#include <bits/stdc++.h>
// created [2020/02/05] 11:35:57
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"

using i32   = int32_t;
using i64   = int64_t;
using u32   = uint32_t;
using u64   = uint64_t;
using uint  = unsigned int;
using usize = std::size_t;
using ll    = long long;
using ull   = unsigned long long;
using ld    = long double;
template<typename T, usize n>
using arr = T (&)[n];
template<typename T, usize n>
using c_arr = const T (&)[n];
template<typename T>
using max_heap = std::priority_queue<T>;
template<typename T>
using min_heap = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template<typename T> constexpr T popcount(const T u) { return u ? static_cast<T>(__builtin_popcountll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T log2p1(const T u) { return u ? static_cast<T>(64 - __builtin_clzll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T msbp1(const T u) { return log2p1(u); }
template<typename T> constexpr T lsbp1(const T u) { return __builtin_ffsll(u); }
template<typename T> constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast<T>(u); }
template<typename T> constexpr bool ispow2(const T u) { return u and (static_cast<u64>(u) & static_cast<u64>(u - 1)) == 0; }
template<typename T> constexpr T ceil2(const T u) { return static_cast<T>(1) << clog(u); }
template<typename T> constexpr T floor2(const T u) { return u == 0 ? static_cast<T>(0) : static_cast<T>(1) << (log2p1(u) - 1); }
template<typename T> constexpr bool btest(const T mask, const usize ind) { return static_cast<bool>((static_cast<u64>(mask) >> ind) & static_cast<u64>(1)); }
template<typename T> void bset(T& mask, const usize ind) { mask |= (static_cast<T>(1) << ind); }
template<typename T> void breset(T& mask, const usize ind) { mask &= ~(static_cast<T>(1) << ind); }
template<typename T> void bflip(T& mask, const usize ind) { mask ^= (static_cast<T>(1) << ind); }
template<typename T> void bset(T& mask, const usize ind, const bool b) { (b ? bset(mask, ind) : breset(mask, ind)); }
template<typename T> constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast<T>(0) : static_cast<T>((static_cast<u64>(mask) << (64 - ind)) >> (64 - ind)); }
template<typename T> bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); }
template<typename T> bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); }
constexpr unsigned int mod                  = 1000000007;
template<typename T> constexpr T inf_v      = std::numeric_limits<T>::max() / 4;
template<typename Real> constexpr Real pi_v = Real{3.141592653589793238462643383279502884};
auto mfp = [](auto&& f) { return [=](auto&&... args) { return f(f, std::forward<decltype(args)>(args)...); }; };

template<typename T>
T in()
{
    T v;
    return std::cin >> v, v;
}
template<typename T, typename Uint, usize n, usize i>
T in_v(typename std::enable_if<(i == n), c_arr<Uint, n>>::type) { return in<T>(); }
template<typename T, typename Uint, usize n, usize i>
auto in_v(typename std::enable_if<(i < n), c_arr<Uint, n>>::type& szs)
{
    const usize s = (usize)szs[i];
    std::vector<decltype(in_v<T, Uint, n, i + 1>(szs))> ans(s);
    for (usize j = 0; j < s; j++) { ans[j] = in_v<T, Uint, n, i + 1>(szs); }
    return ans;
}
template<typename T, typename Uint, usize n>
auto in_v(c_arr<Uint, n> szs) { return in_v<T, Uint, n, 0>(szs); }
template<typename... Types>
auto in_t() { return std::tuple<std::decay_t<Types>...>{in<Types>()...}; }
struct io_init
{
    io_init()
    {
        std::cin.tie(nullptr), std::ios::sync_with_stdio(false);
        std::cout << std::fixed << std::setprecision(20);
    }
    void clear()
    {
        std::cin.tie(), std::ios::sync_with_stdio(true);
    }
} io_setting;

int out() { return 0; }
template<typename T>
int out(const T& v) { return std::cout << v, 0; }
template<typename T>
int out(const std::vector<T>& v)
{
    for (usize i = 0; i < v.size(); i++) {
        if (i > 0) { std::cout << ' '; }
        out(v[i]);
    }
    return 0;
}
template<typename T1, typename T2>
int out(const std::pair<T1, T2>& v) { return out(v.first), std::cout << ' ', out(v.second), 0; }
template<typename T, typename... Args>
int out(const T& v, const Args... args) { return out(v), std::cout << ' ', out(args...), 0; }
template<typename... Args>
int outln(const Args... args) { return out(args...), std::cout << '\n', 0; }
template<typename... Args>
int outel(const Args... args) { return out(args...), std::cout << std::endl, 0; }
#    define SHOW(...) static_cast<void>(0)
constexpr ull TEN(const usize n) { return n == 0 ? 1ULL : TEN(n - 1) * 10ULL; }

template<typename T, typename Uint, usize n, usize i>
auto make_v(typename std::enable_if<(i == n), c_arr<Uint, n>>::type, const T& v = T{}) { return v; }
template<typename T, typename Uint, usize n, usize i>
auto make_v(typename std::enable_if<(i < n), c_arr<Uint, n>>::type szs, const T& v = T{})
{
    const usize s = (usize)szs[i];
    return std::vector<decltype(make_v<T, Uint, n, i + 1>(szs, v))>(s, make_v<T, Uint, n, i + 1>(szs, v));
}
template<typename T, typename Uint, usize n>
auto make_v(c_arr<Uint, n> szs, const T& t = T{}) { return make_v<T, Uint, n, 0>(szs, t); }


template<typename Real>
struct complex
{
    using value_type = Real;
    complex() : real{Real{0}}, imag{Real{0}} {}
    complex(const complex&) = default;
    complex(const Real& theta) : real(std::cos(theta)), imag(std::sin(theta)) {}
    complex(const Real& r, const Real& i) : real{r}, imag{i} {}
    ~complex() = default;
    friend complex operator+(const complex& c) { return c; }
    friend complex operator-(const complex& c) { return complex{-c.real, -c.imag}; }
    friend complex operator+(const complex& c1, const complex& c2) { return complex{c1.real + c2.real, c1.imag + c2.imag}; }
    friend complex operator-(const complex& c1, const complex& c2) { return complex{c1.real - c2.real, c1.imag - c2.imag}; }
    friend complex operator*(const complex& c1, const complex& c2) { return complex{c1.real * c2.real - c1.imag * c2.imag, c1.real * c2.imag + c1.imag * c2.real}; }
    friend complex operator*(const complex& c, const Real& r) { return complex{c.real * r, c.imag * r}; }
    friend complex operator/(complex& c1, complex& c2) { c1* c2.conj() / c2.norm(); }
    friend bool operator==(const complex& c1, const complex& c2) { return c1.real == c2.real and c1.imag == c2.imag; }
    friend bool operator!=(const complex& c1, const complex& c2) { return not(c1 == c2); }
    friend complex& operator+=(complex& c1, const complex& c2) { return c1.real += c2.real, c1.imag += c2.imag, c1; }
    friend complex& operator-=(complex& c1, const complex& c2) { return c1.real += c2.real, c1.imag += c2.imag, c1; }
    friend complex& operator*=(complex& c1, const complex& c2) { return c1 = c1 * c2; }
    friend complex& operator*=(complex& c, const Real& r) { return c = c * r; }
    friend complex& operator/=(complex& c1, const complex& c2) { return c1 = c1 / c2; }
    complex conj() const { return complex{real, -imag}; }
    Real norm() const { return real * real + imag * imag; }
    Real abs() const { return std::sqrt(norm()); }
    Real arg() const { return std::atan2(imag, real); }
    friend std::ostream& operator<<(std::ostream& os, const complex& c) { return os << c.real << "+" << c.imag << "i"; }
    Real real, imag;
};


template<typename T> T gcd(const T& a, const T& b) { return a < 0 ? gcd(-a, b) : b < 0 ? gcd(a, -b) : (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); }
template<typename T> T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; }
template<typename T>
constexpr std::pair<T, T> extgcd(const T a, const T b)
{
    if (b == 0) { return std::pair<T, T>{1, 0}; }
    const auto g = gcd(a, b), da = std::abs(b) / g;
    const auto p = extgcd(b, a % b);
    const auto x = (da + p.second % da) % da, y = (g - a * x) / b;
    return {x, y};
}
template<typename T>
constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; }
template<uint mod_value, bool dynamic = false>
class modint_base
{
public:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, const UInt> mod() { return mod_ref(); }
    template<typename UInt = uint>
    static constexpr std::enable_if_t<not dynamic, const UInt> mod() { return mod_value; }
    template<typename UInt = uint>
    static void set_mod(const std::enable_if_t<dynamic, const UInt> mod) { mod_ref() = mod, inv_ref() = {1, 1}; }
    modint_base() : v{0} {}
    modint_base(const ll val) : v{norm(static_cast<uint>(val % static_cast<ll>(mod()) + static_cast<ll>(mod())))} {}
    modint_base(const modint_base& n) : v{n()} {}
    explicit operator bool() const { return v != 0; }
    bool operator!() const { return not static_cast<bool>(*this); }
    modint_base& operator=(const modint_base& m) { return v = m(), (*this); }
    modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast<ll>(mod()) + static_cast<ll>(mod()))), (*this); }
    friend modint_base operator+(const modint_base& m) { return m; }
    friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); }
    friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); }
    friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); }
    friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast<uint>(static_cast<ll>(m1.v) * static_cast<ll>(m2.v) % static_cast<ll>(mod()))); }
    friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); }
    friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) - val}; }
    friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * inv(val)}; }
    friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast<ll>(m.v) + val}; }
    friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast<ll>(m.v))}; }
    friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; }
    friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; }
    friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; }
    friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; }
    friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; }
    friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; }
    friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; }
    friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; }
    friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); }
    friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; }
    friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; }
    friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); }
    friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); }
    friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); }
    friend std::istream& operator>>(std::istream& is, modint_base& m)
    {
        ll v;
        return is >> v, m = v, is;
    }
    friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); }
    uint operator()() const { return v; }
    static modint_base small_inv(const usize n)
    {
        auto& in = inv_ref();
        if (n < in.size()) { return in[n]; }
        for (usize i = in.size(); i <= n; i++) { in.push_back(-in[modint_base::mod() % i] * (modint_base::mod() / i)); }
        return in.back();
    }
    std::pair<ll, ll> quad() const
    {
        const auto ans = quad_r(v, mod());
        ll x = std::get<0>(ans), y = std::get<1>(ans);
        if (y < 0) { x = -x, y = -y; }
        return {x, y};
    }

private:
    static std::tuple<ll, ll, ll> quad_r(const ll r, const ll p)  // r = x/y (mod p), (x,y,z) s.t. x=yr+pz
    {
        if (std::abs(r) <= 1000) { return {r, 1, 0}; }
        ll nr = p % r, q = p / r;
        if (nr * 2LL >= r) { nr -= r, q++; }
        if (nr * 2LL <= -r) { nr += r, q--; }
        const auto sub = quad_r(nr, r);
        const ll x = std::get<0>(sub), z = std::get<1>(sub), y = std::get<2>(sub);
        return {x, y - q * z, z};
    }

    template<typename UInt = uint>
    static std::enable_if_t<dynamic, UInt&> mod_ref()
    {
        static UInt mod = 0;
        return mod;
    }
    static uint norm(const uint x) { return x < mod() ? x : x - mod(); }
    static modint_base make(const uint x)
    {
        modint_base m;
        return m.v = x, m;
    }
    static modint_base power(modint_base x, ull n)
    {
        modint_base ans = 1;
        for (; n; n >>= 1, x *= x) {
            if (n & 1) { ans *= x; }
        }
        return ans;
    }
    static modint_base inv(const ll v) { return v <= 2000000 ? small_inv(static_cast<usize>(v)) : modint_base{inverse(v, static_cast<ll>(mod()))}; }
    static std::vector<modint_base>& inv_ref()
    {
        static std::vector<modint_base> in{1, 1};
        return in;
    }
    uint v;
};
template<uint mod>
using modint = modint_base<mod, false>;
template<uint id>
using dynamic_modint = modint_base<id, true>;
template<typename Real = double>
class fft
{
private:
    static constexpr usize depth = 30;
    static constexpr Real pi     = pi_v<Real>;
    static void transform(std::vector<complex<Real>>& a, const usize lg, const bool rev)
    {
        static std::vector<complex<Real>> root[depth];
        const usize sz = a.size();
        assert((1UL << lg) == sz);
        if (root[lg].empty()) {
            root[lg].reserve(sz), root[lg].resize(sz);
            for (usize i = 0; i < sz; i++) { root[lg][i] = complex<Real>(pi * Real(2 * i) / Real(sz)); }
        }
        std::vector<complex<Real>> tmp(sz);
        for (usize w = (sz >> 1); w > 0; w >>= 1) {
            for (usize y = 0; y < (sz >> 1); y += w) {
                const complex<Real> r = rev ? root[lg][y].conj() : root[lg][y];
                for (usize x = 0; x < w; x++) {
                    const auto u = a[y << 1 | x], v = a[y << 1 | x | w] * r;
                    tmp[y | x] = u + v, tmp[y | x | (sz >> 1)] = u - v;
                }
            }
            std::swap(tmp, a);
        }
    }

public:
    using value_type = Real;
    fft()            = delete;
    template<typename T = ll, typename I = int>
    static std::vector<T> simple_convolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<Real>> x(sz), y(sz);
        for (usize i = 0; i < a.size(); i++) { x[i] = {(Real)a[i], (Real)0}; }
        for (usize i = 0; i < b.size(); i++) { y[i] = {(Real)b[i], (Real)0}; }
        transform(x, lg, false), transform(y, lg, false);
        for (usize i = 0; i < sz; i++) { x[i] *= y[i]; }
        transform(x, lg, true);
        std::vector<T> ans(need);
        for (usize i = 0; i < need; i++) { ans[i] = (T)std::round(x[i].real / (Real)sz); }
        return ans;
    }
    template<typename T = ll, usize division = 2, typename I = int>
    static std::vector<T> convolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        constexpr usize bitnum = (depth + division - 1) / division;
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<value_type>> x[division], y[division], tmp(sz);
        for (usize i = 0; i < division; i++) {
            x[i].reserve(sz), x[i].resize(sz), y[i].reserve(sz), y[i].resize(sz);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), complex<value_type>{});
            for (usize j = 0; j < a.size(); j++) { tmp[j].real = value_type((a[j] >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            for (usize j = 0; j < b.size(); j++) { tmp[j].imag = value_type((b[j] >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            transform(tmp, lg, false);
            for (usize j = 0; j < sz; j++) { tmp[j] *= value_type(0.5); }
            for (usize j = 0; j < sz; j++) {
                const usize k = j == 0 ? 0UL : sz - j;
                x[i][j] = complex<value_type>{tmp[j].real + tmp[k].real, tmp[j].imag - tmp[k].imag}, y[i][j] = complex<value_type>{tmp[j].imag + tmp[k].imag, -tmp[j].real + tmp[k].real};
            }
        }
        std::vector<complex<value_type>> z[division];
        for (usize i = 0; i < division; i++) { z[i].reserve(sz), z[i].resize(sz); }
        for (usize a = 0; a < division; a++) {
            for (usize b = 0; b < division; b++) {
                for (usize i = 0; i < sz; i++) {
                    if (a + b < division) {
                        z[a + b][i] += x[a][i] * y[b][i];
                    } else {
                        z[a + b - division][i] += x[a][i] * y[b][i] * complex<value_type>(0, 1);
                    }
                }
            }
        }
        for (usize i = 0; i < division; i++) { transform(z[i], lg, true); }
        std::vector<T> ans(need);
        T base = 1;
        for (usize k = 0; k < 2 * division - 1; k++, base *= (1LL << bitnum)) {
            for (usize i = 0; i < need; i++) {
                if (k < division) {
                    ans[i] += base * T(std::round(z[k][i].real / value_type(sz)));
                } else {
                    ans[i] += base * T(std::round(z[k - division][i].imag / value_type(sz)));
                }
            }
        }
        return ans;
    }
    template<uint mod, bool dynamic = false, usize division = 2>
    static std::vector<modint_base<mod, dynamic>> convolute(const std::vector<modint_base<mod, dynamic>>& a, const std::vector<modint_base<mod, dynamic>>& b)
    {
        using mint             = modint_base<mod, dynamic>;
        constexpr usize bitnum = (depth + division - 1) / division;
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<value_type>> x[division], y[division], tmp(sz);
        for (usize i = 0; i < division; i++) {
            x[i].reserve(sz), x[i].resize(sz), y[i].reserve(sz), y[i].resize(sz);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), complex<value_type>{});
            for (usize j = 0; j < a.size(); j++) { tmp[j].real = value_type((a[j]() >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            for (usize j = 0; j < b.size(); j++) { tmp[j].imag = value_type((b[j]() >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            transform(tmp, lg, false);
            for (usize j = 0; j < sz; j++) { tmp[j] *= value_type(0.5); }
            for (usize j = 0; j < sz; j++) {
                const usize k = j == 0 ? 0UL : sz - j;
                x[i][j] = complex<value_type>{tmp[j].real + tmp[k].real, tmp[j].imag - tmp[k].imag}, y[i][j] = complex<value_type>{tmp[j].imag + tmp[k].imag, -tmp[j].real + tmp[k].real};
            }
        }
        std::vector<complex<value_type>> z[division];
        for (usize i = 0; i < division; i++) { z[i].reserve(sz), z[i].resize(sz); }
        for (usize a = 0; a < division; a++) {
            for (usize b = 0; b < division; b++) {
                for (usize i = 0; i < sz; i++) {
                    if (a + b < division) {
                        z[a + b][i] += x[a][i] * y[b][i];
                    } else {
                        z[a + b - division][i] += x[a][i] * y[b][i] * complex<value_type>(0, 1);
                    }
                }
            }
        }
        for (usize i = 0; i < division; i++) { transform(z[i], lg, true); }
        std::vector<mint> ans(need);
        mint base = 1;
        for (usize k = 0; k < 2 * division - 1; k++, base *= (1LL << bitnum)) {
            for (usize i = 0; i < need; i++) {
                if (k < division) {
                    ans[i] += int((base * ll(std::round(z[k][i].real / value_type(sz))))());
                } else {
                    ans[i] += int((base * ll(std::round(z[k - division][i].imag / value_type(sz))))());
                }
            }
        }
        return ans;
    }
};
template<usize bucket_digits = 7, usize fft_division = 2>
class bigint
{
public:
    using bucket_type                         = ll;
    static constexpr bucket_type bucket_value = TEN(bucket_digits);
    bigint() {}
    bigint(const bucket_type x) : sign(x >= 0), d(x == 0 ? 0 : 1, x < 0 ? -x : x) { normalize(); }
    bigint(const bool sign, const std::vector<bucket_type>& d) : sign(sign), d(d) { normalize(); }
    bigint(const bigint& n) : sign(n.sign), d(n.d) {}
    bigint(const std::string& str)
    {
        assert(not str.empty());
        sign = str[0] != '-';
        for (usize i = 0; i < str.size(); i += bucket_digits) {
            const usize r = str.size() - i, l = (sign ? (r >= bucket_digits ? r - bucket_digits : 0) : (r > bucket_digits ? r - bucket_digits : 1));
            if (r > l) { d.push_back(static_cast<bucket_type>(std::stoll(str.substr(l, r - l)))); }
        }
    }
    friend bigint operator+(const bigint& m) { return m; }
    friend bigint operator-(const bigint& m) { return bigint{not m.sign, m.d}; }
    explicit operator bool() const { return not d.empty(); }
    bool operator!() const { return not static_cast<bool>(*this); }
    friend bigint operator+(const bigint& m, const bigint& n)
    {
        if (m.sign == n.sign) {
            usize sz = std::max(m.d.size(), n.d.size());
            std::vector<bucket_type> v(sz, 0);
            for (usize i = 0; i < m.d.size(); i++) { v[i] += m.d[i]; }
            for (usize i = 0; i < n.d.size(); i++) { v[i] += n.d[i]; }
            return bigint{m.sign, v};
        } else {
            usize sz = std::max(m.d.size(), n.d.size());
            std::vector<bucket_type> v(sz, 0);
            if (abs_comp(m, n) == -1) {
                for (usize i = 0; i < m.d.size(); i++) { v[i] -= m.d[i]; }
                for (usize i = 0; i < n.d.size(); i++) { v[i] += n.d[i]; }
                return bigint{n.sign, v};
            } else {
                for (usize i = 0; i < m.d.size(); i++) { v[i] += m.d[i]; }
                for (usize i = 0; i < n.d.size(); i++) { v[i] -= n.d[i]; }
                return bigint{m.sign, v};
            }
        }
    }
    friend bigint operator-(const bigint& m, const bigint& n) { return m + (-n); }
    friend bigint operator*(const bigint& m, const bigint& n) { return bigint{not(m.sign ^ n.sign), fft<>::convolute<bucket_type, fft_division, bucket_type>(m.d, n.d)}; }
    friend bigint operator/(const bigint& m, const bigint& n) { return div(m, n).first; }
    friend bigint operator%(const bigint& m, const bigint& n) { return div(m, n).second; }
    friend bigint operator^(const bigint& m, const bigint& n) { return n == 0 ? bigint{1} : n % 2 == 1 ? (m ^ (n - 1)) * m : (m * m) ^ (n / 2); }
    friend bool operator<(const bigint& m, const bigint& n) { return comp(m, n) == -1; }
    friend bool operator>(const bigint& m, const bigint& n) { return comp(m, n) == 1; }
    friend bool operator==(const bigint& m, const bigint& n) { return comp(m, n) == 0; }
    friend bool operator!=(const bigint& m, const bigint& n) { return not(m == n); }
    friend bool operator<=(const bigint& m, const bigint& n) { return not(m > n); }
    friend bool operator>=(const bigint& m, const bigint& n) { return not(m < n); }
    friend bigint operator<<(const bigint& m, const usize n)  // *(B^n)
    {
        std::vector<bucket_type> ans(m.d.size() + n, 0);
        for (usize i = 0; i < m.d.size(); i++) { ans[i + n] = m.d[i]; }
        return bigint{m.sign, ans};
    }
    friend bigint operator>>(const bigint& m, const usize n)  // /(B^n)
    {
        if (m.d.size() <= n) { return 0; }
        std::vector<bucket_type> ans(m.d.size() - n, 0);
        for (usize i = 0; i < m.d.size() - n; i++) { ans[i] = m.d[i + n]; }
        return bigint{m.sign, ans};
    }
    bigint& operator=(const bigint& n) { return sign = n.sign, d = n.d, *this; }
    bigint& operator=(const bucket_type n) { return *this = bigint{n}; }
    friend bigint& operator+=(bigint& m, const bigint& n) { return m = m + n; }
    friend bigint& operator-=(bigint& m, const bigint& n) { return m = m - n; }
    friend bigint& operator*=(bigint& m, const bigint& n) { return m = m * n; }
    friend bigint& operator/=(bigint& m, const bigint& n) { return m = m / n; }
    friend bigint& operator%=(bigint& m, const bigint& n) { return m = m % n; }
    friend bigint& operator^=(bigint m, const bigint& n) { return m = m ^ n; }
    friend bigint& operator<<=(bigint& m, const usize n) { return m = m << n; }
    friend bigint& operator>>=(bigint& m, const usize n) { return m = m >> n; }
    friend bigint operator+(const bigint& m, const bucket_type n) { return m + bigint{n}; }
    friend bigint operator-(const bigint& m, const bucket_type n) { return m - bigint{n}; }
    friend bigint operator*(const bigint& m, const bucket_type n) { return small_multiply(m, n); }
    friend bigint operator/(const bigint& m, const bucket_type n) { return m / bigint{n}; }
    friend bigint operator%(const bigint& m, const bucket_type n) { return m % bigint{n}; }
    friend bool operator<(const bigint& m, const bucket_type n) { return comp(m, bigint{n}) == -1; }
    friend bool operator>(const bigint& m, const bucket_type n) { return comp(m, bigint{n}) == 1; }
    friend bool operator==(const bigint& m, const bucket_type n) { return comp(m, bigint{n}) == 0; }
    friend bool operator!=(const bigint& m, const bucket_type n) { return not(m == n); }
    friend bool operator<=(const bigint& m, const bucket_type n) { return not(m > n); }
    friend bool operator>=(const bigint& m, const bucket_type n) { return not(m < n); }
    friend bigint& operator+=(const bigint& m, const bucket_type n) { return m += bigint{n}; }
    friend bigint& operator-=(const bigint& m, const bucket_type n) { return m -= bigint{n}; }
    friend bigint& operator*=(const bigint& m, const bucket_type n) { return m *= bigint{n}; }
    friend bigint& operator/=(const bigint& m, const bucket_type n) { return m /= bigint{n}; }
    friend bigint& operator%=(const bigint& m, const bucket_type n) { return m %= bigint{n}; }
    friend bigint operator+(const bucket_type m, const bigint& n) { return bigint{m} + n; }
    friend bigint operator-(const bucket_type m, const bigint& n) { return bigint{m} - n; }
    friend bigint operator*(const bucket_type m, const bigint& n) { return bigint{m} * n; }
    friend bigint operator/(const bucket_type m, const bigint& n) { return bigint{m} / n; }
    friend bigint operator%(const bucket_type m, const bigint& n) { return bigint{m} % n; }
    friend bool operator<(const bucket_type m, const bigint& n) { return comp(bigint{m}, n) == -1; }
    friend bool operator>(const bucket_type m, const bigint& n) { return comp(bigint{m}, n) == 1; }
    friend bool operator==(const bucket_type m, const bigint& n) { return comp(bigint{m}, n) == 0; }
    friend bool operator!=(const bucket_type m, const bigint& n) { return not(m == n); }
    friend bool operator<=(const bucket_type m, const bigint& n) { return not(m > n); }
    friend bool operator>=(const bucket_type m, const bigint& n) { return not(m < n); }
    bigint& operator++() { return *this += bigint{1}; }
    bigint operator++(int) { return std::exchange(*this, *this + bigint{1}); }
    bigint& operator--() { return *this -= bigint{1}; }
    bigint operator--(int) { return std::exchange(*this, *this - bigint{1}); }
    friend std::pair<bigint, bigint> div(bigint m, bigint n)
    {
        assert(not n.d.empty());
        if (abs_comp(m, n) == -1) { return {0, m}; }
        const bool ms = m.sign, ns = n.sign;
        m.sign = n.sign      = true;
        const usize sho_size = m.d.size() - n.d.size();
        std::vector<bucket_type> sho(sho_size + 1);
        for (usize i = 0; i <= sho_size; i++) {
            const usize ind = sho_size - i;
            const auto L    = m >> ind;
            bucket_type inf = -1, sup = bucket_value;
            while (sup - inf > 1) {
                const bucket_type mid                                   = (sup + inf) >> 1;
                (abs_comp(L, small_multiply(n, mid)) == -1 ? sup : inf) = mid;
            }
            sho[ind] = inf, m -= (small_multiply(n, inf) << ind);
        }
        return m.sign = ms, std::pair<bigint, bigint>{bigint{not(ms ^ ns), sho}, m};
    }
    friend std::istream& operator>>(std::istream& is, bigint& n)
    {
        std::string str;
        is >> str;
        return n = bigint{str}, is;
    }
    friend std::ostream& operator<<(std::ostream& os, const bigint& n)
    {
        if (n.d.empty()) { return os << '0'; }
        if (n.sign == false) { os << '-'; }
        const usize sz = n.d.size();
        for (usize i = 0; i < sz; i++) {
            if (i == 0) {
                os << n.d[sz - i - 1];
            } else {
                usize nz = 0;
                for (bucket_type base = 1; n.d[sz - i - 1] >= base; nz++, base *= bucket_type(10)) {}
                for (usize i = 0; i + nz < bucket_digits; i++) { os << '0'; }
                if (n.d[sz - i - 1] != 0LL) { os << n.d[sz - i - 1]; }
            }
        }
        return os;
    }
    friend usize log10p1(const bigint& m)
    {
        if (m.d.empty()) { return 0; }
        assert(m.sign);
        return (m.d.size() - 1) * bucket_digits + log10p1(m.d.back());
    }
    friend bigint abs(const bigint& m)
    {
        auto ans        = m;
        return ans.sign = true, ans;
    }
    long long to_ll() const
    {
        long long ans = 0, base = 1;
        for (usize i = 0; i < d.size(); i++, base *= bucket_value) { ans += base * d[i]; }
        return sign ? ans : -ans;
    }
    std::string to_string() const
    {
        if (d.empty()) { return "0"; }
        std::string ans;
        if (sign == false) { ans.push_back('-'); }
        const usize N = d.size();
        for (usize i = 0; i < N; i++) {
            if (i == 0) {
                ans += std::to_string(d[N - i - 1]);
            } else {
                usize nz = 0;
                for (bucket_type U = 1; d[N - i - 1] >= U; nz++, U *= bucket_type(10)) {}
                for (usize i = 0; i + nz < bucket_digits; i++) { ans.push_back('0'); }
                if (d[N - i - 1] != 0LL) { ans += std::to_string(d[N - i - 1]); }
            }
        }
        return ans;
    }

private:
    friend bigint small_multiply(const bigint& m, bucket_type n)  // [TODO] karatsuba
    {
        const bool s = n < 0 ? not m.sign : m.sign;
        n            = n < 0 ? -n : n;
        assert(n < bucket_value);
        std::vector<bucket_type> ans(m.d.size());
        for (usize i = 0; i < m.d.size(); i++) { ans[i] = m.d[i] * n; }
        return bigint{s, ans};
    }
    friend int comp(const bigint& m, const bigint& n)
    {
        if (m.sign != n.sign) {
            return m.sign ? 1 : -1;
        } else {
            return m.sign ? abs_comp(m, n) : -abs_comp(m, n);
        }
    }
    static usize log10p1(const bucket_type n)
    {
        usize ans = 0;
        for (bucket_type base = 1; base <= n; base *= static_cast<bucket_type>(10), ans++) {}
        return ans;
    }
    friend int abs_comp(const bigint& m, const bigint& n)  // m<n:-1  m=n:0  m>n:1
    {
        if (m.d.size() != n.d.size()) {
            return m.d.size() < n.d.size() ? -1 : 1;
        } else {
            for (usize i = 0; i < m.d.size(); i++) {
                const auto m_back = m.d[m.d.size() - i - 1], n_back = n.d[n.d.size() - i - 1];
                if (m_back != n_back) { return m_back < n_back ? -1 : 1; }
            }
        }
        return 0;
    }
    void normalize()
    {
        for (usize i = 0; i < d.size(); i++) {
            if (d[i] >= bucket_value) {
                if (i + 1 == d.size()) {
                    d.push_back(d[i] / bucket_value);
                } else {
                    d[i + 1] += d[i] / bucket_value;
                }
                d[i] %= bucket_value;
            } else if (d[i] < 0) {
                assert(i + 1 != d.size());
                while (d[i] < 0) { d[i + 1]--, d[i] += bucket_value; }
            }
        }
        for (; not d.empty() and d.back() == 0; d.pop_back()) {}
    }
    bool sign = true;
    std::vector<bucket_type> d;
};
int main()
{
    const auto N = in<bigint<>>();
    auto pow     = mfp([&](auto&& self, const bigint<>& x, const bigint<>& n) -> bigint<> {
        if (n == 0) { return 1; }
        if (n % 2 == 1) {
            return self(self, x, n - 1) * x % N;
        } else {
            return self(self, x * x % N, n / 2);
        }
    });
    if (N % 2 == 0) { return outel('!', N / 2, 2); }
    for (bigint a = 2; a * a <= N; a++) {
        if (N % a == 0) { return outel('!', N / a, a); }
        outel('?', a);
        const auto p = in<bigint<>>();
        if (p % 2 == 1) { continue; }
        const auto b = pow(a, p / 2) + 1;
        if (b != N and b != 1 and N % b == 0) { return outel('!', N / b, b); }
    }
    return 0;
}