from fractions import Fraction
from sys import maxsize
from modp import ModP


class PAdic:
    def __init__(self, p):
        self.p = p
        self.val = ''  # current known value
        self.prec = 0  # current known precision, not containing trailing zeros
        self.order = 0  # order/valuation of number
        pass  # initialize object as subclass perhaps

    def get(self, prec, decimal=True):
        'Return value of number with given precision.'
        while self.prec < prec:
            # update val based on value
            self.val = str(int(self._nextdigit())) + self.val
            self.prec += 1
        if self.order < 0:
            return (
                self.val[:self.order] + ('.' if decimal else '') +
                self.val[self.order:])[-prec-1:]
        return (self.val + self.order * '0')[-prec:]

    def _nextdigit(self):
        'Calculate next digit of p-adic number.'
        raise NotImplementedError

    def getdigit(self, index):
        'Return digit at given index.'
        return int(self.get(index + 1, False)[0])
        # int(
        #    self.get(index+1+int(index < -self.order)
        # )[-int(index < -self.order)])

    # return value with precision up to 32 bits
    def __int__(self):
        return int(self.get(32), self.p)

    def __str__(self):
        return self.get(32)

    # arithmetic operations
    def __neg__(self):
        return PAdicNeg(self.p, self)

    def __add__(self, other):
        return PAdicAdd(self.p, self, other)

    def __radd__(self, other):
        return PAdicAdd(self.p, other, self)

    def __sub__(self, other):
        return PAdicAdd(self.p, self, PAdicNeg(self.p, other))

    def __rsub__(self, other):
        return PAdicAdd(self.p, other, PAdicNeg(self.p, self))

    def __mul__(self, other):
        return PAdicMul(self.p, self, other)

    def __rmul__(self, other):
        return PAdicMul(self.p, other, self)

    # p-adic norm
    def __abs__(self):
        if self.order == maxsize:
            return 0
        numer = denom = 1
        if self.order > 0:
            numer = self.p ** self.order
        if self.order < 0:
            denom = self.p ** self.order
        return Fraction(numer, denom)


class PAdicConst(PAdic):
    def __init__(self, p, value):
        PAdic.__init__(self, p)
        value = Fraction(value)

        # calculate valuation
        if value == 0:
            self.value = value
            self.val = '0'
            self.order = maxsize
            return

        self.order = 0
        while not value.numerator % self.p:
            self.order += 1
            value /= self.p
        while not value.denominator % self.p:
            self.order -= 1
            value *= self.p
        self.value = value

        self.zero = not value

    def get(self, prec, decimal=True):
        if self.zero:
            return '0' * prec
        return PAdic.get(self, prec, decimal)

    def _nextdigit(self):
        'Calculate next digit of p-adic number.'
        rem = ModP(self.p, self.value.numerator) /\
            ModP(self.p, self.value.denominator)
        self.value -= int(rem)
        self.value /= self.p
        return rem


class PAdicAdd(PAdic):
    'Sum of two p-adic numbers.'
    def __init__(self, p, arg1, arg2):
        PAdic.__init__(self, p)
        self.carry = 0
        self.arg1 = arg1
        self.arg2 = arg2
        # might be larger than this
        self.order = self.prec = min(arg1.order, arg2.order)
        arg1.order -= self.order
        arg2.order -= self.order
        # loop until first nonzero digit is found
        self.index = 0
        digit = self._nextdigit()
        while digit == 0:
            self.index += 1
            self.order += 1
            digit = self._nextdigit()
        self.val += str(int(digit))
        self.prec = 1

    def _nextdigit(self):
        s = self.arg1.getdigit(self.index) + \
            self.arg2.getdigit(self.index) + self.carry
        self.carry = s // self.p
        self.index += 1
        return s % self.p


class PAdicNeg(PAdic):
    'Negation of a p-adic number.'
    def __init__(self, p, arg):
        PAdic.__init__(self, p)
        self.arg = arg
        self.order = arg.order

    def _nextdigit(self):
        if self.prec == 0:
            # cannot be p, 0th digit of arg must be nonzero
            return self.p - self.arg.getdigit(0)
        return self.p - 1 - self.arg.getdigit(self.prec)


class PAdicMul(PAdic):
    'Product of two p-adic numbers.'
    def __init__(self, p, arg1, arg2):
        PAdic.__init__(self, p)
        self.carry = 0
        self.arg1 = arg1
        self.arg2 = arg2
        self.order = arg1.order + arg2.order
        self.arg1.order = self.arg2.order = 0  # TODO requires copy
        self.index = 0

    def _nextdigit(self):
        s = sum(
            self.arg1.getdigit(i) * self.arg2.getdigit(self.index - i)
            for i in xrange(self.index + 1)
            ) + self.carry
        self.carry = s // self.p
        self.index += 1
        return s % self.p