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Big Integer(Float)
(Math/bigint.hpp)

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• Last update: 2026-01-30 05:37:33+09:00
• Include: #include "Math/bigint.hpp"

Depends on

• Arbitrary Mod Convolution (Convolution/arbitrary.hpp)
• Number Theoretic Transform (Convolution/ntt.hpp)
• Combination (Math/comb.hpp)
• Fast Division (Math/fastdiv.hpp)
• Miller-Rabin (Math/miller.hpp)
• Modint (Math/modint.hpp)
• Pollard-Rho (Math/pollard.hpp)
• Primitive Function (Math/primitive.hpp)
• Random (Utility/random.hpp)

Code

#pragma once
#include "Convolution/arbitrary.hpp"


template <int D> struct bigint {
    using u128 = __uint128_t;
    static const int B = pow(10, D);
    int sign = 0;
    vector<int> v;
    static int get_D() { return D; }
    static int get_B() { return B; }
    bigint() {}
    bigint(const vector<int> &_v, bool _s = false) : sign(_s), v(_v) {}
    bigint(ll x) {
        if (x < 0)
            x *= -1, sign = 1;
        while (x) {
            v.push_back(x % B);
            x /= B;
        }
    }
    bigint(string s) {
        if (s[0] == '-')
            s.erase(s.begin()), sign = 1;
        int add = 0, cnt = 0, base = 1;
        while (s.size()) {
            if (cnt == D) {
                v.push_back(add);
                cnt = 0;
                add = 0;
                base = 1;
            }
            add = (s.back() - '0') * base + add;
            cnt++;
            base *= 10;
            s.pop_back();
        }
        if (add)
            v.push_back(add);
    }
    bigint operator-() const {
        bigint res = *this;
        res.sign ^= 1;
        return res;
    }
    bigint abs() const {
        bigint res = *this;
        res.sign = 0;
        return res;
    }
    int &operator[](const int i) { return v[i]; }
    int size() const { return v.size(); }
    void norm() {
        rep(i, 0, v.size() - 1) {
            if (v[i] >= 0) {
                v[i + 1] += v[i] / B;
                v[i] %= B;
            } else {
                int c = (-v[i] + B - 1) / B;
                v[i] += c * B;
                v[i + 1] -= c;
            }
        }
        while (!v.empty() and v.back() >= B) {
            int c = v.back() / B;
            v.back() %= B;
            v.push_back(c);
        }
        while (!v.empty() and v.back() == 0)
            v.pop_back();
    }
    string to_str() const {
        string res;
        if (v.empty())
            return "0";
        if (sign)
            res += '-';
        res += to_string(v.back());
        for (int i = v.size() - 2; i >= 0; i--) {
            string add;
            int w = v[i];
            rep(_, 0, D) {
                add += ('0' + (w % 10));
                w /= 10;
            }
            reverse(ALL(add));
            res += add;
        }
        return res;
    }
    friend istream &operator>>(istream &is, bigint<D> &x) {
        string tmp;
        is >> tmp;
        x = bigint(tmp);
        return is;
    }
    friend ostream &operator<<(ostream &os, bigint<D> x) {
        os << x.to_str();
        return os;
    }
    bigint &operator<<=(const int &x) {
        if (!v.empty()) {
            vector<int> add(x, 0);
            v.insert(v.begin(), ALL(add));
        }
        return *this;
    }
    bigint &operator>>=(const int &x) {
        v = vector<int>(v.begin() + min(x, (int)v.size()), v.end());
        return *this;
    }
    bigint &operator+=(const bigint &x) {
        if (sign != x.sign) {
            *this -= (-x);
            return *this;
        }
        if ((int)v.size() < (int)x.size())
            v.resize(x.size(), 0);
        rep(i, 0, x.size()) { v[i] += x.v[i]; }
        norm();
        return *this;
    }
    bigint &operator-=(const bigint &x) {
        if (sign != x.sign) {
            *this += (-x);
            return *this;
        }
        if (abs() < x.abs()) {
            *this = x - (*this);
            sign ^= 1;
            return *this;
        }
        rep(i, 0, x.size()) { v[i] -= x.v[i]; }
        norm();
        return *this;
    }
    bigint &operator*=(const bigint &x) {
        sign ^= x.sign;
        auto v1 = ArbitraryMult<u128>(v, x.v);
        u128 add = 0;
        v.clear();
        v.reserve(v1.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v1.size()) and add == 0)
                break;
            if (i < int(v1.size()))
                add += v1[i];
            v.push_back(add % B);
            add /= B;
        }
        norm();
        return *this;
    }
    bigint div_naive(const bigint &a, const bigint &b) {
        if (SZ(b) == 1)
            return a.div(b.v.back());
        if (a < b)
            return bigint();
        int norm = B / (b.v.back() + 1);
        bigint x = a.mul(norm), y = b.mul(norm);
        int yb = y.v.back();
        bigint quo, rem;
        quo.v.resize(x.size() - y.size() + 1);
        rem.v = {x.v.end() - y.size(), x.v.end()};
        for (int i = x.size() - y.size(); i >= 0; i--) {
            if (rem.size() == y.size()) {
                if (rem >= y) {
                    quo[i] = 1;
                    rem -= y;
                }
            } else if (rem.size() > y.size()) {
                ll rb = ll(rem.v.back()) * B + rem[rem.size() - 2];
                int q = rb / yb;
                bigint yq = y.mul(q);
                while (rem < yq) {
                    q--;
                    yq -= y;
                }
                rem -= yq;
                while (rem >= y) {
                    q++;
                    rem -= y;
                }
                quo[i] = q;
            }
            if (i)
                rem.v.insert(rem.v.begin(), x[i - 1]);
        }
        return quo;
    }
    bigint &operator/=(const bigint &x) {
        bigint a = abs(), b = x.abs();
        sign ^= x.sign;
        if (a < b)
            return *this = bigint();
        if (b.size() == 1)
            return *this = a.div(b.v.back());

        int deg = a.size() - b.size() + 2, k = deg;
        while (k > 64)
            k = (k + 1) >> 1;
        bigint base(1);
        base <<= (b.size() + k);
        bigint inv(div_naive(base, b));

        while (k < deg) {
            bigint y = inv.square();
            y.v.insert(y.v.begin(), 0);
            int l = min(SZ(b), k * 2 + 1);
            bigint pref;
            pref.v = {b.v.end() - l, b.v.end()};
            y *= pref;
            y >>= l;
            inv = inv + inv;
            inv <<= k + 1;
            inv -= y;
            inv.v.erase(inv.v.begin());
            k <<= 1;
        }
        inv >>= (k - deg);

        (*this) = a * inv;
        (*this) >>= int(a.size() + 2);
        bigint mul = (*this) * b;
        while (mul + b <= a) {
            (*this) += bigint(1);
            mul += b;
        }
        while (mul > a) {
            (*this) -= bigint(1);
            mul -= b;
        }
        return *this;
    }
    bigint &operator%=(const bigint &x) {
        bigint div = (*this) / x;
        (*this) -= div * x;
        return *this;
    }
    bigint square() const {
        bigint ret;
        auto v1 = ArbitraryMult<u128>(v, v);
        u128 add = 0;
        ret.v.reserve(v1.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v1.size()) and add == 0)
                break;
            if (i < int(v1.size()))
                add += v1[i];
            ret.v.push_back(add % B);
            add /= B;
        }
        return ret;
    }
    bigint mul(ll x) const {
        bigint res;
        if (x < 0)
            res.sign ^= 1, x *= -1;
        u128 add = 0;
        res.v.reserve(v.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v.size()) and add == 0)
                break;
            if (i < int(v.size()))
                add += ll(v[i]) * x;
            res.v.push_back(add % B);
            add /= B;
        }
        return res;
    }
    bigint div(ll x) const {
        bigint res = *this;
        if (x < 0)
            res.sign ^= 1, x *= -1;
        ll add = 0;
        for (int i = res.v.size() - 1; i >= 0; i--) {
            add = add * B + res.v[i];
            int q = add / x, r = add % x;
            res.v[i] = q, add = r;
        }
        res.norm();
        return res;
    }
    bigint operator<<(const int &x) const { return bigint(*this) <<= x; }
    bigint operator>>(const int &x) const { return bigint(*this) >>= x; }
    bigint operator+(const bigint &x) const { return bigint(*this) += x; }
    bigint operator-(const bigint &x) const { return bigint(*this) -= x; }
    bigint operator*(const bigint &x) const { return bigint(*this) *= x; }
    bigint operator/(const bigint &x) const { return bigint(*this) /= x; }
    bigint operator%(const bigint &x) const { return bigint(*this) %= x; }
    bool operator<(const bigint &x) const {
        if (sign != x.sign)
            return sign > x.sign;
        if ((int)v.size() != (int)x.size()) {
            if (sign)
                return (int)x.size() < (int)v.size();
            else
                return (int)v.size() < (int)x.size();
        }
        for (int i = v.size() - 1; i >= 0; i--)
            if (v[i] != x.v[i]) {
                if (sign)
                    return x.v[i] < v[i];
                else
                    return v[i] < x.v[i];
            }
        return false;
    }
    bool operator>(const bigint &x) const { return x < *this; }
    bool operator<=(const bigint &x) const { return !(*this > x); }
    bool operator>=(const bigint &x) const { return !(*this < x); }
    bool operator==(const bigint &x) const {
        return !(*this < x) and !(*this > x);
    }
    bool operator!=(const bigint &x) const { return !(*this == x); }
};
typedef bigint<9> Bigint;

struct Bigfloat {
    Bigint v;
    int p = 0;
    Bigfloat() {}
    Bigfloat(const ll &_v) { v = Bigint(_v); }
    Bigfloat(const Bigint &_v, int _p = 0) : v(_v), p(_p) {}
    void set(int _p) {
        if (p < _p) {
            v >>= (_p - p);
        } else {
            v <<= (p - _p);
        }
        p = _p;
    }
    Bigint to_int() const {
        if (p < 0)
            return v >> (-p);
        else
            return v << p;
    }
    Bigfloat &operator+=(const Bigfloat &x) {
        if (p > x.p)
            set(x.p), v += x.v;
        else
            v += x.v << (x.p - p);
        return *this;
    }
    Bigfloat &operator-=(const Bigfloat &x) {
        if (p > x.p)
            set(x.p), v -= x.v;
        else
            v -= x.v << (x.p - p);
        return *this;
    }
    Bigfloat square() {
        Bigfloat res = *this;
        res.p *= 2;
        res.v = res.v.square();
        return res;
    }
    Bigfloat mul(ll x) {
        Bigfloat res = *this;
        res.v = v.mul(x);
        return res;
    }
    Bigfloat div(ll x) {
        Bigfloat res = *this;
        res.v = v.div(x);
        return res;
    }
    Bigfloat &operator*=(const Bigfloat &x) {
        p += x.p;
        v *= x.v;
        return *this;
    }
    Bigfloat &operator/=(const Bigfloat &x) {
        p -= x.p;
        v /= x.v;
        return *this;
    }
    Bigfloat operator+(const Bigfloat &x) const { return Bigfloat(*this) += x; }
    Bigfloat operator-(const Bigfloat &x) const { return Bigfloat(*this) -= x; }
    Bigfloat operator*(const Bigfloat &x) const { return Bigfloat(*this) *= x; }
    Bigfloat operator/(const Bigfloat &x) const { return Bigfloat(*this) /= x; }
    string to_str() {
        string res = v.abs().to_str();
        int d = Bigint::get_D();
        if (p * d > 0)
            res += string(p * d, '0');
        else if (-p * d >= 1 and -p * d < (int)res.size()) {
            res = res.substr(0, (int)res.size() + p * d) + '.' +
                  res.substr((int)res.size() + p * d);
        } else if (-p * d >= (int)res.size())
            res = "0." + string(-p * d - (int)res.size(), '0') + res;
        if (v.sign) {
            res.insert(res.begin(), '-');
        }
        return res;
    }
    friend ostream &operator<<(ostream &os, Bigfloat x) {
        os << x.to_str();
        return os;
    }
};

Bigfloat sqrt(ll n, int d) {
    Bigfloat res(Bigint((ll)sqrt(1LL * Bigint::get_B() * Bigint::get_B() / n)),
                 -1),
        pre;
    int cur = 1;
    while (pre.v != res.v) {
        cur = min(cur << 1, d);
        pre = res;
        Bigfloat add = Bigfloat(1) - res.square().mul(n);
        add.set(-cur);
        res += (res * add).div(2);
        res.set(-cur);
    }
    return res.mul(n);
}

/**
 * @brief Big Integer(Float)
 */

#line 2 "Math/fastdiv.hpp"

struct FastDiv {
    using u64 = uint64_t;
    using u128 = __uint128_t;
    constexpr FastDiv() : m(), s(), x() {}
    constexpr FastDiv(int _m)
        : m(_m), s(__lg(m - 1)), x(((u128(1) << (s + 64)) + m - 1) / m) {}
    constexpr int get() {
        return m;
    }
    constexpr friend u64 operator/(u64 n, const FastDiv &d) {
        return (u128(n) * d.x >> d.s) >> 64;
    }
    constexpr friend int operator%(u64 n, const FastDiv &d) {
        return n - n / d * d.m;
    }
    constexpr pair<u64, int> divmod(u64 n) const {
        u64 q = n / (*this);
        return {q, n - q * m};
    }
    int m, s;
    u64 x;
};

struct FastDiv64 {
    using u64 = uint64_t;
    using u128 = __uint128_t;
    u128 mod, mh, ml;
    explicit FastDiv64(u64 mod = 1) : mod(mod) {
        u128 m = u128(-1) / mod;
        if (m * mod + mod == u128(0))
            ++m;
        mh = m >> 64;
        ml = m & u64(-1);
    }
    u64 umod() const {
        return mod;
    }
    u64 modulo(u128 x) {
        u128 z = (x & u64(-1)) * ml;
        z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64);
        z = (x >> 64) * mh + (z >> 64);
        x -= z * mod;
        return x < mod ? x : x - mod;
    }
    u64 mul(u64 a, u64 b) {
        return modulo(u128(a) * b);
    }
};

/**
 * @brief Fast Division
 */
#line 2 "Math/miller.hpp"

struct m64 {
    using i64 = int64_t;
    using u64 = uint64_t;
    using u128 = __uint128_t;

    static u64 mod;
    static u64 r;
    static u64 n2;

    static u64 get_r() {
        u64 ret = mod;
        rep(_,0,5) ret *= 2 - mod * ret;
        return ret;
    }

    static void set_mod(u64 m) {
        assert(m < (1LL << 62));
        assert((m & 1) == 1);
        mod = m;
        n2 = -u128(m) % m;
        r = get_r();
        assert(r * mod == 1);
    }
    static u64 get_mod() { return mod; }

    u64 a;
    m64() : a(0) {}
    m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){};

    static u64 reduce(const u128 &b) {
        return (b + u128(u64(b) * u64(-r)) * mod) >> 64;
    }
    u64 get() const {
        u64 ret = reduce(a);
        return ret >= mod ? ret - mod : ret;
    }
    m64 &operator*=(const m64 &b) {
        a = reduce(u128(a) * b.a);
        return *this;
    }
    m64 operator*(const m64 &b) const { return m64(*this) *= b; }
    bool operator==(const m64 &b) const {
        return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
    }
    bool operator!=(const m64 &b) const {
        return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
    }
    m64 pow(u128 n) const {
        m64 ret(1), mul(*this);
        while (n > 0) {
        if (n & 1) ret *= mul;
        mul *= mul;
        n >>= 1;
        }
        return ret;
    }
};
typename m64::u64 m64::mod, m64::r, m64::n2;

bool Miller(ll n){
    if(n<2 or (n&1)==0)return (n==2);
    m64::set_mod(n);
    ll d=n-1; while((d&1)==0)d>>=1;
    vector<ll> seeds;
    if(n<(1<<30))seeds={2, 7, 61};
    else seeds={2, 325, 9375, 28178, 450775, 9780504};
    for(auto& x:seeds){
        if(n<=x)break;
        ll t=d;
        m64 y=m64(x).pow(t);
        while(t!=n-1 and y!=1 and y!=n-1){
            y*=y;
            t<<=1;
        }
        if(y!=n-1 and (t&1)==0)return 0;
    } return 1;
}

/**
 * @brief Miller-Rabin
 */
#line 2 "Utility/random.hpp"

namespace Random {
mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count());
using u64 = unsigned long long;
u64 get() {
    return randgen();
}
template <typename T> T get(T L) { // [0,L]

    return get() % (L + 1);
}
template <typename T> T get(T L, T R) { // [L,R]

    return get(R - L) + L;
}
double uniform() {
    return double(get(1000000000)) / 1000000000;
}
string str(int n) {
    string ret;
    rep(i, 0, n) ret += get('a', 'z');
    return ret;
}
template <typename Iter> void shuffle(Iter first, Iter last) {
    if (first == last)
        return;
    int len = 1;
    for (auto it = first + 1; it != last; it++) {
        len++;
        int j = get(0, len - 1);
        if (j != len - 1)
            iter_swap(it, first + j);
    }
}
template <typename T> vector<T> select(int n, T L, T R) { // [L,R]

    if (n * 2 >= R - L + 1) {
        vector<T> ret(R - L + 1);
        iota(ALL(ret), L);
        shuffle(ALL(ret));
        ret.resize(n);
        return ret;
    } else {
        unordered_set<T> used;
        vector<T> ret;
        while (SZ(used) < n) {
            T x = get(L, R);
            if (!used.count(x)) {
                used.insert(x);
                ret.push_back(x);
            }
        }
        return ret;
    }
}

void relabel(int n, vector<pair<int, int>> &es) {
    shuffle(ALL(es));
    vector<int> ord(n);
    iota(ALL(ord), 0);
    shuffle(ALL(ord));
    for (auto &[u, v] : es)
        u = ord[u], v = ord[v];
}
template <bool directed, bool multi, bool self>
vector<pair<int, int>> genGraph(int n, int m) {
    vector<pair<int, int>> cand, es;
    rep(u, 0, n) rep(v, 0, n) {
        if (!self and u == v)
            continue;
        if (!directed and u > v)
            continue;
        cand.push_back({u, v});
    }
    if (m == -1)
        m = get(SZ(cand));
    // chmin(m, SZ(cand));

    vector<int> ord;
    if (multi)
        rep(_, 0, m) ord.push_back(get(SZ(cand) - 1));
    else {
        ord = select(m, 0, SZ(cand) - 1);
    }
    for (auto &i : ord)
        es.push_back(cand[i]);
    relabel(n, es);
    return es;
}
vector<pair<int, int>> genTree(int n) {
    vector<pair<int, int>> es;
    rep(i, 1, n) es.push_back({get(i - 1), i});
    relabel(n, es);
    return es;
}
}; // namespace Random


/**
 * @brief Random
 */
#line 4 "Math/pollard.hpp"

vector<ll> Pollard(ll n) {
    if (n <= 1)
        return {};
    if (Miller(n))
        return {n};
    if ((n & 1) == 0) {
        vector<ll> v = Pollard(n >> 1);
        v.push_back(2);
        return v;
    }
    for (ll x = 2, y = 2, d;;) {
        ll c = Random::get(2LL, n - 1);
        do {
            x = (__int128_t(x) * x + c) % n;
            y = (__int128_t(y) * y + c) % n;
            y = (__int128_t(y) * y + c) % n;
            d = __gcd(x - y + n, n);
        } while (d == 1);
        if (d < n) {
            vector<ll> lb = Pollard(d), rb = Pollard(n / d);
            lb.insert(lb.end(), ALL(rb));
            return lb;
        }
    }
}

vector<pair<ll, int>> Pollard2(ll n) {
    auto ps = Pollard(n);
    sort(ALL(ps));
    using P = pair<ll, int>;
    vector<P> pes;
    for (auto &p : ps) {
        if (pes.empty() or pes.back().first != p) {
            pes.push_back({p, 1});
        } else {
            pes.back().second++;
        }
    }
    return pes;
}

vector<ll> EnumDivisors(ll n) {
    auto pes = Pollard2(n);
    vector<ll> ret;
    auto rec = [&](auto &rec, int id, ll d) -> void {
        if (id == SZ(pes)) {
            ret.push_back(d);
            return;
        }
        rec(rec, id + 1, d);
        rep(e, 0, pes[id].second) {
            d *= pes[id].first;
            rec(rec, id + 1, d);
        }
    };
    rec(rec, 0, 1);
    sort(ALL(ret));
    return ret;
}

/**
 * @brief Pollard-Rho
 */
#line 4 "Math/primitive.hpp"

ll mpow(ll a, ll t, ll m) {
    ll res = 1;
    FastDiv64 im(m);
    while (t) {
        if (t & 1)
            res = im.modulo(__int128_t(res) * a);
        a = im.modulo(__int128_t(a) * a);
        t >>= 1;
    }
    return res;
}
ll minv(ll a, ll m) {
    ll b = m, u = 1, v = 0;
    while (b) {
        ll t = a / b;
        a -= t * b;
        swap(a, b);
        u -= t * v;
        swap(u, v);
    }
    u = (u % m + m) % m;
    return u;
}
ll getPrimitiveRoot(ll p) {
    vector<ll> ps = Pollard(p - 1);
    sort(ALL(ps));
    rep(x, 1, inf) {
        for (auto &q : ps) {
            if (mpow(x, (p - 1) / q, p) == 1)
                goto fail;
        }
        return x;
    fail:;
    }
    assert(0);
}
template <typename T> T extgcd(T a, T b, T &p, T &q) {
    if (b == 0) {
        p = 1;
        q = 0;
        return a;
    }
    T d = extgcd(b, a % b, q, p);
    q -= a / b * p;
    return d;
}
template <typename T> pair<T, T> crt(vector<T> vs, vector<T> ms) {
    T V = vs[0], M = ms[0];
    rep(i, 1, vs.size()) {
        T p, q, v = vs[i], m = ms[i];
        if (M < m)
            swap(M, m), swap(V, v);
        T d = extgcd(M, m, p, q);
        if ((v - V) % d != 0)
            return {0, -1};
        T md = m / d, tmp = (v - V) / d % md * p % md;
        V += M * tmp;
        M *= md;
    }
    V = (V % M + M) % M;
    return {V, M};
}
ll garner(vector<ll> vs, vector<ll> p, int mod) {
    int sz = SZ(vs);
    vector<ll> kp(sz + 1), rmul(sz + 1, 1);
    p.push_back(mod);
    rep(i, 0, sz) {
        ll x = (vs[i] - kp[i]) * minv(rmul[i], p[i]) % p[i];
        if (x < 0)
            x += p[i];
        rep(j, i + 1, sz + 1) {
            kp[j] += rmul[j] * x;
            kp[j] %= p[j];
            rmul[j] *= p[i];
            rmul[j] %= p[j];
        }
    }
    return kp.back();
}

ll ModLog(ll a, ll b, ll p) {
    ll g = 1;
    for (ll t = p; t; t >>= 1)
        g = g * a % p;
    g = __gcd(g, p);
    ll t = 1, c = 0;
    for (; t % g; c++) {
        if (t == b)
            return c;
        t = t * a % p;
    }
    if (b % g)
        return -1;
    t /= g, b /= g;
    ll n = p / g, h = 0, gs = 1;
    for (; h * h < n; h++)
        gs = gs * a % n;
    unordered_map<ll, ll> bs;
    for (ll s = 0, e = b; s < h; bs[e] = ++s)
        e = e * a % n;
    for (ll s = 0, e = t; s < n;) {
        e = e * gs % n, s += h;
        if (bs.count(e)) {
            return c + s - bs[e];
        }
    }
    return -1;
}

ll TonelliShanks(ll a, ll p) {
    a %= p;
    if (a == 0)
        return 0;
    if (p == 2)
        return a;
    if (mpow(a, (p - 1) >> 1, p) != 1)
        return -1;
    ll b = 1;
    while (mpow(b, (p - 1) >> 1, p) == 1)
        b = Random::get(1LL, p - 1);

    ll q = p - 1, k = 0;
    while (q % 2 == 0) {
        q >>= 1;
        k++;
    }
    ll x = mpow(a, (q + 1) >> 1, p);
    b = mpow(b, q, p);
    k -= 2;
    while (mpow(x, 2, p) != a) {
        ll err = minv(a, p) * mpow(x, 2, p) % p;
        if (mpow(err, 1 << k, p) != 1)
            x = x * b % p;
        b = mpow(b, 2, p);
        k--;
    }
    return x;
}

ll mod_root(ll k, ll a, ll m) {
    if (a == 0)
        return k ? 0 : -1;
    if (m == 2)
        return a & 1;
    k %= m - 1;
    ll g = gcd(k, m - 1);
    if (mpow(a, (m - 1) / g, m) != 1)
        return -1;
    a = mpow(a, minv(k / g, (m - 1) / g), m);
    FastDiv64 im(m);

    auto _subroot = [&](ll p, int e, ll a) -> ll { // x^(p^e)==a(mod m)

        ll q = m - 1;
        int s = 0;
        while (q % p == 0) {
            q /= p;
            s++;
        }
        int d = s - e;
        ll pe = mpow(p, e, m),
           res = mpow(a, ((pe - 1) * minv(q, pe) % pe * q + 1) / pe, m), c = 1;
        while (mpow(c, (m - 1) / p, m) == 1)
            c++;
        c = mpow(c, q, m);
        map<ll, ll> mp;
        ll v = 1, block = sqrt(d * p) + 1,
           bs = mpow(c, mpow(p, s - 1, m - 1) * block % (m - 1), m);
        rep(i, 0, block + 1) mp[v] = i, v = im.modulo(i128(v) * bs);
        ll gs = minv(mpow(c, mpow(p, s - 1, m - 1), m), m);
        rep(i, 0, d) {
            ll err = im.modulo(i128(a) * minv(mpow(res, pe, m), m));
            ll pos = mpow(err, mpow(p, d - 1 - i, m - 1), m);
            rep(j, 0, block + 1) {
                if (mp.count(pos)) {
                    res = im.modulo(i128(res) *
                                    mpow(c,
                                         (block * mp[pos] + j) *
                                             mpow(p, i, m - 1) % (m - 1),
                                         m));
                    break;
                }
                pos = im.modulo(i128(pos) * gs);
            }
        }
        return res;
    };

    for (ll d = 2; d * d <= g; d++)
        if (g % d == 0) {
            int sz = 0;
            while (g % d == 0) {
                g /= d;
                sz++;
            }
            a = _subroot(d, sz, a);
        }
    if (g > 1)
        a = _subroot(g, 1, a);
    return a;
}

ull floor_root(ull a, ull k) {
    if (a <= 1 or k == 1)
        return a;
    if (k >= 64)
        return 1;
    if (k == 2)
        return sqrtl(a);
    constexpr ull LIM = -1;
    if (a == LIM)
        a--;
    auto mul = [&](ull &x, const ull &y) {
        if (x <= LIM / y)
            x *= y;
        else
            x = LIM;
    };
    auto pw = [&](ull x, ull t) -> ull {
        ull y = 1;
        while (t) {
            if (t & 1)
                mul(y, x);
            mul(x, x);
            t >>= 1;
        }
        return y;
    };
    ull ret = (k == 3 ? cbrt(a) - 1 : pow(a, nextafter(1 / double(k), 0)));
    while (pw(ret + 1, k) <= a)
        ret++;
    return ret;
}

/**
 * @brief Primitive Function
 */
#line 3 "Convolution/ntt.hpp"

template <typename T> struct NTT {
    static constexpr int rank2 = __builtin_ctzll(T::get_mod() - 1);
    std::array<T, rank2 + 1> root;  // root[i]^(2^i) == 1

    std::array<T, rank2 + 1> iroot; // root[i] * iroot[i] == 1


    std::array<T, std::max(0, rank2 - 2 + 1)> rate2;
    std::array<T, std::max(0, rank2 - 2 + 1)> irate2;

    std::array<T, std::max(0, rank2 - 3 + 1)> rate3;
    std::array<T, std::max(0, rank2 - 3 + 1)> irate3;

    NTT() {
        T g = getPrimitiveRoot(T::get_mod());
        root[rank2] = g.pow((T::get_mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        {
            T prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 2; i++) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            T prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 3; i++) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }

    void ntt(std::vector<T> &a, bool type = 0) {
        int n = int(a.size());
        int h = __builtin_ctzll((unsigned int)n);
        a.resize(1 << h);

        if (type) {
            int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed

            while (len) {
                if (len == 1) {
                    int p = 1 << (h - len);
                    T irot = 1;
                    for (int s = 0; s < (1 << (len - 1)); s++) {
                        int offset = s << (h - len + 1);
                        for (int i = 0; i < p; i++) {
                            auto l = a[i + offset];
                            auto r = a[i + offset + p];
                            a[i + offset] = l + r;
                            a[i + offset + p] =
                                ((unsigned long long)(T::get_mod() + l.v -
                                                      r.v) *
                                 irot.v) %
                                T::get_mod();
                        }
                        if (s + 1 != (1 << (len - 1)))
                            irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
                    }
                    len--;
                } else {
                    // 4-base

                    int p = 1 << (h - len);
                    T irot = 1, iimag = iroot[2];
                    for (int s = 0; s < (1 << (len - 2)); s++) {
                        T irot2 = irot * irot;
                        T irot3 = irot2 * irot;
                        int offset = s << (h - len + 2);
                        for (int i = 0; i < p; i++) {
                            auto a0 = 1ULL * a[i + offset + 0 * p].v;
                            auto a1 = 1ULL * a[i + offset + 1 * p].v;
                            auto a2 = 1ULL * a[i + offset + 2 * p].v;
                            auto a3 = 1ULL * a[i + offset + 3 * p].v;

                            auto a2na3iimag =
                                1ULL * T((T::get_mod() + a2 - a3) * iimag.v).v;

                            a[i + offset] = (a0 + a1 + a2 + a3) % T::get_mod();
                            a[i + offset + 1 * p] =
                                (a0 + (T::get_mod() - a1) + a2na3iimag) *
                                irot.v % T::get_mod();
                            a[i + offset + 2 * p] =
                                (a0 + a1 + (T::get_mod() - a2) +
                                 (T::get_mod() - a3)) *
                                irot2.v % T::get_mod();
                            a[i + offset + 3 * p] =
                                (a0 + (T::get_mod() - a1) +
                                 (T::get_mod() - a2na3iimag)) *
                                irot3.v % T::get_mod();
                        }
                        if (s + 1 != (1 << (len - 2)))
                            irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
                    }
                    len -= 2;
                }
            }
            T e = T(n).inv();
            for (auto &x : a)
                x *= e;
        } else {
            int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed

            while (len < h) {
                if (h - len == 1) {
                    int p = 1 << (h - len - 1);
                    T rot = 1;
                    for (int s = 0; s < (1 << len); s++) {
                        int offset = s << (h - len);
                        for (int i = 0; i < p; i++) {
                            auto l = a[i + offset];
                            auto r = a[i + offset + p] * rot;
                            a[i + offset] = l + r;
                            a[i + offset + p] = l - r;
                        }
                        if (s + 1 != (1 << len))
                            rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
                    }
                    len++;
                } else {
                    // 4-base

                    int p = 1 << (h - len - 2);
                    T rot = 1, imag = root[2];
                    for (int s = 0; s < (1 << len); s++) {
                        T rot2 = rot * rot;
                        T rot3 = rot2 * rot;
                        int offset = s << (h - len);
                        for (int i = 0; i < p; i++) {
                            auto mod2 = 1ULL * T::get_mod() * T::get_mod();
                            auto a0 = 1ULL * a[i + offset].v;
                            auto a1 = 1ULL * a[i + offset + p].v * rot.v;
                            auto a2 = 1ULL * a[i + offset + 2 * p].v * rot2.v;
                            auto a3 = 1ULL * a[i + offset + 3 * p].v * rot3.v;
                            auto a1na3imag =
                                1ULL * T(a1 + mod2 - a3).v * imag.v;
                            auto na2 = mod2 - a2;
                            a[i + offset] = (a0 + a2 + a1 + a3) % T::get_mod();
                            a[i + offset + 1 * p] =
                                (a0 + a2 + (2 * mod2 - (a1 + a3))) %
                                T::get_mod();
                            a[i + offset + 2 * p] =
                                (a0 + na2 + a1na3imag) % T::get_mod();
                            a[i + offset + 3 * p] =
                                (a0 + na2 + (mod2 - a1na3imag)) % T::get_mod();
                        }
                        if (s + 1 != (1 << len))
                            rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
                    }
                    len += 2;
                }
            }
        }
    }
    vector<T> mult(const vector<T> &a, const vector<T> &b) {
        if (a.empty() or b.empty())
            return vector<T>();
        int as = a.size(), bs = b.size();
        int n = as + bs - 1;
        assert(n <= (1 << rank2));
        if (as <= 30 or bs <= 30) {
            if (as > 30)
                return mult(b, a);
            vector<T> res(n);
            rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
            return res;
        }
        int m = 1;
        while (m < n)
            m <<= 1;
        vector<T> res(m);
        rep(i, 0, as) res[i] = a[i];
        ntt(res);
        if (a == b)
            rep(i, 0, m) res[i] *= res[i];
        else {
            vector<T> c(m);
            rep(i, 0, bs) c[i] = b[i];
            ntt(c);
            rep(i, 0, m) res[i] *= c[i];
        }
        ntt(res, 1);
        res.resize(n);
        return res;
    }
};

/**
 * @brief Number Theoretic Transform
 */
#line 2 "Math/comb.hpp"

template <typename T> T Inv(ll n) {
    static int md;
    static vector<T> buf({0, 1});
    if (md != T::get_mod()) {
        md = T::get_mod();
        buf = vector<T>({0, 1});
    }
    assert(n > 0);
    n %= md;
    while (SZ(buf) <= n) {
        int k = SZ(buf), q = (md + k - 1) / k;
        buf.push_back(buf[k * q - md] * q);
    }
    return buf[n];
}

template <typename T> T Fact(ll n, bool inv = 0) {
    static int md;
    static vector<T> buf({1, 1}), ibuf({1, 1});
    if (md != T::get_mod()) {
        md = T::get_mod();
        buf = ibuf = vector<T>({1, 1});
    }
    assert(n >= 0 and n < md);
    while (SZ(buf) <= n) {
        buf.push_back(buf.back() * SZ(buf));
        ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
    }
    return inv ? ibuf[n] : buf[n];
}

template <typename T> T nPr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
    if (n < 0 || n < r || r < 0)
        return 0;
    return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
// sum = n, r tuples
template <typename T> T nHr(int n, int r, bool inv = 0) {
    return nCr<T>(n + r - 1, r - 1, inv);
}
// [x^n]C(x)^k
template <typename T> T Catalan(int n, int k) {
    if (k == 0)
        return (n == 0 ? 1 : 0);
    return T(k) * Inv<T>(n * 2 + k) * nCr<T>(n * 2 + k, n);
}
// sum = n, a nonzero tuples and b tuples
template <typename T> T choose(int n, int a, int b) {
    if (n == 0)
        return !a;
    return nCr<T>(n + b - 1, a + b - 1);
}

/**
 * @brief Combination
 */
#line 3 "Math/modint.hpp"

template <unsigned mod = 1000000007> struct fp {
    static_assert(mod < uint(1) << 31);
    unsigned v;
    static constexpr int get_mod() {
        return mod;
    }
    constexpr unsigned inv() const {
        assert(v != 0);
        int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
        while (y > 0) {
            t = x / y;
            x -= t * y, p -= t * q;
            tmp = x, x = y, y = tmp;
            tmp = p, p = q, q = tmp;
        }
        if (p < 0)
            p += mod;
        return p;
    }
    constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
    fp operator-() const {
        return fp() - *this;
    }
    fp pow(ull t) {
        fp res = 1, b = *this;
        while (t) {
            if (t & 1)
                res *= b;
            b *= b;
            t >>= 1;
        }
        return res;
    }
    fp &operator+=(const fp &x) {
        if ((v += x.v) >= mod)
            v -= mod;
        return *this;
    }
    fp &operator-=(const fp &x) {
        if ((v += mod - x.v) >= mod)
            v -= mod;
        return *this;
    }
    fp &operator*=(const fp &x) {
        v = ull(v) * x.v % mod;
        return *this;
    }
    fp &operator/=(const fp &x) {
        if (x.v < 15000000) {
            return *this *= Inv<fp>(x.v);
        }
        v = ull(v) * x.inv() % mod;
        return *this;
    }
    fp operator+(const fp &x) const {
        return fp(*this) += x;
    }
    fp operator-(const fp &x) const {
        return fp(*this) -= x;
    }
    fp operator*(const fp &x) const {
        return fp(*this) *= x;
    }
    fp operator/(const fp &x) const {
        return fp(*this) /= x;
    }
    bool operator==(const fp &x) const {
        return v == x.v;
    }
    bool operator!=(const fp &x) const {
        return v != x.v;
    }
    friend istream &operator>>(istream &is, fp &x) {
        return is >> x.v;
    }
    friend ostream &operator<<(ostream &os, const fp &x) {
        return os << x.v;
    }
};

template <unsigned mod> void rd(fp<mod> &x) {
    fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
    fastio::wt(x.v);
}

/**
 * @brief Modint
 */
#line 4 "Convolution/arbitrary.hpp"

using M1 = fp<469762049>;  // 2^26

using M2 = fp<1811939329>; // 2^26

using M3 = fp<2013265921>; // 2^27

NTT<M1> N1;
NTT<M2> N2;
NTT<M3> N3;
constexpr uint r_12 = M2(M1::get_mod()).inv();
constexpr uint r_13 = M3(M1::get_mod()).inv();
constexpr uint r_23 = M3(M2::get_mod()).inv();
constexpr uint r_1323 = M3(ll(r_13) * r_23).v;
constexpr ll w1 = M1::get_mod();
constexpr ll w2 = ll(w1) * M2::get_mod();
template <typename T>
vector<T> ArbitraryMultint(const vector<int> &a, const vector<int> &b) {
    if (a.empty() or b.empty())
        return vector<T>();
    int n = a.size() + b.size() - 1;
    vector<T> res(n);
    if (min(a.size(), b.size()) <= 60) {
        rep(i, 0, a.size()) rep(j, 0, b.size()) res[i + j] += T(a[i]) * b[j];
        return res;
    }
    vector<ll> vals[3];
    vector<M1> a1(ALL(a)), b1(ALL(b)), c1 = N1.mult(a1, b1);
    vector<M2> a2(ALL(a)), b2(ALL(b)), c2 = N2.mult(a2, b2);
    vector<M3> a3(ALL(a)), b3(ALL(b)), c3 = N3.mult(a3, b3);
    for (M1 x : c1)
        vals[0].push_back(x.v);
    for (M2 x : c2)
        vals[1].push_back(x.v);
    for (M3 x : c3)
        vals[2].push_back(x.v);
    rep(i, 0, n) {
        ll p = vals[0][i];
        ll q = (vals[1][i] + M2::get_mod() - p) * r_12 % M2::get_mod();
        ll r = ((vals[2][i] + M3::get_mod() - p) * r_1323 +
                (M3::get_mod() - q) * r_23) %
               M3::get_mod();
        res[i] = (T(r) * w2 + q * w1 + p);
    }
    return res;
}

template <typename T>
vector<T> ArbitraryMult(const vector<T> &a, const vector<T> &b) {
    vector<int> A, B;
    for (auto &x : a)
        A.push_back(x.v);
    for (auto &x : b)
        B.push_back(x.v);
    return ArbitraryMultint<T>(A, B);
}

/**
 * @brief Arbitrary Mod Convolution
 */
#line 3 "Math/bigint.hpp"

template <int D> struct bigint {
    using u128 = __uint128_t;
    static const int B = pow(10, D);
    int sign = 0;
    vector<int> v;
    static int get_D() { return D; }
    static int get_B() { return B; }
    bigint() {}
    bigint(const vector<int> &_v, bool _s = false) : sign(_s), v(_v) {}
    bigint(ll x) {
        if (x < 0)
            x *= -1, sign = 1;
        while (x) {
            v.push_back(x % B);
            x /= B;
        }
    }
    bigint(string s) {
        if (s[0] == '-')
            s.erase(s.begin()), sign = 1;
        int add = 0, cnt = 0, base = 1;
        while (s.size()) {
            if (cnt == D) {
                v.push_back(add);
                cnt = 0;
                add = 0;
                base = 1;
            }
            add = (s.back() - '0') * base + add;
            cnt++;
            base *= 10;
            s.pop_back();
        }
        if (add)
            v.push_back(add);
    }
    bigint operator-() const {
        bigint res = *this;
        res.sign ^= 1;
        return res;
    }
    bigint abs() const {
        bigint res = *this;
        res.sign = 0;
        return res;
    }
    int &operator[](const int i) { return v[i]; }
    int size() const { return v.size(); }
    void norm() {
        rep(i, 0, v.size() - 1) {
            if (v[i] >= 0) {
                v[i + 1] += v[i] / B;
                v[i] %= B;
            } else {
                int c = (-v[i] + B - 1) / B;
                v[i] += c * B;
                v[i + 1] -= c;
            }
        }
        while (!v.empty() and v.back() >= B) {
            int c = v.back() / B;
            v.back() %= B;
            v.push_back(c);
        }
        while (!v.empty() and v.back() == 0)
            v.pop_back();
    }
    string to_str() const {
        string res;
        if (v.empty())
            return "0";
        if (sign)
            res += '-';
        res += to_string(v.back());
        for (int i = v.size() - 2; i >= 0; i--) {
            string add;
            int w = v[i];
            rep(_, 0, D) {
                add += ('0' + (w % 10));
                w /= 10;
            }
            reverse(ALL(add));
            res += add;
        }
        return res;
    }
    friend istream &operator>>(istream &is, bigint<D> &x) {
        string tmp;
        is >> tmp;
        x = bigint(tmp);
        return is;
    }
    friend ostream &operator<<(ostream &os, bigint<D> x) {
        os << x.to_str();
        return os;
    }
    bigint &operator<<=(const int &x) {
        if (!v.empty()) {
            vector<int> add(x, 0);
            v.insert(v.begin(), ALL(add));
        }
        return *this;
    }
    bigint &operator>>=(const int &x) {
        v = vector<int>(v.begin() + min(x, (int)v.size()), v.end());
        return *this;
    }
    bigint &operator+=(const bigint &x) {
        if (sign != x.sign) {
            *this -= (-x);
            return *this;
        }
        if ((int)v.size() < (int)x.size())
            v.resize(x.size(), 0);
        rep(i, 0, x.size()) { v[i] += x.v[i]; }
        norm();
        return *this;
    }
    bigint &operator-=(const bigint &x) {
        if (sign != x.sign) {
            *this += (-x);
            return *this;
        }
        if (abs() < x.abs()) {
            *this = x - (*this);
            sign ^= 1;
            return *this;
        }
        rep(i, 0, x.size()) { v[i] -= x.v[i]; }
        norm();
        return *this;
    }
    bigint &operator*=(const bigint &x) {
        sign ^= x.sign;
        auto v1 = ArbitraryMult<u128>(v, x.v);
        u128 add = 0;
        v.clear();
        v.reserve(v1.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v1.size()) and add == 0)
                break;
            if (i < int(v1.size()))
                add += v1[i];
            v.push_back(add % B);
            add /= B;
        }
        norm();
        return *this;
    }
    bigint div_naive(const bigint &a, const bigint &b) {
        if (SZ(b) == 1)
            return a.div(b.v.back());
        if (a < b)
            return bigint();
        int norm = B / (b.v.back() + 1);
        bigint x = a.mul(norm), y = b.mul(norm);
        int yb = y.v.back();
        bigint quo, rem;
        quo.v.resize(x.size() - y.size() + 1);
        rem.v = {x.v.end() - y.size(), x.v.end()};
        for (int i = x.size() - y.size(); i >= 0; i--) {
            if (rem.size() == y.size()) {
                if (rem >= y) {
                    quo[i] = 1;
                    rem -= y;
                }
            } else if (rem.size() > y.size()) {
                ll rb = ll(rem.v.back()) * B + rem[rem.size() - 2];
                int q = rb / yb;
                bigint yq = y.mul(q);
                while (rem < yq) {
                    q--;
                    yq -= y;
                }
                rem -= yq;
                while (rem >= y) {
                    q++;
                    rem -= y;
                }
                quo[i] = q;
            }
            if (i)
                rem.v.insert(rem.v.begin(), x[i - 1]);
        }
        return quo;
    }
    bigint &operator/=(const bigint &x) {
        bigint a = abs(), b = x.abs();
        sign ^= x.sign;
        if (a < b)
            return *this = bigint();
        if (b.size() == 1)
            return *this = a.div(b.v.back());

        int deg = a.size() - b.size() + 2, k = deg;
        while (k > 64)
            k = (k + 1) >> 1;
        bigint base(1);
        base <<= (b.size() + k);
        bigint inv(div_naive(base, b));

        while (k < deg) {
            bigint y = inv.square();
            y.v.insert(y.v.begin(), 0);
            int l = min(SZ(b), k * 2 + 1);
            bigint pref;
            pref.v = {b.v.end() - l, b.v.end()};
            y *= pref;
            y >>= l;
            inv = inv + inv;
            inv <<= k + 1;
            inv -= y;
            inv.v.erase(inv.v.begin());
            k <<= 1;
        }
        inv >>= (k - deg);

        (*this) = a * inv;
        (*this) >>= int(a.size() + 2);
        bigint mul = (*this) * b;
        while (mul + b <= a) {
            (*this) += bigint(1);
            mul += b;
        }
        while (mul > a) {
            (*this) -= bigint(1);
            mul -= b;
        }
        return *this;
    }
    bigint &operator%=(const bigint &x) {
        bigint div = (*this) / x;
        (*this) -= div * x;
        return *this;
    }
    bigint square() const {
        bigint ret;
        auto v1 = ArbitraryMult<u128>(v, v);
        u128 add = 0;
        ret.v.reserve(v1.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v1.size()) and add == 0)
                break;
            if (i < int(v1.size()))
                add += v1[i];
            ret.v.push_back(add % B);
            add /= B;
        }
        return ret;
    }
    bigint mul(ll x) const {
        bigint res;
        if (x < 0)
            res.sign ^= 1, x *= -1;
        u128 add = 0;
        res.v.reserve(v.size() + 3);
        for (int i = 0;; i++) {
            if (i >= int(v.size()) and add == 0)
                break;
            if (i < int(v.size()))
                add += ll(v[i]) * x;
            res.v.push_back(add % B);
            add /= B;
        }
        return res;
    }
    bigint div(ll x) const {
        bigint res = *this;
        if (x < 0)
            res.sign ^= 1, x *= -1;
        ll add = 0;
        for (int i = res.v.size() - 1; i >= 0; i--) {
            add = add * B + res.v[i];
            int q = add / x, r = add % x;
            res.v[i] = q, add = r;
        }
        res.norm();
        return res;
    }
    bigint operator<<(const int &x) const { return bigint(*this) <<= x; }
    bigint operator>>(const int &x) const { return bigint(*this) >>= x; }
    bigint operator+(const bigint &x) const { return bigint(*this) += x; }
    bigint operator-(const bigint &x) const { return bigint(*this) -= x; }
    bigint operator*(const bigint &x) const { return bigint(*this) *= x; }
    bigint operator/(const bigint &x) const { return bigint(*this) /= x; }
    bigint operator%(const bigint &x) const { return bigint(*this) %= x; }
    bool operator<(const bigint &x) const {
        if (sign != x.sign)
            return sign > x.sign;
        if ((int)v.size() != (int)x.size()) {
            if (sign)
                return (int)x.size() < (int)v.size();
            else
                return (int)v.size() < (int)x.size();
        }
        for (int i = v.size() - 1; i >= 0; i--)
            if (v[i] != x.v[i]) {
                if (sign)
                    return x.v[i] < v[i];
                else
                    return v[i] < x.v[i];
            }
        return false;
    }
    bool operator>(const bigint &x) const { return x < *this; }
    bool operator<=(const bigint &x) const { return !(*this > x); }
    bool operator>=(const bigint &x) const { return !(*this < x); }
    bool operator==(const bigint &x) const {
        return !(*this < x) and !(*this > x);
    }
    bool operator!=(const bigint &x) const { return !(*this == x); }
};
typedef bigint<9> Bigint;

struct Bigfloat {
    Bigint v;
    int p = 0;
    Bigfloat() {}
    Bigfloat(const ll &_v) { v = Bigint(_v); }
    Bigfloat(const Bigint &_v, int _p = 0) : v(_v), p(_p) {}
    void set(int _p) {
        if (p < _p) {
            v >>= (_p - p);
        } else {
            v <<= (p - _p);
        }
        p = _p;
    }
    Bigint to_int() const {
        if (p < 0)
            return v >> (-p);
        else
            return v << p;
    }
    Bigfloat &operator+=(const Bigfloat &x) {
        if (p > x.p)
            set(x.p), v += x.v;
        else
            v += x.v << (x.p - p);
        return *this;
    }
    Bigfloat &operator-=(const Bigfloat &x) {
        if (p > x.p)
            set(x.p), v -= x.v;
        else
            v -= x.v << (x.p - p);
        return *this;
    }
    Bigfloat square() {
        Bigfloat res = *this;
        res.p *= 2;
        res.v = res.v.square();
        return res;
    }
    Bigfloat mul(ll x) {
        Bigfloat res = *this;
        res.v = v.mul(x);
        return res;
    }
    Bigfloat div(ll x) {
        Bigfloat res = *this;
        res.v = v.div(x);
        return res;
    }
    Bigfloat &operator*=(const Bigfloat &x) {
        p += x.p;
        v *= x.v;
        return *this;
    }
    Bigfloat &operator/=(const Bigfloat &x) {
        p -= x.p;
        v /= x.v;
        return *this;
    }
    Bigfloat operator+(const Bigfloat &x) const { return Bigfloat(*this) += x; }
    Bigfloat operator-(const Bigfloat &x) const { return Bigfloat(*this) -= x; }
    Bigfloat operator*(const Bigfloat &x) const { return Bigfloat(*this) *= x; }
    Bigfloat operator/(const Bigfloat &x) const { return Bigfloat(*this) /= x; }
    string to_str() {
        string res = v.abs().to_str();
        int d = Bigint::get_D();
        if (p * d > 0)
            res += string(p * d, '0');
        else if (-p * d >= 1 and -p * d < (int)res.size()) {
            res = res.substr(0, (int)res.size() + p * d) + '.' +
                  res.substr((int)res.size() + p * d);
        } else if (-p * d >= (int)res.size())
            res = "0." + string(-p * d - (int)res.size(), '0') + res;
        if (v.sign) {
            res.insert(res.begin(), '-');
        }
        return res;
    }
    friend ostream &operator<<(ostream &os, Bigfloat x) {
        os << x.to_str();
        return os;
    }
};

Bigfloat sqrt(ll n, int d) {
    Bigfloat res(Bigint((ll)sqrt(1LL * Bigint::get_B() * Bigint::get_B() / n)),
                 -1),
        pre;
    int cur = 1;
    while (pre.v != res.v) {
        cur = min(cur << 1, d);
        pre = res;
        Bigfloat add = Bigfloat(1) - res.square().mul(n);
        add.set(-cur);
        res += (res * add).div(2);
        res.set(-cur);
    }
    return res.mul(n);
}

/**
 * @brief Big Integer(Float)
 */

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