// [https://gist.github.com/ar-pa/957297fb3f88996ead11 <- https://codeforces.com/blog/entry/22566?locale=ru <- google:‘c++ bigint’]
#include <iomanip> // for `std::setw()` [in GCC]

class BigInt
{
	static const int base = 1000000000;
	static const int base_digits = 9;
	std::vector<int> a;
	int sign;
	friend class Fraction;
	/*<arpa>*/
	int size(){
		if(a.empty())return 0;
		int ans=int(a.size()-1)*base_digits;
		int ca=a.back();
		while(ca)
			ans++,ca/=10;
		return ans;
	}
public:
	friend BigInt pow(const BigInt &t, const BigInt &v){
		BigInt ans=1,a=t,b=v;
		while(!b.isZero()){
			if(b%2)
				ans*=a;
			a*=a,b/=2;
		}
		return ans;
	}
	friend BigInt pow(BigInt base, BigInt exponent, const BigInt &modulus) {
		BigInt result = 1;

		while (exponent != 0) {
			if (exponent % 2 != 0)
				result = (result * base) % modulus;
			exponent /= 2;
			base = (base * base) % modulus;
		}
		return result;
	}
	operator String() const {
		std::stringstream ss;
		ss << *this;
		std::string s;
		ss >> s;
		return String(s.begin(), s.end());
	}
	/*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(int v) {
		*this = v;
	}

	BigInt(const String &s) {
		read(s);
	}
	BigInt(const String &s, int base) {
		*this = to_int_t<BigInt>(s, base);
	}

	const BigInt &operator=(const BigInt &v) {
		sign = v.sign;
		a = v.a;
		return *this;
	}

	const BigInt &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);
		return *this;
	}
	const BigInt &operator=(int v) {
		*this = (long long)v;
		return *this;
	}

	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+(int v) const {
		return *this + BigInt(v);
	}
	friend BigInt operator+(int v, const BigInt &b) {
		return b + v;
	}
	BigInt operator+(Int64 v) const {
		return *this + BigInt(v);
	}
	friend BigInt operator+(Int64 v, const BigInt &b) {
		return b + 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;
	}
	friend BigInt operator*(int v, const BigInt &b) {
		return b * v;
	}

	void operator*=(long long v) {
		if (v < 0)
			sign = -sign, v = -v;
		if(v > base){
			*this = *this * (v / base) * base + *this * (v % base);
			return ;
		}
		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 BigInt operator*(long long v, const BigInt &b) {
		return b * v;
	}

	friend std::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 = (int)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 std::make_pair(q, r / norm);
	}

	BigInt operator/(const BigInt &v) const {
		return divmod(*this, v).first;
	}
	friend BigInt idiv(const BigInt &a, const BigInt &b) {
		return a / b;
	}
	friend BigInt idiv(const BigInt &a, int b) {
		return a / b;
	}
	friend BigInt idiv(const BigInt &a, Int64 b) {
		return a / BigInt(b);
	}

	BigInt operator%(const BigInt &v) const {
		return divmod(*this, v).second;
	}
	friend BigInt mod(const BigInt &a, const BigInt &b) {
		return a % b;
	}
	friend BigInt mod(const BigInt &a, int b) {
		return a % b;
	}
	friend BigInt mod(const BigInt &a, Int64 b) {
		return a % BigInt(b);
	}
	friend BigInt nmod(const BigInt &x, const BigInt &y) {
		auto r = x % y; if (r < 0) r += y; return r;
	}

	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 = (int)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++(int) {
		*this = *this + 1;
	}
	void operator--(int) {
		*this = *this - 1;
	}
	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 = (int)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;
	}
	friend BigInt abs(const BigInt &b) {
		return b.abs();
	}

	long long longValue() const {
		long long res = 0;
		for (int i = (int)a.size() - 1; i >= 0; i--)
			res = res * base + a[i];
		return res * sign;
	}
	friend Int to_int(const BigInt &n) {
		return Int(n.longValue());
	}
	friend Int64 to_int64(const BigInt &n) {
		return n.longValue();
	}

	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] == u'-' || s[pos] == u'+')) {
			if (s[pos] == u'-')
				sign = -sign;
			++pos;
		}
		for (int i = (int)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 std::ostream& operator<<(std::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 << std::setw(base_digits) << std::setfill('0') << v.a[i];
		return stream;
	}

	static std::vector<int> convert_base(const std::vector<int> &a, int old_digits, int new_digits) {
		std::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;
		std::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 std::vector<long long> vll;

	static vll karatsubaMultiply(const vll &a, const vll &b) {
		int n = (int)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 {
		std::vector<int> a6 = convert_base(this->a, base_digits, 6);
		std::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;
	}
};

Char::Char(const BigInt &b)
{
	long long c = b.longValue();
	if (in(c, range_ee((long long)0, (long long)0xFFFF)))
		code = char16_t(c);
	else
		throw AssertionError();
}

Int pow(Int base, Int exponent, const Int modulus)
{
	Int result = 1;

	while (exponent != 0) {
		if (exponent % 2 != 0)
			result = (result * base) % modulus;
		exponent /= 2;
		base = (base * base) % modulus;
	}
	return result;
}

#ifndef INT_IS_INT64
Int pow(Int64 base, Int exponent, const Int modulus)
{
	Int result = 1;

	while (exponent != 0) {
		if (exponent % 2 != 0)
			result = Int((result * base) % modulus);
		exponent /= 2;
		base = (base * base) % modulus;
	}
	return result;
}

Int pow(Int64 base, Int64 exponent, const Int modulus)
{
	Int result = 1;

	while (exponent != 0) {
		if (exponent % 2 != 0)
			result = Int((result * base) % modulus);
		exponent /= 2;
		base = (base * base) % modulus;
	}
	return result;
}
#endif

namespace bits {
int length(BigInt v)
{
	int result = 0;
	while (v > 0) {
		v /= 2;
		result++;
	}
	return result;
}}


class Fraction
{
public:
	BigInt numerator, denominator;

	explicit Fraction(const BigInt &numerator) : numerator(numerator), denominator(1) {}

	Fraction(const BigInt &numer, const BigInt &denom) : numerator(numer), denominator(denom) {
		if (denominator.isZero())
			throw AssertionError();

		if (denominator.sign == -1) {
			numerator.sign   = -numerator.sign;
			denominator.sign = -denominator.sign;
		}

		BigInt d = gcd(numerator, denominator);
		if (d != 1) {
			numerator /= d;
			denominator /= d;
		}
	}

	Fraction operator+(const Fraction &other) const {
		return Fraction(numerator * other.denominator + denominator * other.numerator, denominator * other.denominator);
	}

	Fraction operator-(const Fraction &other) const {
		return Fraction(numerator * other.denominator - denominator * other.numerator, denominator * other.denominator);
	}

	Fraction operator*(const Fraction &other) const {
		return Fraction(numerator * other.numerator, denominator * other.denominator);
	}

	Fraction operator/(const Fraction &other) const {
		return Fraction(numerator * other.denominator, denominator * other.numerator);
	}

	void operator+=(const Fraction &other) {*this = *this + other;}
	void operator-=(const Fraction &other) {*this = *this - other;}
	void operator*=(const Fraction &other) {*this = *this * other;}
	void operator/=(const Fraction &other) {*this = *this / other;}

	bool operator==(const Fraction &other) const {
		return numerator == other.numerator && denominator == other.denominator;
	}
	bool operator!=(const Fraction &other) const {
		return !(*this == other);
	}

	bool operator<(const Fraction &other) const {
		return numerator * other.denominator < other.numerator * denominator;
	}

	operator String() const {
		if (denominator == 1)
			return String(numerator);
		else
			return String(numerator) & u'/'_C & String(denominator);
	}

	explicit operator double() const {
		return to_float(numerator) / to_float(denominator);
	}
};

inline double to_float(const Fraction &f) { return (double)f; }
