modulus

    for _ in range(_lowerCamelCase	):
        # These track the position within the cycle detection logic.
        A__    =      seed
        A__    =      seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            A__    =      rand_fn(_lowerCamelCase						,	_lowerCamelCase						,	_lowerCamelCase	)
            A__    =      rand_fn(_lowerCamelCase						,	_lowerCamelCase						,	_lowerCamelCase	)
            A__    =      rand_fn(_lowerCamelCase						,	_lowerCamelCase						,	_lowerCamelCase	)

            # At some point both the tortoise and the hare will enter a cycle whose
            # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
            # the tortoise and hare will end up on the same value modulo ``p``.
            # We can detect when this happens because the position difference between
            # the tortoise and the hare will share a common divisor with ``num``.
            A__    =      gcd(hare - tortoise						,	_lowerCamelCase	)

            if divisor == 1:
                # No common divisor yet, just keep searching.
                continue
            else:
                # We found a common divisor!
                if divisor == num:
                    # Unfortunately, the divisor is ``num`` itself and is useless.
                    break
                else:
                    # The divisor is a nontrivial factor of ``num``!
                    return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
        A__    =      hare

        # The new step value for the random function can just be incremented.
        # At first the results will be similar to what the old function would
        # have produced, but the value will quickly diverge after a bit.
        step += 1

    # We haven't found a divisor within the requested number of attempts.
    # We were unlucky or ``num`` itself is actually prime.
    return None


if __name__ == "__main__":
  import argparse

  __lowerCAmelCase							:     Optional[int]		     =argparse.ArgumentParser()
  parser.add_argument(
      "num",
      type=int,
      help="The value to find a divisor of",
  )
  parser.add_argument(
      "--attempts",
      type=int,
      default=3,
      help="The number of attempts before giving up",
  )
  __lowerCAmelCase							:     Optional[int]		     =parser.parse_args()

  __lowerCAmelCase							:     Dict		     =pollard_rho(args.num, attempts=args.attempts)
  if divisor is None:
    print(f"""{args.num} is probably prime""")
  else:
    __lowerCAmelCase							:     Optional[Any]		     =args.num // divisor
    print(f"""{args.num} = {divisor} * {quotient}""")



