import sys
sys.setrecursionlimit(10000)

# python3 compatibility
try:
    foo = long
except:
    long = int

# The prime modulus of the field
field_modulus = 21888242871839275222246405745257275088696311157297823662689037894645226208583
# See, it's prime!
assert pow(2, field_modulus, field_modulus) == 2

# The modulus of the polynomial in this representation of FQ12
FQ12_modulus_coeffs = [82, 0, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0] # Implied + [1]

# Extended euclidean algorithm to find modular inverses for
# integers
def inv(a, n):
    if a == 0:
        return 0
    lm, hm = 1, 0
    low, high = a % n, n
    while low > 1:
        r = high//low
        nm, new = hm-lm*r, high-low*r
        lm, low, hm, high = nm, new, lm, low
    return lm % n

# A class for field elements in FQ. Wrap a number in this class,
# and it becomes a field element.
class FQ():
    def __init__(self, n):
        if isinstance(n, self.__class__):
            self.n = n.n
        else:
            self.n = n % field_modulus
        assert isinstance(self.n, (int, long))

    def __add__(self, other):
        on = other.n if isinstance(other, FQ) else other
        return FQ((self.n + on) % field_modulus)

    def __mul__(self, other):
        on = other.n if isinstance(other, FQ) else other
        return FQ((self.n * on) % field_modulus)

    def __rmul__(self, other):
        return self * other

    def __radd__(self, other):
        return self + other

    def __rsub__(self, other):
        on = other.n if isinstance(other, FQ) else other
        return FQ((on - self.n) % field_modulus)

    def __sub__(self, other):
        on = other.n if isinstance(other, FQ) else other
        return FQ((self.n - on) % field_modulus)

    def __div__(self, other):
        on = other.n if isinstance(other, FQ) else other
        assert isinstance(on, (int, long))
        return FQ(self.n * inv(on, field_modulus) % field_modulus)

    def __truediv__(self, other):
        return self.__div__(other)

    def __rdiv__(self, other):
        on = other.n if isinstance(other, FQ) else other
        assert isinstance(on, (int, long)), on
        return FQ(inv(self.n, field_modulus) * on % field_modulus)

    def __rtruediv__(self, other):
        return self.__rdiv__(other)

    def __pow__(self, other):
        if other == 0:
            return FQ(1)
        elif other == 1:
            return FQ(self.n)
        elif other % 2 == 0:
            return (self * self) ** (other // 2)
        else:
            return ((self * self) ** int(other // 2)) * self

    def __eq__(self, other):
        if isinstance(other, FQ):
            return self.n == other.n
        else:
            return self.n == other

    def __ne__(self, other):
        return not self == other

    def __neg__(self):
        return FQ(-self.n)

    def __repr__(self):
        return repr(self.n)

    @classmethod
    def one(cls):
        return cls(1)

    @classmethod
    def zero(cls):
        return cls(0)

# Check that the field works fine
assert FQ(2) * FQ(2) == FQ(4)
assert FQ(2) / FQ(7) + FQ(9) / FQ(7) == FQ(11) / FQ(7)
assert FQ(2) * FQ(7) + FQ(9) * FQ(7) == FQ(11) * FQ(7)
assert FQ(9) ** field_modulus == FQ(9)

# Utility methods for polynomial math
def deg(p):
    d = len(p) - 1
    while p[d] == 0 and d:
        d -= 1
    return d

def poly_rounded_div(a, b):
    dega = deg(a)
    degb = deg(b)
    temp = [x for x in a]
    o = [0 for x in a]
    for i in range(dega - degb, -1, -1):
        o[i] += temp[degb + i] / b[degb]
        for c in range(degb + 1):
            temp[c + i] -= o[c]
    return o[:deg(o)+1]

# A class for elements in polynomial extension fields
class FQP():
    def __init__(self, coeffs, modulus_coeffs): 
        assert len(coeffs) == len(modulus_coeffs)
        self.coeffs = [FQ(c) for c in coeffs]
        # The coefficients of the modulus, without the leading [1]
        self.modulus_coeffs = modulus_coeffs
        # The degree of the extension field
        self.degree = len(self.modulus_coeffs)

    def __add__(self, other):
        assert isinstance(other, self.__class__)
        return self.__class__([x+y for x,y in zip(self.coeffs, other.coeffs)])

    def __sub__(self, other):
        assert isinstance(other, self.__class__)
        return self.__class__([x-y for x,y in zip(self.coeffs, other.coeffs)])

    def __mul__(self, other):
        if isinstance(other, (FQ, int, long)):
            return self.__class__([c * other for c in self.coeffs])
        else:
            assert isinstance(other, self.__class__)
            b = [FQ(0) for i in range(self.degree * 2 - 1)]
            for i in range(self.degree):
                for j in range(self.degree):
                    b[i + j] += self.coeffs[i] * other.coeffs[j]
            while len(b) > self.degree:
                exp, top = len(b) - self.degree - 1, b.pop()
                for i in range(self.degree):
                    b[exp + i] -= top * FQ(self.modulus_coeffs[i])
            return self.__class__(b)

    def __rmul__(self, other):
        return self * other

    def __div__(self, other):
        if isinstance(other, (FQ, int, long)):
            return self.__class__([c / other for c in self.coeffs])
        else:
            assert isinstance(other, self.__class__)
            return self * other.inv()

    def __truediv__(self, other):
        return self.__div__(other)

    def __pow__(self, other):
        if other == 0:
            return self.__class__([1] + [0] * (self.degree - 1))
        elif other == 1:
            return self.__class__(self.coeffs)
        elif other % 2 == 0:
            return (self * self) ** (other // 2)
        else:
            return ((self * self) ** int(other // 2)) * self

    # Extended euclidean algorithm used to find the modular inverse
    def inv(self):
        lm, hm = [1] + [0] * self.degree, [0] * (self.degree + 1)
        low, high = self.coeffs + [0], self.modulus_coeffs + [1]
        while deg(low):
            r = poly_rounded_div(high, low)
            r += [0] * (self.degree + 1 - len(r))
            nm = [x for x in hm]
            new = [x for x in high]
            assert len(lm) == len(hm) == len(low) == len(high) == len(nm) == len(new) == self.degree + 1
            for i in range(self.degree + 1):
                for j in range(self.degree + 1 - i):
                    nm[i+j] -= lm[i] * r[j]
                    new[i+j] -= low[i] * r[j]
            lm, low, hm, high = nm, new, lm, low
        return self.__class__(lm[:self.degree]) / low[0]

    def __repr__(self):
        return repr(self.coeffs)

    def __eq__(self, other):
        assert isinstance(other, self.__class__)
        for c1, c2 in zip(self.coeffs, other.coeffs):
            if c1 != c2:
                return False
        return True

    def __ne__(self, other):
        return not self == other

    def __neg__(self):
        return self.__class__([-c for c in self.coeffs])

    @classmethod
    def one(cls):
        return cls([1] + [0] * (cls.degree - 1))

    @classmethod
    def zero(cls):
        return cls([0] * cls.degree)

# The quadratic extension field
class FQ2(FQP):
    def __init__(self, coeffs):
        self.coeffs = [FQ(c) for c in coeffs]
        self.modulus_coeffs = [1, 0]
        self.degree = 2
        self.__class__.degree = 2

x = FQ2([1, 0])
f = FQ2([1, 2])
fpx = FQ2([2, 2])
one = FQ2.one()

# Check that the field works fine
assert x + f == fpx
assert f / f == one
assert one / f + x / f == (one + x) / f
assert one * f + x * f == (one + x) * f
assert x ** (field_modulus ** 2 - 1) == one

# The 12th-degree extension field
class FQ12(FQP):
    def __init__(self, coeffs):
        self.coeffs = [FQ(c) for c in coeffs]
        self.modulus_coeffs = FQ12_modulus_coeffs
        self.degree = 12
        self.__class__.degree = 12

x = FQ12([1] + [0] * 11)
f = FQ12([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
fpx = FQ12([2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
one = FQ12.one()

# Check that the field works fine
assert x + f == fpx
assert f / f == one
assert one / f + x / f == (one + x) / f
assert one * f + x * f == (one + x) * f
# This check takes too long
# assert x ** (field_modulus ** 12 - 1) == one