#define _USE_MATH_DEFINES
#define _CRT_SECURE_NO_WARNINGS
#pragma GCC optimize("O3")
#pragma GCC target("avx2")
#pragma GCC optimize("unroll-loops")
//#pragma GCC optimize ("Ofast")
#include "bits/stdc++.h"
#define rep(i, n) for (int i = 0; i < (n); ++i)
#define FOR(i, m, n) for (int i = (m); i < (n); ++i)
#define rrep(i, n) for (int i = (n)-1; i >= 0; --i)
#define rfor(i, m, n) for (int i = (m); i >= (n); --i)
#define unless(c) if (!(c))
#define sz(x) ((int)(x).size())
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define range_it(a, l, r) (a).begin() + (l), (a).begin() + (r)

using namespace std;
using ll = long long;
using LD = long double;
using VB = vector<bool>;
using VVB = vector<VB>;
using VI = vector<int>;
using VVI = vector<VI>;
using VL = vector<ll>;
using VVL = vector<VL>;
using VS = vector<string>;
using VD = vector<LD>;
using PII = pair<int, int>;
using VP = vector<PII>;
using PLL = pair<ll, ll>;
using VPL = vector<PLL>;
template <class T> using PQ = priority_queue<T>;
template <class T> using PQS = priority_queue<T, vector<T>, greater<T>>;
constexpr int inf = (int)1e9;
constexpr ll inf_ll = (ll)1e18, MOD = 1000000007;
constexpr LD PI = M_PI, EPS = 1e-12;

// --- input --- //
#ifdef _WIN32
#define getchar_unlocked _getchar_nolock
#define putchar_unlocked _putchar_nolock
#define fwrite_unlocked fwrite
#define fflush_unlocked fflush
#endif
class Input {
	static int gc() {
		return getchar_unlocked();
	}
	template <class T> static void i(T& v) {
		cin >> v;
	}
	static void i(char& v) {
		while (isspace(v = gc()))
			;
	}
	static void i(bool& v) {
		v = in<char>() != '0';
	}
	static void i(string& v) {
		v.clear();
		char c;
		for (i(c); !isspace(c); c = gc()) v += c;
	}
	static void i(int& v) {
		bool neg = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void i(long long& v) {
		bool neg = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void i(double& v) {
		double dp = 1;
		bool neg = false, adp = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c) || c == '.'; c = gc()) {
			if (c == '.')
				adp = true;
			else if (adp)
				v += (c - '0') * (dp *= 0.1);
			else
				v = v * 10 + (c - '0');
		}
		if (neg) v = -v;
	}
	static void i(long double& v) {
		long double dp = 1;
		bool neg = false, adp = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c) || c == '.'; c = gc()) {
			if (c == '.')
				adp = true;
			else if (adp)
				v += (c - '0') * (dp *= 0.1);
			else
				v = v * 10 + (c - '0');
		}
		if (neg) v = -v;
	}
	template <class T, class U> static void i(pair<T, U>& v) {
		i(v.first);
		i(v.second);
	}
	template <class T> static void i(vector<T>& v) {
		for (auto& e : v) i(e);
	}
	template <size_t N = 0, class T> static void input_tuple(T& v) {
		if constexpr (N < tuple_size_v<T>) {
			i(get<N>(v));
			input_tuple<N + 1>(v);
		}
	}
	template <class... T> static void i(tuple<T...>& v) {
		input_tuple(v);
	}
	struct InputV {
		int n, m;
		InputV(int _n) : n(_n), m(0) {}
		InputV(const pair<int, int>& nm) : n(nm.first), m(nm.second) {}
		template <class T> operator vector<T>() {
			vector<T> v(n);
			i(v);
			return v;
		}
		template <class T> operator vector<vector<T>>() {
			vector<vector<T>> v(n, vector<T>(m));
			i(v);
			return v;
		}
	};

public:
	static string get_line() {
		string v;
		char c;
		for (i(c); c != '\n' && c != '\0'; c = gc()) v += c;
		return v;
	}
	template <class T> static T in() {
		T v;
		i(v);
		return v;
	}
	template <class T> operator T() const {
		return in<T>();
	}
	int operator--(int) const {
		return in<int>() - 1;
	}
	InputV operator[](int n) const {
		return InputV(n);
	}
	InputV operator[](const pair<int, int>& n) const {
		return InputV(n);
	}
	void operator()() const {}
	template <class H, class... T> void operator()(H&& h, T&&... t) const {
		i(h);
		operator()(forward<T>(t)...);
	}

private:
	template <template <class...> class, class...> struct Multiple;
	template <template <class...> class V, class Head, class... Tail> struct Multiple<V, Head, Tail...> {
		template <class... Args> using vec = V<vector<Head>, Args...>;
		using type = typename Multiple<vec, Tail...>::type;
	};
	template <template <class...> class V> struct Multiple<V> { using type = V<>; };
	template <class... T> using multiple_t = typename Multiple<tuple, T...>::type;
	template <size_t N = 0, class T> void in_multiple(T& t) const {
		if constexpr (N < tuple_size_v<T>) {
			auto& vec = get<N>(t);
			using V = typename remove_reference_t<decltype(vec)>::value_type;
			vec.push_back(i<V>());
			in_multiple<N + 1>(t);
		}
	}

public:
	template <class... T> auto multiple(int H) const {
		multiple_t<T...> res;
		while (H--) in_multiple(res);
		return res;
	}
} in;
#define input(T) Input::in<T>()
#define INT input(int)
#define LL input(ll)
#define STR input(string)
#define inputs(T, ...) \
	T __VA_ARGS__;     \
	in(__VA_ARGS__)
#define ini(...) inputs(int, __VA_ARGS__)
#define inl(...) inputs(ll, __VA_ARGS__)
#define ins(...) inputs(string, __VA_ARGS__)

// --- output --- //
struct BoolStr {
	const char *t, *f;
	BoolStr(const char* _t, const char* _f) : t(_t), f(_f) {}
} Yes("Yes", "No"), yes("yes", "no"), YES("YES", "NO"), Int("1", "0");
struct DivStr {
	const char *d, *l;
	DivStr(const char* _d, const char* _l) : d(_d), l(_l) {}
} spc(" ", "\n"), no_spc("", "\n"), end_line("\n", "\n"), comma(",", "\n"), no_endl(" ", "");
class Output {
	BoolStr B{Yes};
	DivStr D{spc};
	void p(int v) const {
		if (v < 0) putchar_unlocked('-'), v = -v;
		char b[10];
		int i = 0;
		while (v) b[i++] = '0' + v % 10, v /= 10;
		if (!i) b[i++] = '0';
		while (i--) putchar_unlocked(b[i]);
	}
	void p(ll v) const {
		if (v < 0) putchar_unlocked('-'), v = -v;
		char b[20];
		int i = 0;
		while (v) b[i++] = '0' + v % 10, v /= 10;
		if (!i) b[i++] = '0';
		while (i--) putchar_unlocked(b[i]);
	}
	void p(bool v) const {
		p(v ? B.t : B.f);
	}
	void p(char v) const {
		putchar_unlocked(v);
	}
	void p(const char* v) const {
		fwrite_unlocked(v, 1, strlen(v), stdout);
	}
	void p(double v) const {
		printf("%.20f", v);
	}
	void p(LD v) const {
		printf("%.20Lf", v);
	}
	template <class T> void p(const T& v) const {
		cout << v;
	}
	template <class T, class U> void p(const pair<T, U>& v) const {
		p(v.first);
		p(D.d);
		p(v.second);
	}
	template <class T> void p(const vector<T>& v) const {
		rep(i, sz(v)) {
			if (i) p(D.d);
			p(v[i]);
		}
	}
	template <class T> void p(const vector<vector<T>>& v) const {
		rep(i, sz(v)) {
			if (i) p(D.l);
			p(v[i]);
		}
	}

public:
	Output& operator()() {
		p(D.l);
		return *this;
	}
	template <class H> Output& operator()(H&& h) {
		p(h);
		p(D.l);
		return *this;
	}
	template <class H, class... T> Output& operator()(H&& h, T&&... t) {
		p(h);
		p(D.d);
		return operator()(forward<T>(t)...);
	}
	template <class It> Output& range(const It& l, const It& r) {
		for (It i = l; i != r; i++) {
			if (i != l) p(D.d);
			p(*i);
		}
		p(D.l);
		return *this;
	}
	template <class T> Output& range(const T& a) {
		range(a.begin(), a.end());
		return *this;
	}
	template <class... T> void exit(T&&... t) {
		operator()(forward<T>(t)...);
		std::exit(EXIT_SUCCESS);
	}
	Output& flush() {
		fflush_unlocked(stdout);
		return *this;
	}
	Output& set(const BoolStr& b) {
		B = b;
		return *this;
	}
	Output& set(const DivStr& d) {
		D = d;
		return *this;
	}
	Output& set(const char* t, const char* f) {
		B = BoolStr(t, f);
		return *this;
	}
} out;

// --- step --- //
template <class T> struct Step {
	class It {
		T a, b, c;

	public:
		constexpr It() : a(T()), b(T()), c(T()) {}
		constexpr It(T _b, T _c, T _s) : a(_b), b(_c), c(_s) {}
		constexpr It& operator++() {
			--b;
			a += c;
			return *this;
		}
		constexpr It operator++(int) {
			It tmp = *this;
			--b;
			a += c;
			return tmp;
		}
		constexpr const T& operator*() const {
			return a;
		}
		constexpr const T* operator->() const {
			return &a;
		}
		constexpr bool operator==(const It& i) const {
			return b == i.b;
		}
		constexpr bool operator!=(const It& i) const {
			return !(b == i.b);
		}
		constexpr T start() const {
			return a;
		}
		constexpr T count() const {
			return b;
		}
		constexpr T step() const {
			return c;
		}
	};
	constexpr Step(T b, T c, T s) : be(b, c, s) {}
	constexpr It begin() const {
		return be;
	}
	constexpr It end() const {
		return en;
	}
	constexpr T start() const {
		return be.start();
	}
	constexpr T count() const {
		return be.count();
	}
	constexpr T step() const {
		return be.step();
	}
	constexpr T sum() const {
		return start() * count() + step() * (count() * (count() - 1) / 2);
	}
	operator vector<T>() const {
		return as_vector();
	}
	template <class F> void each(const F& f) const {
		for (T i : *this) f(i);
	}
	auto as_vector() const {
		vector<T> res;
		res.reserve(count());
		each([&](T i) { res.push_back(i); });
		return res;
	}
	template <class F, class U = invoke_result_t<F, T>> auto map(const F& f) const {
		vector<U> res;
		res.reserve(count());
		each([&](T i) { res.push_back(f(i)); });
		return res;
	}
	template <class F> auto select(const F& f) const {
		vector<T> res;
		each([&](T i) {
			if (f(i)) res.push_back(i);
		});
		return res;
	}
	template <class F> int count_if(const F& f) const {
		int res = 0;
		each([&](T i) {
			if (f(i)) ++res;
		});
		return res;
	}
	template <class F> optional<T> find_if(const F& f) const {
		for (T i : *this)
			if (f(i)) return i;
		return nullopt;
	}
	template <class F> auto max_by(const F& f) const {
		auto v = map(f);
		return *max_element(v.begin(), v.end());
	}
	template <class F> auto min_by(const F& f) const {
		auto v = map(f);
		return *min_element(v.begin(), v.end());
	}
	template <class F> bool all_of(const F& f) const {
		for (T i : *this)
			if (!f(i)) return false;
		return true;
	}
	template <class F> bool any_of(const F& f) const {
		for (T i : *this)
			if (f(i)) return true;
		return false;
	}
	template <class F, class U = invoke_result_t<F, T>> auto sum(const F& f) const {
		U res = 0;
		each([&](T i) { res += static_cast<U>(f(i)); });
		return res;
	}
	using value_type = T;
	using iterator = It;

private:
	It be, en;
};
template <class T> inline constexpr auto step(T a) {
	return Step<T>(0, a, 1);
}
template <class T> inline constexpr auto step(T a, T b) {
	return Step<T>(a, b - a, 1);
}
template <class T> inline constexpr auto step(T a, T b, T c) {
	return Step<T>(a, a < b ? (b - a - 1) / c + 1 : 0, c);
}

// --- functions --- //
inline namespace {
	template <class T> inline void Sort(T& a) {
		sort(all(a));
	}
	template <class T> inline void RSort(T& a) {
		sort(rall(a));
	}
	template <class T, class F> inline void Sort(T& a, const F& f) {
		sort(all(a), f);
	}
	template <class T, class F> inline void RSort(T& a, const F& f) {
		sort(rall(a), f);
	}
	template <class T> inline T Sorted(T a) {
		Sort(a);
		return a;
	}
	template <class T> inline T RSorted(T a) {
		RSort(a);
		return a;
	}
	template <class T, class F> inline T Sorted(T& a, const F& f) {
		Sort(a, f);
		return a;
	}
	template <class T, class F> inline T RSorted(T& a, const F& f) {
		RSort(a, f);
		return a;
	}
	template <class T, class F> inline void SortBy(T& a, const F& f) {
		sort(all(a), [&](const auto& x, const auto& y) { return f(x) < f(y); });
	}
	template <class T, class F> inline void RSortBy(T& a, const F& f) {
		sort(rall(a), [&](const auto& x, const auto& y) { return f(x) < f(y); });
	}
	template <class T> inline void Reverse(T& a) {
		reverse(all(a));
	}
	template <class T> inline void Unique(T& a) {
		a.erase(unique(all(a)), a.end());
	}
	template <class T> inline void Uniq(T& a) {
		Sort(a);
		Unique(a);
	}
	template <class T> inline void Rotate(T& a, int left) {
		rotate(a.begin(), a.begin() + left, a.end());
	}
	template <class T> inline T Reversed(T a) {
		Reverse(a);
		return a;
	}
	template <class T> inline T Uniqued(T a) {
		Unique(a);
		return a;
	}
	template <class T> inline T Uniqed(T a) {
		Uniq(a);
		return a;
	}
	template <class T> inline T Rotated(T a, int left) {
		Rotate(a, left);
		return a;
	}
	template <class T> inline auto Max(const T& a) {
		return *max_element(all(a));
	}
	template <class T> inline auto Min(const T& a) {
		return *min_element(all(a));
	}
	template <class T> inline int MaxPos(const T& a) {
		return max_element(all(a)) - a.begin();
	}
	template <class T> inline int MinPos(const T& a) {
		return min_element(all(a)) - a.begin();
	}
	template <class T, class F> inline auto MaxBy(const T& a, const F& f) {
		return *max_element(all(a), [&](const auto& x, const auto& y) { return f(x) < f(y); });
	}
	template <class T, class F> inline auto MinBy(const T& a, const F& f) {
		return *min_element(all(a), [&](const auto& x, const auto& y) { return f(x) < f(y); });
	}
	template <class T, class U> inline int Count(const T& a, const U& v) {
		return count(all(a), v);
	}
	template <class T, class F> inline int CountIf(const T& a, const F& f) {
		return count_if(all(a), f);
	}
	template <class T, class U> inline int Find(const T& a, const U& v) {
		return find(all(a), v) - a.begin();
	}
	template <class T, class F> inline int FindIf(const T& a, const F& f) {
		return find_if(all(a), f) - a.begin();
	}
	template <class T, class U = typename T::value_type> inline U Sum(const T& a) {
		return accumulate(all(a), U());
	}
	template <class T, class U> inline bool Includes(const T& a, const U& v) {
		return find(all(a), v) != a.end();
	}
	template <class T, class F> inline auto Sum(const T& v, const F& f) {
		return accumulate(next(v.begin()), v.end(), f(*v.begin()), [&](auto a, auto b) { return a + f(b); });
	}
	template <class T, class U> inline int Lower(const T& a, const U& v) {
		return lower_bound(all(a), v) - a.begin();
	}
	template <class T, class U> inline int Upper(const T& a, const U& v) {
		return upper_bound(all(a), v) - a.begin();
	}
	template <class T, class F> inline void RemoveIf(T& a, const F& f) {
		a.erase(remove_if(all(a), f), a.end());
	}
	template <class F> inline auto Vector(size_t size, const F& f) {
		vector<invoke_result_t<F, size_t>> res(size);
		for (size_t i = 0; i < size; ++i) res[i] = f(i);
		return res;
	}
	template <class T> inline auto Grid(size_t h, size_t w, const T& v = T()) {
		return vector<vector<T>>(h, vector<T>(w, v));
	}
	template <class T> inline auto Slice(const T& v, size_t i, size_t len) {
		return i < v.size() ? T(v.begin() + i, v.begin() + min(i + len, v.size())) : T();
	}
	template <class T, class F> inline auto Each(const T& v, F&& f) {
		for (auto& i : v) f(i);
	}
	template <class T, class F> inline auto Select(const T& v, const F& f) {
		T res;
		for (const auto& e : v)
			if (f(e)) res.push_back(e);
		return res;
	}
	template <class T, class F> inline auto Map(const T& v, F&& f) {
		vector<invoke_result_t<F, typename T::value_type>> res(v.size());
		size_t i = 0;
		for (const auto& e : v) res[i++] = f(e);
		return res;
	}
	template <class T, class F> inline auto MapIndex(const T& v, const F& f) {
		vector<invoke_result_t<F, size_t, typename T::value_type>> res(v.size());
		size_t i = 0;
		for (auto it = v.begin(); it != v.end(); ++it, ++i) res[i] = f(i, *it);
		return res;
	}
	template <class T, class F> inline auto TrueIndex(const T& v, const F& f) {
		vector<size_t> res;
		for (size_t i = 0; i < v.size(); ++i)
			if (f(v[i])) res.push_back(i);
		return res;
	}
	template <class T, class U = typename T::value_type> inline auto Indexed(const T& v) {
		vector<pair<U, int>> res(v.size());
		for (int i = 0; i < (int)v.size(); ++i) res[i] = make_pair(static_cast<U>(v[i]), i);
		return res;
	}
	inline auto operator*(string s, size_t n) {
		string res;
		for (size_t i = 0; i < n; ++i) res += s;
		return res;
	}
	template <class T> inline auto& operator<<(vector<T>& v, const vector<T>& v2) {
		v.insert(v.end(), all(v2));
		return v;
	}
	template <class T> inline T Ceil(T n, T m) {
		return (n + m - 1) / m;
	}
	template <class T> inline T Ceil2(T n, T m) {
		return Ceil(n, m) * m;
	}
	template <class T> inline T Tri(T n) {
		return (n & 1) ? (n + 1) / 2 * n : n / 2 * (n + 1);
	}
	template <class T> inline T nC2(T n) {
		return (n & 1) ? (n - 1) / 2 * n : n / 2 * (n - 1);
	}
	template <class T> inline T Mid(const T& l, const T& r) {
		return l + (r - l) / 2;
	}
	inline int pop_count(int n) {
		return bitset<32>(n).count();
	}
	inline int pop_count(ll n) {
		return bitset<64>(n).count();
	}
	template <class T> inline bool chmax(T& a, const T& b) {
		if (a < b) {
			a = b;
			return true;
		}
		return false;
	}
	template <class T> inline bool chmin(T& a, const T& b) {
		if (a > b) {
			a = b;
			return true;
		}
		return false;
	}
	template <class T> inline bool inRange(const T& v, const T& min, const T& max) {
		return min <= v && v < max;
	}
	template <class T> inline bool isSquere(T n) {
		T s = sqrt(n);
		return s * s == n || (s + 1) * (s + 1) == n;
	}
	template <class T = ll> inline T BIT(int b) {
		return T(1) << b;
	}
	template <class T, class U = typename T::value_type> inline U Gcdv(const T& v) {
		return accumulate(next(v.begin()), v.end(), U(*v.begin()), gcd<U, U>);
	}
	template <class T, class U = typename T::value_type> inline U Lcmv(const T& v) {
		return accumulate(next(v.begin()), v.end(), U(*v.begin()), lcm<U, U>);
	}
	template <class T> inline T Pow(T a, T n) {
		T r = 1;
		while (n > 0) {
			if (n & 1) r *= a;
			a *= a;
			n /= 2;
		}
		return r;
	}
	template <class T> inline T Powmod(T a, T n, T m = MOD) {
		T r = 1;
		while (n > 0) {
			if (n & 1)
				r = r * a % m, n--;
			else
				a = a * a % m, n /= 2;
		}
		return r;
	}
}  // namespace

// --- dump --- //
#if __has_include("dump.hpp")
#include "dump.hpp"
#elif __has_include("/home/yuruhiya/contest/dump.hpp")
#include "/home/yuruhiya/contest/dump.hpp"
#else
#define dump(...) ((void)0)
#endif

// ---------------------------------------------------------------- //

template <int MOD> struct modint {
	using T = long long;
	T n;
	constexpr modint(const T x = 0) : n(x % MOD) {
		if (n < 0) n += MOD;
	}
	constexpr int get_mod() const {
		return MOD;
	}
	constexpr modint operator+() const {
		return *this;
	}
	constexpr modint operator-() const {
		return n ? MOD - n : 0;
	}
	constexpr modint& operator++() {
		if (MOD <= ++n) n = 0;
		return *this;
	}
	constexpr modint& operator--() {
		if (n <= 0) n = MOD;
		n--;
		return *this;
	}
	constexpr modint operator++(int) {
		modint t = *this;
		++*this;
		return t;
	}
	constexpr modint operator--(int) {
		modint t = *this;
		--*this;
		return t;
	}
	constexpr modint next() const {
		return ++modint(*this);
	}
	constexpr modint pred() const {
		return --modint(*this);
	}
	constexpr modint operator+(const modint& m) const {
		return modint(*this) += m;
	}
	constexpr modint operator-(const modint& m) const {
		return modint(*this) -= m;
	}
	constexpr modint operator*(const modint& m) const {
		return modint(*this) *= m;
	}
	constexpr modint operator/(const modint& m) const {
		return modint(*this) /= m;
	}
	constexpr modint& operator+=(const modint& m) {
		n += m.n;
		if (n >= MOD) n -= MOD;
		return *this;
	}
	constexpr modint& operator-=(const modint& m) {
		n -= m.n;
		if (n < 0) n += MOD;
		return *this;
	}
	constexpr modint& operator*=(const modint& m) {
		n = n * m.n % MOD;
		return *this;
	}
	constexpr modint& operator/=(const modint& m) {
		T a = m.n, b = MOD, u = 1, v = 0;
		while (b) {
			T t = a / b;
			a -= t * b;
			swap(a, b);
			u -= t * v;
			swap(u, v);
		}
		n = n * u % MOD;
		if (n < 0) n += MOD;
		return *this;
	}
	constexpr bool operator==(const modint& m) const {
		return n == m.n;
	}
	constexpr bool operator!=(const modint& m) const {
		return n != m.n;
	}
	constexpr modint pow(T m) const {
		modint t = n, res = 1;
		while (m > 0) {
			if (m & 1) res *= t;
			t *= t;
			m >>= 1;
		}
		return res;
	}
	constexpr modint operator^(T m) const {
		return pow(m);
	}
	friend ostream& operator<<(ostream& os, const modint<MOD>& m) {
		return os << m.n;
	}
	friend istream& operator>>(istream& is, modint<MOD>& m) {
		return is >> m.n;
	}
};
using mint = modint<1000000007>;
using VM = vector<mint>;
inline mint operator""_m(unsigned long long n) {
	return n;
}

template <class T> struct Matrix {
	vector<vector<T>> A;

public:
	static Matrix I(size_t n) {
		Matrix A(n, n);
		for (size_t i = 0; i < n; ++i) {
			A[i][i] = 1;
		}
		return A;
	}
	Matrix() {}
	Matrix(size_t h, size_t w, const T& x = T()) : A(h, vector<T>(w, x)) {}
	Matrix(const vector<vector<T>>& _A) : A(_A) {}
	size_t height() const {
		return A.size();
	}
	size_t width() const {
		return A.front().size();
	}
	const vector<T>& operator[](int i) const {
		return A[i];
	}
	vector<T>& operator[](int i) {
		return A[i];
	}
	const vector<vector<T>>& operator*() const {
		return A;
	}
	Matrix& operator+=(const Matrix& B) {
		assert(height() == B.height() && width() == B.width());
		for (size_t i = 0; i < height(); ++i) {
			for (size_t j = 0; j < width(); ++j) {
				A[i][j] += B[i][j];
			}
		}
		return *this;
	}
	Matrix& operator-=(const Matrix& B) {
		assert(height() == B.height() && width() == B.width());
		for (size_t i = 0; i < height(); ++i) {
			for (size_t j = 0; j < width(); ++j) {
				A[i][j] -= B[i][j];
			}
		}
		return *this;
	}
	Matrix& operator*=(const Matrix& B) {
		size_t n = B.width();
		assert(width() == B.height());
		vector<vector<T>> C(height(), vector<T>(n, 0));
		for (size_t i = 0; i < height(); i++) {
			for (size_t j = 0; j < n; j++) {
				for (size_t k = 0; k < width(); k++) {
					C[i][j] += A[i][k] * B[k][j];
				}
			}
		}
		A.swap(C);
		return *this;
	}
	Matrix& operator^=(long long k) {
		Matrix B = Matrix::I(height());
		while (k > 0) {
			if (k & 1) {
				B *= *this;
			}
			*this *= *this;
			k >>= 1;
		}
		A.swap(B.A);
		return *this;
	}
	Matrix operator+(const Matrix& B) const {
		return Matrix(*this) += B;
	}
	Matrix operator-(const Matrix& B) const {
		return Matrix(*this) -= B;
	}
	Matrix operator*(const Matrix& B) const {
		return Matrix(*this) *= B;
	}
	Matrix operator^(const long long k) const {
		return Matrix(*this) ^= k;
	}
	Matrix pow(long long k) const {
		return *this ^ k;
	}
};

int main() {
	int k = in, m = in;
	ll n = in;
	VVI a = in[{m, 3}];
	rep(i, m) rep(j, 3) a[i][j]--;

	Matrix<mint> A(m, m);
	rep(i, m) {
		rep(j, m) {
			if (a[i][0] == a[j][1] && a[i][1] == a[j][2]) {
				A[i][j] = 1;
			}
		}
	}
	Matrix<mint> B(m, 1);
	rep(i, m) {
		if (a[i][0] == 0) {
			B[i][0] = 1;
		}
	}

	A ^= n - 3;
	A *= B;

	mint ans = 0;
	rep(i, m) {
		if (a[i][2] == 0) {
			ans += A[i][0];
		}
	}
	out(ans);
}