Bigint

#include <bits/stdc++.h>
using namespace std;
/*
  ######################################################################
  #######################   THE   BIG   INT   ##########################
*/
const int base = 1000000000;
const int base_digits = 9;
struct bigint {
    vector<int> a;
    int sign;
    /*<arpa>*/
    int size(){
    if(a.empty())return 0;
    int ans=(a.size()-1)*base_digits;
    int ca=a.back();
    while(ca)
        ans++,ca/=10;
    return ans;
    }
    bigint operator ^(const bigint &v){
    bigint ans=1,a=*this,b=v;
    while(!b.isZero()){
        if(b%2)
        ans*=a;
        a*=a,b/=2;
    }
    return ans;
    }
    string to_string(){
    stringstream ss;
    ss << *this;
    string s;
    ss >> s;
    return s;
    }
    int sumof(){
    string s = to_string();
    int ans = 0;
    for(auto c : s)  ans += c - '0';
    return ans;
    }
    /*</arpa>*/
    bigint() :
    sign(1) {
    }

    bigint(long long v) {
    *this = v;
    }

    bigint(const string &s) {
    read(s);
    }

    void operator=(const bigint &v) {
    sign = v.sign;
    a = v.a;
    }

    void operator=(long long v) {
    sign = 1;
    a.clear();
    if (v < 0)
        sign = -1, v = -v;
    for (; v > 0; v = v / base)
        a.push_back(v % base);
    }

    bigint operator+(const bigint &v) const {
    if (sign == v.sign) {
        bigint res = v;

        for (int i = 0, carry = 0; i < (int) max(a.size(), v.a.size()) || carry; ++i) {
        if (i == (int) res.a.size())
            res.a.push_back(0);
        res.a[i] += carry + (i < (int) a.size() ? a[i] : 0);
        carry = res.a[i] >= base;
        if (carry)
            res.a[i] -= base;
        }
        return res;
    }
    return *this - (-v);
    }

    bigint operator-(const bigint &v) const {
    if (sign == v.sign) {
        if (abs() >= v.abs()) {
        bigint res = *this;
        for (int i = 0, carry = 0; i < (int) v.a.size() || carry; ++i) {
            res.a[i] -= carry + (i < (int) v.a.size() ? v.a[i] : 0);
            carry = res.a[i] < 0;
            if (carry)
            res.a[i] += base;
        }
        res.trim();
        return res;
        }
        return -(v - *this);
    }
    return *this + (-v);
    }

    void operator*=(int v) {
    if (v < 0)
        sign = -sign, v = -v;
    for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) {
        if (i == (int) a.size())
        a.push_back(0);
        long long cur = a[i] * (long long) v + carry;
        carry = (int) (cur / base);
        a[i] = (int) (cur % base);
        //asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base));
    }
    trim();
    }

    bigint operator*(int v) const {
    bigint res = *this;
    res *= v;
    return res;
    }

    void operator*=(long long v) {
    if (v < 0)
        sign = -sign, v = -v;
    for (int i = 0, carry = 0; i < (int) a.size() || carry; ++i) {
        if (i == (int) a.size())
        a.push_back(0);
        long long cur = a[i] * (long long) v + carry;
        carry = (int) (cur / base);
        a[i] = (int) (cur % base);
        //asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base));
    }
    trim();
    }

    bigint operator*(long long v) const {
    bigint res = *this;
    res *= v;
    return res;
    }

    friend pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
    int norm = base / (b1.a.back() + 1);
    bigint a = a1.abs() * norm;
    bigint b = b1.abs() * norm;
    bigint q, r;
    q.a.resize(a.a.size());

    for (int i = a.a.size() - 1; i >= 0; i--) {
        r *= base;
        r += a.a[i];
        int s1 = r.a.size() <= b.a.size() ? 0 : r.a[b.a.size()];
        int s2 = r.a.size() <= b.a.size() - 1 ? 0 : r.a[b.a.size() - 1];
        int d = ((long long) base * s1 + s2) / b.a.back();
        r -= b * d;
        while (r < 0)
        r += b, --d;
        q.a[i] = d;
    }

    q.sign = a1.sign * b1.sign;
    r.sign = a1.sign;
    q.trim();
    r.trim();
    return make_pair(q, r / norm);
    }

    bigint operator/(const bigint &v) const {
    return divmod(*this, v).first;
    }

    bigint operator%(const bigint &v) const {
    return divmod(*this, v).second;
    }

    void operator/=(int v) {
    if (v < 0)
        sign = -sign, v = -v;
    for (int i = (int) a.size() - 1, rem = 0; i >= 0; --i) {
        long long cur = a[i] + rem * (long long) base;
        a[i] = (int) (cur / v);
        rem = (int) (cur % v);
    }
    trim();
    }

    bigint operator/(int v) const {
    bigint res = *this;
    res /= v;
    return res;
    }

    int operator%(int v) const {
    if (v < 0)
        v = -v;
    int m = 0;
    for (int i = a.size() - 1; i >= 0; --i)
        m = (a[i] + m * (long long) base) % v;
    return m * sign;
    }

    void operator+=(const bigint &v) {
    *this = *this + v;
    }
    void operator-=(const bigint &v) {
    *this = *this - v;
    }
    void operator*=(const bigint &v) {
    *this = *this * v;
    }
    void operator/=(const bigint &v) {
    *this = *this / v;
    }

    bool operator<(const bigint &v) const {
    if (sign != v.sign)
        return sign < v.sign;
    if (a.size() != v.a.size())
        return a.size() * sign < v.a.size() * v.sign;
    for (int i = a.size() - 1; i >= 0; i--)
        if (a[i] != v.a[i])
        return a[i] * sign < v.a[i] * sign;
    return false;
    }

    bool operator>(const bigint &v) const {
    return v < *this;
    }
    bool operator<=(const bigint &v) const {
    return !(v < *this);
    }
    bool operator>=(const bigint &v) const {
    return !(*this < v);
    }
    bool operator==(const bigint &v) const {
    return !(*this < v) && !(v < *this);
    }
    bool operator!=(const bigint &v) const {
    return *this < v || v < *this;
    }

    void trim() {
    while (!a.empty() && !a.back())
        a.pop_back();
    if (a.empty())
        sign = 1;
    }

    bool isZero() const {
    return a.empty() || (a.size() == 1 && !a[0]);
    }

    bigint operator-() const {
    bigint res = *this;
    res.sign = -sign;
    return res;
    }

    bigint abs() const {
    bigint res = *this;
    res.sign *= res.sign;
    return res;
    }

    long long longValue() const {
    long long res = 0;
    for (int i = a.size() - 1; i >= 0; i--)
        res = res * base + a[i];
    return res * sign;
    }

    friend bigint gcd(const bigint &a, const bigint &b) {
    return b.isZero() ? a : gcd(b, a % b);
    }
    friend bigint lcm(const bigint &a, const bigint &b) {
    return a / gcd(a, b) * b;
    }

    void read(const string &s) {
    sign = 1;
    a.clear();
    int pos = 0;
    while (pos < (int) s.size() && (s[pos] == '-' || s[pos] == '+')) {
        if (s[pos] == '-')
        sign = -sign;
        ++pos;
    }
    for (int i = s.size() - 1; i >= pos; i -= base_digits) {
        int x = 0;
        for (int j = max(pos, i - base_digits + 1); j <= i; j++)
        x = x * 10 + s[j] - '0';
        a.push_back(x);
    }
    trim();
    }

    friend istream& operator>>(istream &stream, bigint &v) {
    string s;
    stream >> s;
    v.read(s);
    return stream;
    }

    friend ostream& operator<<(ostream &stream, const bigint &v) {
    if (v.sign == -1)
        stream << '-';
    stream << (v.a.empty() ? 0 : v.a.back());
    for (int i = (int) v.a.size() - 2; i >= 0; --i)
        stream << setw(base_digits) << setfill('0') << v.a[i];
    return stream;
    }

    static vector<int> convert_base(const vector<int> &a, int old_digits, int new_digits) {
    vector<long long> p(max(old_digits, new_digits) + 1);
    p[0] = 1;
    for (int i = 1; i < (int) p.size(); i++)
        p[i] = p[i - 1] * 10;
    vector<int> res;
    long long cur = 0;
    int cur_digits = 0;
    for (int i = 0; i < (int) a.size(); i++) {
        cur += a[i] * p[cur_digits];
        cur_digits += old_digits;
        while (cur_digits >= new_digits) {
        res.push_back(int(cur % p[new_digits]));
        cur /= p[new_digits];
        cur_digits -= new_digits;
        }
    }
    res.push_back((int) cur);
    while (!res.empty() && !res.back())
        res.pop_back();
    return res;
    }

    typedef vector<long long> vll;

    static vll karatsubaMultiply(const vll &a, const vll &b) {
    int n = a.size();
    vll res(n + n);
    if (n <= 32) {
        for (int i = 0; i < n; i++)
        for (int j = 0; j < n; j++)
            res[i + j] += a[i] * b[j];
        return res;
    }

    int k = n >> 1;
    vll a1(a.begin(), a.begin() + k);
    vll a2(a.begin() + k, a.end());
    vll b1(b.begin(), b.begin() + k);
    vll b2(b.begin() + k, b.end());

    vll a1b1 = karatsubaMultiply(a1, b1);
    vll a2b2 = karatsubaMultiply(a2, b2);

    for (int i = 0; i < k; i++)
        a2[i] += a1[i];
    for (int i = 0; i < k; i++)
        b2[i] += b1[i];

    vll r = karatsubaMultiply(a2, b2);
    for (int i = 0; i < (int) a1b1.size(); i++)
        r[i] -= a1b1[i];
    for (int i = 0; i < (int) a2b2.size(); i++)
        r[i] -= a2b2[i];

    for (int i = 0; i < (int) r.size(); i++)
        res[i + k] += r[i];
    for (int i = 0; i < (int) a1b1.size(); i++)
        res[i] += a1b1[i];
    for (int i = 0; i < (int) a2b2.size(); i++)
        res[i + n] += a2b2[i];
    return res;
    }

    bigint operator*(const bigint &v) const {
    vector<int> a6 = convert_base(this->a, base_digits, 6);
    vector<int> b6 = convert_base(v.a, base_digits, 6);
    vll a(a6.begin(), a6.end());
    vll b(b6.begin(), b6.end());
    while (a.size() < b.size())
        a.push_back(0);
    while (b.size() < a.size())
        b.push_back(0);
    while (a.size() & (a.size() - 1))
        a.push_back(0), b.push_back(0);
    vll c = karatsubaMultiply(a, b);
    bigint res;
    res.sign = sign * v.sign;
    for (int i = 0, carry = 0; i < (int) c.size(); i++) {
        long long cur = c[i] + carry;
        res.a.push_back((int) (cur % 1000000));
        carry = (int) (cur / 1000000);
    }
    res.a = convert_base(res.a, 6, base_digits);
    res.trim();
    return res;
    }
};
/*
  #######################   THE   BIG   INT   ##########################
  ######################################################################
*/