/**
 * code generated by JHelper
 * More info: https://github.com/AlexeyDmitriev/JHelper
 * @author
 */

#include <cstdlib>
#include <ctime>
#include <iostream>
#include <fstream>

/**
 *  This solution uses cpplib (prewritten code library)
 *  More info: https://github.com/agul/algolib/tree/master/cpplib
 */


#pragma comment(linker, "/STACK:66777216")
#pragma warning(disable : 4996)
#define _USE_MATH_DEFINES
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
#include <functional>
#include <iomanip>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <string>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
#pragma hdrstop












// todo[c++17]: replace with std::disjunction_v
template <bool... Bools>
struct Disjunction : std::false_type {};

template <bool First, bool... Others>
// todo[c++14] use bool_constant
struct Disjunction<First, Others...> : std::integral_constant<bool, First || Disjunction<Others...>::value> {};

#if __cplusplus >= 201703L
template <bool... Bools>
constexpr bool DisjunctionV = Disjunction<Bools...>::value;

static_assert(!DisjunctionV<>, "");
static_assert(!DisjunctionV<false, false>, "");
static_assert(DisjunctionV<true>, "");
static_assert(DisjunctionV<false, true>, "");
static_assert(DisjunctionV<true, false>, "");
static_assert(DisjunctionV<false, false, true, true>, "");
#endif


#include <iterator>


template<typename T>
constexpr auto has_begin(int) -> decltype(std::begin(std::declval<T>()), true) {
	return true;
}

template<typename T>
constexpr bool has_begin(...) {
	return false;
}

template <typename T>
using is_container = std::integral_constant<bool, has_begin<T>(0)>;






template <typename T, typename... Types>
// todo[c++17] use is_same_v
// todo[c++17] use variable template
using is_one_of = Disjunction<std::is_same<T, Types>::value...>;

//todo[c++14] use remove_cv_t
//todo[c++17] use is_integral_v
//todo[c++17] use  variable template
template <typename T>
using is_sane_integer = std::integral_constant<
		bool,
		std::is_integral<T>::value && !is_one_of<typename std::remove_cv<T>::type, char, signed char, unsigned char, bool, char16_t, char32_t, wchar_t>::value
>;


std::string to_string(const std::string& s) {
	return '"' + s + '"';
}

std::string to_string(const char* s) {
	return to_string(std::string(s));
}

std::string to_string(const char ch) {
	return {'\'', ch, '\''};
}

std::string to_string(const bool b) {
	return (b ? "true" : "false");
}

template<typename T, typename std::enable_if<Disjunction<is_sane_integer<T>::value, std::is_floating_point<T>::value>::value>::type* = nullptr>
std::string to_string(const T value) {
	return std::to_string(value);
}

template<typename A, typename B>
std::string to_string(const std::pair<A, B>&);

template<typename T, typename std::enable_if<is_container<T>::value>::type* = nullptr>
std::string to_string(const T& container) {
	bool is_first = true;
	std::string res = "{";
	for (const auto & x : container) {
		if (!is_first) {
			res += ", ";
		}
		is_first = false;
		res += to_string(x);
	}
	res += "}";
	return res;
}

template<typename A, typename B>
std::string to_string(const std::pair<A, B>& p) {
	return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

std::vector<std::string> vec_splitter(std::string s) {
	s += ',';
	std::vector<std::string> res;
	while (!s.empty()) {
		const size_t pos = s.find(',');
		res.emplace_back(s.substr(0, pos));
		s = s.substr(pos + 1);
	}
	return res;
}

void debug_out(
		std::vector<std::string> __attribute__ ((unused)) args,
		__attribute__ ((unused)) int idx,
		__attribute__ ((unused)) int LINE_NUM)
{
	std::cerr << std::endl;
}

template<typename Head, typename... Tail>
void debug_out(std::vector<std::string> args, int idx, int LINE_NUM, Head H, Tail... T) {
	std::stringstream ss;
	ss << to_string(H);
	if (idx > 0) {
		std::cerr << ", ";
	} else {
		std::cerr << "Line(" << LINE_NUM << ") ";
	}
	const std::string& arg = args[idx];
	const std::string value = ss.str();
	if (arg == value) {
		std::cerr << arg;
	} else {
		std::cerr << arg << " = " << value;
	}
	debug_out(args, idx + 1, LINE_NUM, T...);
}

#ifdef HOME
#define debug(...) debug_out(vec_splitter(#__VA_ARGS__), 0, __LINE__, __VA_ARGS__)
#else
#define debug(...) 42
#endif





namespace std {

template<typename T>
std::istream& operator>>(std::istream& in, std::vector<T>& vec) {
	for (auto& it : vec) {
		in >> it;
	}
	return in;
}


template<typename T, typename U>
std::istream& operator>>(std::istream& in, std::pair<T, U>& rhs) {
	in >> rhs.first >> rhs.second;
	return in;
}

}  // namespace std

void assign_files_input_txt() {
	freopen("input.txt", "r", stdin);
	freopen("output.txt", "w", stdout);
}








#include <cstddef>


template<typename T>
class IntegerIterator : public std::iterator<std::input_iterator_tag, T, std::ptrdiff_t, T*, T> {
public:
	using value_type = T;

	constexpr explicit IntegerIterator(const value_type value) : value_(value) {}

	IntegerIterator& operator++() {
		++value_;
		return *this;
	}

	IntegerIterator operator++(int) {
		IntegerIterator copy = *this;
		++value_;
		return copy;
	}

	IntegerIterator& operator--() {
		--value_;
		return *this;
	}

	IntegerIterator operator--(int) {
		IntegerIterator copy = *this;
		--value_;
		return copy;
	}

	constexpr value_type operator*() const {
		return value_;
	}

	constexpr bool operator ==(const IntegerIterator rhs) const {
		return value_ == rhs.value_;
	}

	constexpr bool operator !=(const IntegerIterator rhs) const {
		return !(*this == rhs);
	}

	constexpr value_type operator +(const IntegerIterator rhs) const {
		return value_ + rhs.value_;
	}

	IntegerIterator& operator +=(const IntegerIterator rhs) const {
		value_ += rhs.value_;
		return *this;
	}

	constexpr value_type operator -(const IntegerIterator rhs) const {
		return value_ - rhs.value_;
	}

	IntegerIterator& operator -=(const IntegerIterator rhs) const {
		value_ -= rhs.value_;
		return *this;
	}

private:
	value_type value_;
};


template<typename T>
class IntegerRange {
public:
	using const_iterator = IntegerIterator<T>;

	constexpr IntegerRange(const T begin, const T end) : begin_(begin), end_(end) {}

	constexpr const_iterator begin() const {
		return const_iterator(begin_);
	}

	constexpr const_iterator end() const {
		return const_iterator(end_);
	}

	constexpr T size() const {
		return end_ - begin_;
	}

private:
	T begin_;
	T end_;
};





template<typename T>
class ReversedRange {
public:
	using const_iterator = typename std::decay<T>::type::const_iterator;
	using reverse_iterator = std::reverse_iterator<const_iterator>;

	explicit constexpr ReversedRange(const T& iterable) : begin_(iterable.begin()), end_(iterable.end()) {}

	constexpr reverse_iterator begin() const {
		return reverse_iterator(end_);
	}

	constexpr reverse_iterator end() const {
		return reverse_iterator(begin_);
	}

private:
	const_iterator begin_;
	const_iterator end_;
};

template<typename T>
ReversedRange<T> reversed(T&& iterable) {
	return ReversedRange<T>{iterable};
}


template<typename T>
constexpr IntegerRange<T> range(const T to) {
	return IntegerRange<T>(0, to);
}

template<typename T>
constexpr IntegerRange<T> range(const T from, const T to) {
	return IntegerRange<T>(from, to);
}

template<typename T>
constexpr IntegerRange<T> inclusiveRange(const T to) {
	return IntegerRange<T>(0, to + 1);
}

template<typename T>
constexpr IntegerRange<T> inclusiveRange(const T from, const T to) {
	return IntegerRange<T>(from, to + 1);
}

template<typename T>
constexpr ReversedRange<IntegerRange<T>> downrange(const T from) {
	return reversed(range(from));
}

template<typename T>
constexpr ReversedRange<IntegerRange<T>> downrange(const T from, const T to) {
	return reversed(range(to, from));
}

template<typename T>
constexpr ReversedRange<IntegerRange<T>> inclusiveDownrange(const T from) {
	return reversed(range(from + 1));
}

template<typename T>
constexpr ReversedRange<IntegerRange<T>> inclusiveDownrange(const T from, const T to) {
	return reversed(range(to, from + 1));
}

template<typename R>
void sort(R& range) {
	std::sort(range.begin(), range.end());
}

template<typename R, typename Comp>
void sort(R& range, Comp comp) {
	std::sort(range.begin(), range.end(), comp);
}

template<typename R>
void reverse(R& range) {
	std::reverse(range.begin(), range.end());
}

template<typename R, typename T>
auto lower_bound(const R& range, const T& value) -> decltype(range.begin()) {
	return std::lower_bound(range.begin(), range.end(), value);
}

template<typename R, typename T, typename Comp>
auto lower_bound(const R& range, const T& value, Comp comp) -> decltype(range.begin()) {
	return std::lower_bound(range.begin(), range.end(), value, comp);
}

template<typename R, typename T>
auto upper_bound(const R& range, const T& value) -> decltype(range.begin()) {
	return std::upper_bound(range.begin(), range.end(), value);
}

template<typename R, typename T, typename Comp>
auto upper_bound(const R& range, const T& value, Comp comp) -> decltype(range.begin()) {
	return std::upper_bound(range.begin(), range.end(), value, comp);
}

template<typename R>
auto min_element(const R& range) -> decltype(range.begin()) {
	return std::min_element(range.begin(), range.end());
}

template<typename R>
auto max_element(const R& range) -> decltype(range.begin()) {
	return std::max_element(range.begin(), range.end());
}

template<typename R>
bool next_permutation(R& range) {
	return std::next_permutation(range.begin(), range.end());
}

template<typename T>
void unique(std::vector<T>& range) {
	range.erase(std::unique(range.begin(), range.end()), range.end());
}

template<typename T>
void dedup(std::vector<T>& range) {
	sort(range);
	unique(range);
}

template<typename R>
R sorted(R range) {
	sort(range);
	return range;
}

template<typename R, typename Comp>
R sorted(R range, Comp comp) {
	sort(range, comp);
	return range;
}



static constexpr long double PI = 3.14159265358979323846;
static constexpr long double EPS = 1e-9;
static constexpr int32_t INF = 0x3f3f3f3f;
static constexpr int64_t LINF = 0x3f3f3f3f3f3f3f3fLL;

static constexpr int32_t DX[8] = { -1, 0, 1, 0, -1, -1, 1, 1 };
static constexpr int32_t DY[8] = { 0, 1, 0, -1, -1, 1, 1, -1 };

static constexpr uint32_t kDayMonth[12] = { 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 };

enum Direction : uint8_t {
	Up,
	Right,
	Down,
	Left,
	UpLeft,
	UpRight,
	DownRight,
	DownLeft,
	North = Up,
	East = Right,
	South = Down,
	West = Left,
	NorthWest = UpLeft,
	NorthEast = UpRight,
	SouthEast = DownRight,
	SouthWest = DownLeft,
	Unknown = 255
};


#include <cstdint>

















namespace Random {

inline std::chrono::system_clock::rep current_timestamp() {
	return std::chrono::system_clock::now().time_since_epoch().count();
}

inline uint64_t get_new_rand_seed() {
	return static_cast<uint64_t>(current_timestamp());
}

inline uint64_t get_rand_seed() {
	static const uint64_t kRandSeed = get_new_rand_seed();
	return kRandSeed;
}

using Engine = std::mt19937_64;

template<typename T>
using Distribution = std::uniform_int_distribution<T>;

inline Engine& random_engine() {
	static Engine engine(get_rand_seed());
	return engine;
}

template<typename T>
inline T get(const T to) {
	Distribution<T> distribution(0, to);
	return distribution(random_engine());
}

template<typename T>
inline T get(const T from, const T to) {
	return get(to - from) + from;
}

template<typename T = int32_t>
inline T get() {
	return get(static_cast<T>(0), std::numeric_limits<T>::max());
}

}  // namespace Random


template<typename T, typename std::enable_if<std::is_integral<T>::value>::type* = nullptr>
class SafeIntegralHash {
public:
	const T kRandomValue = static_cast<T>(Random::current_timestamp());

	T operator ()(const T& value) const {
		return value ^ kRandomValue;
	}
};

template<typename Key, typename Value, typename std::enable_if<std::is_integral<Key>::value>::type* = nullptr>
using SafeUnorderedMap = std::unordered_map<Key, Value, SafeIntegralHash<Key>>;

template<typename Key, typename Value, typename std::enable_if<std::is_integral<Key>::value>::type* = nullptr>
using SafeUnorderedMultiMap = std::unordered_multimap<Key, Value, SafeIntegralHash<Key>>;

template<typename Key, typename std::enable_if<std::is_integral<Key>::value>::type* = nullptr>
using SafeUnorderedSet = std::unordered_set<Key, SafeIntegralHash<Key>>;

template<typename Key, typename std::enable_if<std::is_integral<Key>::value>::type* = nullptr>
using SafeUnorderedMultiSet = std::unordered_multiset<Key, SafeIntegralHash<Key>>;





namespace std {

#ifndef STD_HASH_PAIR
template<typename T, typename U>
struct hash<std::pair<T, U>> {
	std::size_t operator()(const std::pair<T, U>& pair) const {
		return pair.first * 877117 + pair.second;
	}
};
#endif

}  // namespace std







template<typename T, typename std::enable_if<std::is_floating_point<T>::value>::type* = nullptr>
inline bool is_equal_to_zero(const T& a) {
	return abs(a) < EPS;
}

template<typename T, typename std::enable_if<!std::is_floating_point<T>::value>::type* = nullptr>
inline bool is_equal_to_zero(const T& a) {
	return a == 0;
}



template<typename T>
inline bool umin(T& a, const T& b) {
	return (b < a ? a = b, true : false);
}

template<typename T>
inline bool umax(T& a, const T& b) {
	return (a < b ? a = b, true : false);
}

template<typename T, size_t N>
struct MakeVector {
	template<typename... Args,
		typename R = std::vector<decltype(MakeVector<T, N - 1>::make_vector(std::declval<Args>()...)) >>
		static R make_vector(size_t first, Args... sizes) {
		auto inner = MakeVector<T, N - 1>::make_vector(sizes...);
		return R(first, inner);
	}
};

template<typename T>
struct MakeVector<T, 1> {
	/// caide keep
	template<typename R = std::vector<T>>
	static R make_vector(std::size_t size, const T& value) {
		return R(size, value);
	}
};

template<typename T, typename... Args>
auto make_vector(Args... args) -> decltype(MakeVector<T, sizeof...(Args)-1>::make_vector(args...)) {
	return MakeVector<T, sizeof...(Args)-1>::make_vector(args...);
}

template<typename T>
inline bool is_leap_year(const T year) {
	return year % 400 == 0 || (year % 100 != 0 && (year & 3) == 0);
}

inline size_t get_days_in_month(const size_t month, const size_t year) {
	return kDayMonth[month] + (month == 1 && is_leap_year(year) ? 1 : 0);
}

#define X first
#define Y second
#define all(a) a.begin(), a.end()
#define rall(a) a.rbegin(), a.rend()





#ifdef _MSC_VER
#define LLD "%I64d"
#define LLU "%I64u"
#else
#define LLD "%lld"
#define LLU "%llu"
#endif

using ll = int64_t;
using ull = uint64_t;
using uint = uint32_t;
using ld = long double;

using pii = std::pair<int, int>;
using pll = std::pair<ll, ll>;
using pli = std::pair<ll, int>;

using vi = std::vector<int>;
using vvi = std::vector<vi>;
using vp = std::vector<pii>;








template<typename T>
inline T gcd(T a, T b) {
	while (b != 0) {
		a %= b;
		std::swap(a, b);
	}
	return a;
}

template<typename T>
inline T lcm(const T& a, const T& b) {
    return a / gcd(a, b) * b;
}

template<typename T>
constexpr inline T sqr(const T& x) {
    return x * x;
}

template<typename T>
T extended_gcd(const T a, const T b, T& x, T& y) {
    if (a == 0) {
        x = 0;
        y = 1;
        return b;
    }
    const T p = b / a;
    const T g = extended_gcd(b - p * a, a, y, x);
    x -= p * y;
    return g;
}

// |x|, |y| <= max(|a|, |b|, |c|)
template<typename T>
bool solve_diophantine(const T a, const T b, const T c, T& x, T& y, T& g) {
    if (a == 0 && b == 0) {
        if (c == 0) {
            x = 0;
            y = 0;
            g = 0;
            return true;
        }
        return false;
    }
    if (a == 0) {
        if (c % b == 0) {
            x = 0;
            y = c / b;
            g = std::abs(b);
            return true;
        }
        return false;
    }
    if (b == 0) {
        if (c % a == 0) {
            x = c / a;
            y = 0;
            g = std::abs(a);
            return true;
        }
        return false;
    }
    g = extended_gcd(a, b, x, y);
    if (c % g != 0) {
        return false;
    }
    T adjusted_c = c;
    const T dx = adjusted_c / a;
    adjusted_c -= dx * a;
    const T dy = adjusted_c / b;
    adjusted_c -= dy * b;
    x = dx + static_cast<T>(static_cast<__int128>(x) * (adjusted_c / g) % b);
    y = dy + static_cast<T>(static_cast<__int128>(y) * (adjusted_c / g) % a);
    g = std::abs(g);
    return true;
}

template<typename T, typename std::enable_if<std::is_floating_point<T>::value>::type* = nullptr>
size_t solve_quadratic_equation(const T a, const T b, const T c, T& x1, T& x2) {
    const T D = sqr(b) - 4 * a * c;
    if (D < -EPS) {
        return 0;
    }
    if (is_equal_to_zero(D)) {
        const T x = -b / (a * 2);
        x1 = x;
        return 1;
    }
    x1 = (-b + std::sqrt(D)) / (a * 2);
    x2 = (-b - std::sqrt(D)) / (a * 2);
    return 2;
}

template<typename T>
T factorial(const size_t n) {
	T ret = 1;
	for (size_t i = 2; i <= n; ++i) {
		ret *= i;
	}
	return ret;
}

template<typename T, typename U>
inline T binpow(T a, U b) {
	static_assert(std::is_integral<U>::value, "Degree must be integral. For real degree use pow().");
	T ret = 1;
	while (b != 0) {
		if ((b & 1) != 0) {
			ret *= a;
		}
		a *= a;
		b >>= 1;
	}
	return ret;
}

template<typename T, typename U, typename Q>
inline T binpow(T a, U b, Q mod) {
	static_assert(std::is_integral<U>::value, "Degree must be integral. For real degree use pow().");
	long long ret = 1;
	a %= mod;
	while (b != 0) {
		if ((b & 1) != 0) {
			ret = ret * a % mod;
		}
		a = a * a % mod;
		b >>= 1;
	}
	return ret % mod;
}

template<typename T>
T digit_sum(const T n) {
	T res = 0;
	T tmp = abs(n);
	while (tmp > 0) {
		res += tmp % 10;
		tmp /= 10;
	}
	return res;
}

template<typename T>
std::vector<size_t> digits(const T n) {
	if (n == 0) {
		return{ 0 };
	}
	std::vector<size_t> res;
	T tmp = abs(n);
	while (tmp > 0) {
		res.emplace_back(tmp % 10);
		tmp /= 10;
	}
	return res;
}

template<typename T>
size_t digit_count(const T n) {
	return digits(n).size();
}

template<typename T>
std::vector<T> divisors_vector(const T n) {
	const auto approximate_divisors_count = static_cast<size_t>(std::cbrt(n));
	std::vector<T> ans;
	ans.reserve(approximate_divisors_count);
	const auto floored_square_root = static_cast<T>(sqrt(n));
	for (T i = 1; i <= floored_square_root; ++i) {
		if (n % i == 0) {
			ans.emplace_back(i);
			if (n / i != i) {
				ans.emplace_back(n / i);
			}
		}
	}
	return ans;
}

template<typename T, typename size_type = std::size_t>
std::vector<T> divisors_vector(const std::vector<std::pair<T, size_type>>& prime_divisors, const bool sorted = false) {
    static std::vector<T> buffer;

    size_t divisors_count = 1;
    for (const auto& prime_divisor : prime_divisors) {
        divisors_count *= prime_divisor.second + 1;
    }

    std::vector<T> factors = {1};
    factors.reserve(divisors_count);
    if (sorted) {
        buffer.resize(divisors_count);
    }

    for (const auto& prime_divisor : prime_divisors) {
        const T prime_div = prime_divisor.first;
        const size_t exponent = prime_divisor.second;

        const size_t prev_size = factors.size();
        for (size_t i = 0; i < exponent * prev_size; ++i) {
            factors.emplace_back(factors[factors.size() - prev_size] * prime_div);
        }

        if (sorted && factors[prev_size - 1] > prime_div) {
            for (size_t section = prev_size; section < factors.size(); section *= 2) {
                for (size_t i = 0; i + section < factors.size(); i += 2 * section) {
                    const size_t length = std::min(2 * section, factors.size() - i);
                    std::merge(factors.cbegin() + i,
                               factors.cbegin() + i + section,
                               factors.cbegin() + i + section,
                               factors.cbegin() + i + length,
                               buffer.begin());
                    std::copy(buffer.cbegin(), buffer.cbegin() + length, factors.begin() + i);
                }
            }
        }
    }
    return factors;
}

template<typename T>
T euler_function(T n) {
	T res = n;
	for (T i = 2; i * i <= n; ++i) {
		if (n % i == 0) {
			while (n % i == 0) {
				n /= i;
			}
			res -= res / i;
		}
	}
	if (n > 1) {
		res -= res / n;
	}
	return res;
}

template<typename T, typename U>
T inverse_element(const T n, const U mod)
// inverse element for prime mod
{
	return binpow(static_cast<long long>(n), mod - 2, mod);
}

inline void inverseElementForSegment(int r[], const int n, const int mod)
// inverse element for prime mod for numbers [1; mod - 1]
{
	r[1] = 1;
	for (int i = 2; i < n; ++i) {
		r[i] = (mod - static_cast<long long>(mod) / i * r[mod % i] % mod) % mod;
	}
}

template<typename T>
T inverse_element_comp_mod(const T n, const T mod)
// inverse element for composite mod using formula inv(n) = n^(phi(mod) - 1)
{
	return binpow(n, euler_function(mod) - 1, mod);
}

template<typename T>
void binomial_coefficients(const size_t n, std::vector<std::vector<T>>* c) {
	std::vector<std::vector<T>> result(n, std::vector<T>(n));
	for (size_t i = 0; i < n; ++i) {
		result[i][0] = 1;
		result[i][i] = 1;
		for (size_t j = 1; j < i; ++j) {
			result[i][j] = result[i - 1][j - 1] + result[i - 1][j];
		}
	}
	c->swap(result);
}

template<class T> std::string toRoman(T n) {
	const int ValsCount = 13;
	const int Vals[] = { 1, 4, 5, 9, 10, 40, 50, 90, 100, 400, 500, 900, 1000 };
	const char * Digits[] = { "I", "IV", "V", "IX", "X", "XL", "L", "XC", "C", "CD", "D", "CM", "M" };
	std::string res = "";
	int x = ValsCount - 1;
	while (n) {
		while (Vals[x] > n) {
			--x;
		}
		n -= Vals[x];
		res += Digits[x];
	}
	return res;
}

template<typename T>
std::vector<T> calc_powers(const T base, const size_t n) {
	std::vector<T> powers(n + 1);
	powers[0] = 1;
	for (size_t i = 1; i < powers.size(); ++i) {
		powers[i] = powers[i - 1] * base;
	}
	return powers;
}

template<typename T>
std::vector<T> calc_factorial(const size_t n) {
	std::vector<T> factorial(n + 1);
	factorial[0] = 1;
	for (size_t i = 1; i < factorial.size(); ++i) {
		factorial[i] = factorial[i - 1] * i;
	}
	return factorial;
}

inline uint32_t abs(const uint32_t x) {
	return x;
}

inline uint64_t abs(const uint64_t x) {
	return x;
}







template<typename T>
inline bool is_prime(const T& n)
// Straightforward check in O(sqrt(N))
{
	if (n < 2) {
		return false;
	}
	const T floored_square_root = static_cast<T>(sqrt(n));
	for (T i = 2; i <= floored_square_root; ++i) {
		if (n % i == 0) {
			return false;
		}
	}
	return true;
}

template<typename T>
inline T next_prime(const T& n) {
	for (T candidate = n; ; ++candidate) {
		if (is_prime(candidate)) {
			return candidate;
		}
	}
}

template<typename T>
inline T random_prime(const T from, const T to)
// note that resulting prime is not necessary from [from, to] interval
// [from, to] sets an interval to select a random integer to get next_prime()
{
	return next_prime(Random::get(from, to));
}

template<typename T = uint32_t>
inline T random_big_prime(const T from = 1000000000, const T to = INF) {
	return random_prime(from, to);
}

inline std::vector<bool> eratosthenes_sieve(const size_t n) {
	if (n < 2) {
		return std::vector<bool>(n, false);
	}
	std::vector<bool> prime(n, true);
	prime[0] = false;
	prime[1] = false;
	for (size_t i = 4; i < n; i += 2) {
		prime[i] = false;
	}
	for (size_t i = 3; i * i < n; i += 2) {
		if (prime[i]) {
			const size_t delta = i * 2;
			for (size_t j = i * i; j < n; j += delta) {
				prime[j] = false;
			}
		}
	}
	return prime;
}

template<typename T = uint32_t>
inline std::vector<T> primes_vector(const size_t n) {
	if (n < 2) {
		return {};
	}
	const std::vector<bool> prime = eratosthenes_sieve(n);
	std::vector<T> result = {2};
	for (size_t i = 3; i < n; i += 2) {
		if (prime[i]) {
			result.emplace_back(i);
		}
	}
	return result;
}

template<typename T = uint32_t>
inline std::vector<T> min_prime_div_vector(const size_t n) {
	std::vector<T> result(n, 0);
	result[0] = 1;
	result[1] = 1;
	for (size_t i = 2; i < n; i += 2) {
		result[i] = 2;
	}
	size_t index = 3;
	while (index * index < n) {
		if (result[index] == 0) {
			result[index] = index;
			const size_t delta = index * 2;
			for (size_t j = index * index; j < n; j += delta) {
				if (result[j] == 0) {
					result[j] = index;
				}
			}
		}
		index += 2;
	}
	while (index < n) {
		if (result[index] == 0) {
			result[index] = index;
		}
		++index;
	}
	return result;
}

template<typename T, typename size_type = std::size_t>
std::vector<std::pair<T, size_type>> get_prime_divisors_vector(const T n, const std::vector<T>& min_prime_div_vector) {
	std::vector<std::pair<T, size_type>> prime_divisors;
	T cur = n;
	while (cur > 1) {
		const T min_div = min_prime_div_vector[cur];
		if (!prime_divisors.empty() && prime_divisors.back().first == min_div) {
			++prime_divisors.back().second;
		} else {
			prime_divisors.emplace_back(min_div, 1);
		}
		cur /= min_div;
	}
	return prime_divisors;
}

template<typename T, typename size_type = std::size_t>
std::vector<std::pair<T, size_type>> get_prime_divisors_vector(const T n) {
	std::vector<std::pair<T, size_type>> prime_divisors;
	T cur = n;
	for (T i = 2; static_cast<int64_t>(i) * i <= cur; ++i) {
		if (cur % i == 0) {
			size_type degree = 0;
			while (cur % i == 0) {
				++degree;
				cur /= i;
			}
			prime_divisors.emplace_back(i, degree);
		}
	}
	if (cur > 1) {
		prime_divisors.emplace_back(cur, 1);
	}
	return prime_divisors;
}


class DKolichestvoPar {
public:
	static constexpr bool kWriteCaseNumber = false;
	static constexpr bool kMultiTest = true;

	const int kMaxn = 22222222;

	void solve(std::istream& in, std::ostream& out) {
        ll c, d, x;
        in >> c >> d >> x;

        const auto x_prime_divs = get_prime_divisors_vector<int>(x, min_prime_div);
        auto divs = divisors_vector(x_prime_divs);
        ll ans = 0;
        for (ll div : divs) {
            const ll cg = x + d * div;
            if (cg % c != 0) {
                continue;
            }
            const ll l = cg / c;
            if (l % div != 0) {
                continue;
            }
            // debug(div, l, cnt_divs);
            ans += (1 << cnt_divs[l / div]);
        }
        out << ans << std::endl;
	}

	std::vector<int> min_prime_div;
	std::vector<int> cnt_divs;

	DKolichestvoPar() : cnt_divs(kMaxn, 0) {
	    min_prime_div = min_prime_div_vector<int>(kMaxn);
        cnt_divs[1] = 0;
        for (int i : range(2, kMaxn)) {
            const int prev = i / min_prime_div[i];
            cnt_divs[i] = cnt_divs[prev];
            if (min_prime_div[prev] != min_prime_div[i]) {
                ++cnt_divs[i];
            }
        }
	}


	static constexpr bool kUseCustomChecker = false;
	bool check(std::istringstream in, std::istringstream jury, std::istringstream out) {
		return true;
	}


	static constexpr size_t kGeneratedTestsCount = 0;
	static constexpr bool kStopAfterFirstFail = false;
	static void generate_test(std::ostream& test) {}

};


int main() {
	DKolichestvoPar solver;
	std::istream& in(std::cin);
	std::ostream& out(std::cout);

	std::ios_base::sync_with_stdio(false);
	std::cin.tie(nullptr);
	srand(time(nullptr));
	out << std::fixed << std::setprecision(16);
	if (DKolichestvoPar::kMultiTest) {
		size_t tests_count;
		in >> tests_count;
		for (size_t test_id : inclusiveRange<size_t>(1, tests_count)) {
			if (DKolichestvoPar::kWriteCaseNumber) {
				out << "Case #" << test_id << ": ";
			}
			solver.solve(in, out);
		}
	} else {
		solver.solve(in, out);
	}
	return 0;
}
