結果

問題 No.8046 yukicoderの過去問
コンテスト
ユーザー kyon2326
提出日時 2020-12-15 19:07:10
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 386 ms / 2,000 ms
コード長 22,632 bytes
コンパイル時間 3,578 ms
コンパイル使用メモリ 228,256 KB
最終ジャッジ日時 2025-01-17 01:08:25
ジャッジサーバーID
（参考情報） 		judge5 / judge5
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ファイルパターン 結果
other AC * 9
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ソースコード

diff # 

#include <bits/stdc++.h>
using namespace std;
/*
#include <atcoder/all>
using namespace atcoder;
*/
/*
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
using bll = boost::multiprecision::cpp_int;
using bdouble = boost::multiprecision::number<boost::multiprecision::cpp_dec_float<100>>;
using namespace boost::multiprecision;
*/
#ifdef LOCAL_TEST
	#define BOOST_STACKTRACE_USE_ADDR2LINE
	#define BOOST_STACKTRACE_ADDR2LINE_LOCATION /usr/local/opt/binutils/bin/addr2line
	#define _GNU_SOURCE 1
	#include <boost/stacktrace.hpp>
	namespace std {
		template<typename T> class dvector : public std::vector<T> {
		public:
			dvector() : std::vector<T>() {}
			explicit dvector(size_t n, const T& value = T()) : std::vector<T>(n, value) {}
			dvector(const std::vector<T>& v) : std::vector<T>(v) {}
			dvector(const std::initializer_list<T> il) : std::vector<T>(il) {}
			dvector(const std::string::iterator first, const std::string::iterator last) : std::vector<T>(first, last) {}
			dvector(const typename std::vector<T>::iterator first, const typename std::vector<T>::iterator last) : std::vector<T>(first, last) {}
			dvector(const typename std::vector<T>::reverse_iterator first, const typename std::vector<T>::reverse_iterator last) : std::vector<T>(first, last) {}
			dvector(const typename std::vector<T>::const_iterator first, const typename std::vector<T>::const_iterator last) : std::vector<T>(first, last) {}
			dvector(const typename std::vector<T>::const_reverse_iterator first, const typename std::vector<T>::const_reverse_iterator last) : std::vector<T>(first, last) {}
			T& operator[](size_t n) {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
			const T& operator[](size_t n) const {
				if (this->size() <= n) { std::cerr << boost::stacktrace::stacktrace() << '\n' << "vector::_M_range_check: __n (which is " << n << ") >= this->size() (which is " << this->size() << ")" << '\n'; } return this->at(n);
			}
		};
	}
	class dbool {
	private:
		bool boolvalue;
	public:
		dbool() : boolvalue(false) {}
		dbool(bool b) : boolvalue(b) {}
		operator bool&() { return boolvalue; }
		operator const bool&() const { return boolvalue; }
	};
	#define vector dvector
	#define bool dbool
	class SIGFPE_exception : std::exception {};
	class SIGSEGV_exception : std::exception {};
	void catch_SIGFPE([[maybe_unused]] int e) { std::cerr << boost::stacktrace::stacktrace() << '\n'; throw SIGFPE_exception(); }
	void catch_SIGSEGV([[maybe_unused]] int e) { std::cerr << boost::stacktrace::stacktrace() << '\n'; throw SIGSEGV_exception(); }
	signed convertedmain();
	signed main() { signal(SIGFPE, catch_SIGFPE); signal(SIGSEGV, catch_SIGSEGV); return convertedmain(); }
	#define main() convertedmain()
#endif
#ifdef LOCAL_DEV
	template<typename T1, typename T2> std::ostream& operator<<(std::ostream& s, const std::pair<T1, T2>& p) {
		return s << "(" << p.first << ", " << p.second << ")"; }
	template <typename T, size_t N> std::ostream& operator<<(std::ostream& s, const std::array<T, N>& a) {
		s << "{ "; for (size_t i = 0; i < N; ++i){ s << a[i] << "\t"; } s << "}"; return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::set<T>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::multiset<T>& se) {
		s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}"; return s; }
	template<typename T1, typename T2> std::ostream& operator<<(std::ostream& s, const std::map<T1, T2>& m) {
		s << "{\n"; for (auto itr = m.begin(); itr != m.end(); ++itr){ s << "\t" << (*itr).first << " : " << (*itr).second << "\n"; } s << "}"; return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::deque<T>& v) {
		for (size_t i = 0; i < v.size(); ++i){ s << v[i]; if (i < v.size() - 1) s << "\t"; } return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::vector<T>& v) {
		for (size_t i = 0; i < v.size(); ++i){ s << v[i]; if (i < v.size() - 1) s << "\t"; } return s; }
	template<typename T> std::ostream& operator<<(std::ostream& s, const std::vector<std::vector<T>>& vv) {
		s << "\\\n"; for (size_t i = 0; i < vv.size(); ++i){ s << vv[i] << "\n"; } return s; }
	void debug_impl() { std::cerr << '\n'; }
	template<typename Head, typename... Tail> void debug_impl(Head head, Tail... tail) { std::cerr << " " << head << (sizeof...(tail) ? "," : ""); debug_impl(tail...); }
	#define debug(...) do { std::cerr << ":" << __LINE__ << " (" << #__VA_ARGS__ << ") ="; debug_impl(__VA_ARGS__); } while (false)
#else
	#define debug(...) do {} while (false)
#endif
//#define int long long
using ll = long long;
//constexpr int INF = (ll)1e9 + 7; //INT_MAX = (1<<31)-1 = 2147483647
constexpr ll INF = (ll)1e18; //INT64_MAX = (1LL<<63)-1 = 9223372036854775807
constexpr ll MOD = (ll)1e9 + 7; //primitive root = 5
//constexpr ll MOD = 998244353; //primitive root = 3
constexpr double EPS = 1e-9;
constexpr ll dx[4] = {1, 0, -1, 0};
constexpr ll dy[4] = {0, 1, 0, -1};
constexpr ll dx8[8] = {1, 0, -1, 0, 1, 1, -1, -1};
constexpr ll dy8[8] = {0, 1, 0, -1, 1, -1, 1, -1};
#define rep(i, n)   for(ll i=0, i##_length=(n); i< i##_length; ++i)
#define repeq(i, n) for(ll i=1, i##_length=(n); i<=i##_length; ++i)
#define rrep(i, n)   for(ll i=(n)-1; i>=0; --i)
#define rrepeq(i, n) for(ll i=(n)  ; i>=1; --i)
#define all(v) (v).begin(), (v).end()
#define rall(v) (v).rbegin(), (v).rend()
void p() { std::cout << '\n'; }
template<typename Head, typename... Tail> void p(Head head, Tail... tail) { std::cout << head << (sizeof...(tail) ? " " : ""); p(tail...); }
template<typename T> inline void pv(std::vector<T>& v) { for(ll i=0, N=v.size(); i<N; i++) std::cout << v[i] << " \n"[i==N-1]; }
template<typename T> inline bool chmax(T& a, T b) { return a < b && (a = b, true); }
template<typename T> inline bool chmin(T& a, T b) { return a > b && (a = b, true); }
template<typename T> inline void uniq(std::vector<T>& v) { v.erase(std::unique(v.begin(), v.end()), v.end()); }
template<typename T> inline ll sz(T& v) { return v.size(); }

/*-----8<-----template-----8<-----*/

map<ll,ll> inv_cache;
struct Modint{
	unsigned long long num = 0;
	constexpr Modint() noexcept {}
	//constexpr Modint(const Modint &x) noexcept : num(x.num){}
	inline constexpr operator ll() const noexcept { return num; }
	inline constexpr Modint& operator+=(Modint x) noexcept { num += x.num; if(num >= MOD) num -= MOD; return *this; }
	inline constexpr Modint& operator++() noexcept { if(num == MOD - 1) num = 0; else num++; return *this; }
	inline constexpr Modint operator++(int) noexcept { Modint ans(*this); operator++(); return ans; }
	inline constexpr Modint operator-() const noexcept { return Modint(0) -= *this; }
	inline constexpr Modint& operator-=(Modint x) noexcept { if(num < x.num) num += MOD; num -= x.num; return *this; }
	inline constexpr Modint& operator--() noexcept { if(num == 0) num = MOD - 1; else num--; return *this; }
	inline constexpr Modint operator--(int) noexcept { Modint ans(*this); operator--(); return ans; }
	inline constexpr Modint& operator*=(Modint x) noexcept { num = (unsigned long long)(num) * x.num % MOD; return *this; }
	inline Modint& operator/=(Modint x) noexcept { return operator*=(x.inv()); }
	template<class T> constexpr Modint(T x) noexcept {
		using U = typename conditional<sizeof(T) >= 4, T, int>::type;
		U y = x; y %= U(MOD); if(y < 0) y += MOD; num = (unsigned long long)(y);
	}
	template<class T> inline constexpr Modint operator+(T x) const noexcept { return Modint(*this) += x; }
	template<class T> inline constexpr Modint& operator+=(T x) noexcept { return operator+=(Modint(x)); }
	template<class T> inline constexpr Modint operator-(T x) const noexcept { return Modint(*this) -= x; }
	template<class T> inline constexpr Modint& operator-=(T x) noexcept { return operator-=(Modint(x)); }
	template<class T> inline constexpr Modint operator*(T x) const noexcept { return Modint(*this) *= x; }
	template<class T> inline constexpr Modint& operator*=(T x) noexcept { return operator*=(Modint(x)); }
	template<class T> inline constexpr Modint operator/(T x) const noexcept { return Modint(*this) /= x; }
	template<class T> inline constexpr Modint& operator/=(T x) noexcept { return operator/=(Modint(x)); }
	inline Modint inv() const noexcept { return inv_cache.count(num) ? inv_cache[num] : inv_cache[num] = inv_calc(); }
	inline constexpr ll inv_calc() const noexcept { ll x = 0, y = 0; extgcd(num, MOD, x, y); return x; }
	static inline constexpr ll extgcd(ll a, ll b, ll &x, ll &y) noexcept { ll g = a; x = 1; y = 0; if(b){ g = extgcd(b, a % b, y, x); y -= a / b * x; } return g; }
	inline constexpr Modint pow(ll x) const noexcept { Modint ans = 1, cnt = x>=0 ? *this : Modint(1) / *this; while(x){ if(x & 1) ans *= cnt; cnt *= cnt; x /= 2; } return ans; }
	static inline constexpr ll get_mod() { return MOD; }
};
std::istream& operator>>(std::istream& is, Modint& x){ ll a; is>>a; x = a; return is; }
inline constexpr Modint operator""_M(unsigned long long x) noexcept { return Modint(x); }
std::vector<Modint> fac(1, 1), inv(1, 1);
inline void reserve(size_t a){
	if(fac.size() >= a) return;
	if(a < fac.size() * 2) a = fac.size() * 2;
	if(a >= MOD) a = MOD;
	fac.reserve(a);
	while(fac.size() < a) fac.push_back(fac.back() * Modint(fac.size()));
	inv.resize(fac.size());
	inv.back() = fac.back().inv();
	for(ll i = inv.size() - 1; !inv[i - 1]; i--) inv[i - 1] = inv[i] * i;
}
inline Modint factorial(ll n){ if(n < 0) return 0; reserve(n + 1); return fac[n]; }
inline Modint nPk(ll n, ll r){
    if(r < 0 || n < r) return 0;
    if(n >> 24){ Modint ans = 1; for(ll i = 0; i < r; i++) ans *= n--; return ans; }
    reserve(n + 1); return fac[n] * inv[n - r];
}
inline Modint nCk(ll n, ll r){ if(r < 0 || n < r) return 0; r = min(r, n - r); reserve(r + 1); return inv[r] * nPk(n, r); }
inline Modint nHk(ll n, ll r){ return nCk(n + r - 1, n - 1); } //n種類のものから重複を許してr個選ぶ=玉r個と仕切りn-1個
inline Modint catalan(ll n){ reserve(n * 2 + 1); return fac[n * 2] * inv[n] * inv[n + 1]; }

////

//高速フーリエ変換
//計算量 O((n+m)log(n+m))
namespace FastFourierTransform {
	using real = double;

	struct C {
		real x, y;
		C() : x(0), y(0) {}
		C(real x, real y) : x(x), y(y) {}
		inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
		inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
		inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); }
		inline C conj() const { return C(x, -y); }
	};

	int base = 1;
	vector< C > rts = {{0, 0},
						{1, 0}};
	vector< int > rev = {0, 1};

	void ensure_base(int nbase) {
		if(nbase <= base) return;
		rev.resize(1 << nbase);
		rts.resize(1 << nbase);
		for(int i = 0; i < (1 << nbase); i++) {
			rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
		}
		while(base < nbase) {
			real angle = M_PI * 2.0 / (1 << (base + 1));
			for(int i = 1 << (base - 1); i < (1 << base); i++) {
				rts[i << 1] = rts[i];
				real angle_i = angle * (2 * i + 1 - (1 << base));
				rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
			}
			++base;
		}
	}

	void fft(vector< C > &a, int n) {
		assert((n & (n - 1)) == 0);
		int zeros = __builtin_ctz(n);
		ensure_base(zeros);
		int shift = base - zeros;
		for(int i = 0; i < n; i++) {
			if(i < (rev[i] >> shift)) {
				swap(a[i], a[rev[i] >> shift]);
			}
		}
		for(int k = 1; k < n; k <<= 1) {
			for(int i = 0; i < n; i += 2 * k) {
				for(int j = 0; j < k; j++) {
					C z = a[i + j + k] * rts[j + k];
					a[i + j + k] = a[i + j] - z;
					a[i + j] = a[i + j] + z;
				}
			}
		}
	}

	template<typename T>
	vector< ll > multiply(vector< T > &a, vector< T > &b) {
		int need = (int) a.size() + (int) b.size() - 1;
		int nbase = 1;
		while((1 << nbase) < need) nbase++;
		ensure_base(nbase);
		int sz = 1 << nbase;
		vector< C > fa(sz);
		for(int i = 0; i < sz; i++) {
			real x = (i < (int) a.size() ? a[i] : 0);
			real y = (i < (int) b.size() ? b[i] : 0);
			fa[i] = C(x, y);
		}
		fft(fa, sz);
		C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
		for(int i = 0; i <= (sz >> 1); i++) {
			int j = (sz - i) & (sz - 1);
			C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
			fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
			fa[i] = z;
		}
		for(int i = 0; i < (sz >> 1); i++) {
			C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
			C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
			fa[i] = A0 + A1 * s;
		}
		fft(fa, sz >> 1);
		vector< ll > ret(need);
		for(int i = 0; i < need; i++) {
			ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
		}
		return ret;
	}
};
//任意mod畳み込み(Arbitrary-Mod-Convolution)
template< typename T >
struct ArbitraryModConvolution {
	using real = FastFourierTransform::real;
	using C = FastFourierTransform::C;

	ArbitraryModConvolution() = default;

	vector< T > multiply(const vector< T > &a, const vector< T > &b, int need = -1) {
		if(need == -1) need = a.size() + b.size() - 1;
		int nbase = 0;
		while((1 << nbase) < need) nbase++;
		FastFourierTransform::ensure_base(nbase);
		int sz = 1 << nbase;
		vector< C > fa(sz);
		for(int i = 0; i < (int)a.size(); i++) {
			fa[i] = C(a[i].num & ((1 << 15) - 1), a[i].num >> 15);
		}
		fft(fa, sz);
		vector< C > fb(sz);
		if(a == b) {
			fb = fa;
		} else {
			for(int i = 0; i < (int)b.size(); i++) {
				fb[i] = C(b[i].num & ((1 << 15) - 1), b[i].num >> 15);
			}
			fft(fb, sz);
		}
		real ratio = 0.25 / sz;
		C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
		for(int i = 0; i <= (sz >> 1); i++) {
			int j = (sz - i) & (sz - 1);
			C a1 = (fa[i] + fa[j].conj());
			C a2 = (fa[i] - fa[j].conj()) * r2;
			C b1 = (fb[i] + fb[j].conj()) * r3;
			C b2 = (fb[i] - fb[j].conj()) * r4;
			if(i != j) {
				C c1 = (fa[j] + fa[i].conj());
				C c2 = (fa[j] - fa[i].conj()) * r2;
				C d1 = (fb[j] + fb[i].conj()) * r3;
				C d2 = (fb[j] - fb[i].conj()) * r4;
				fa[i] = c1 * d1 + c2 * d2 * r5;
				fb[i] = c1 * d2 + c2 * d1;
			}
			fa[j] = a1 * b1 + a2 * b2 * r5;
			fb[j] = a1 * b2 + a2 * b1;
		}
		fft(fa, sz);
		fft(fb, sz);
		vector< T > ret(need);
		for(int i = 0; i < need; i++) {
			ll aa = llround(fa[i].x);
			ll bb = llround(fb[i].x);
			ll cc = llround(fa[i].y);
			aa = ll(T(aa)), bb = ll(T(bb)), cc = ll(T(cc));
			ret[i] = T(aa + (bb << 15) + (cc << 30));
		}
		return ret;
	}
};

//形式的冪級数
//https://ei1333.github.io/luzhiled/snippets/math/formal-power-series.html
template< typename T >
struct FormalPowerSeries : vector< T > {
	using vector< T >::vector;
	using P = FormalPowerSeries;
	using MULT = function< P(P, P) >;

	static MULT &get_mult() {
		static MULT mult = nullptr;
		return mult;
	}

	static void set_fft(MULT f) {
		get_mult() = f;
	}

	template< typename E >
	FormalPowerSeries(const vector< E > &x) : vector< T >(begin(x), end(x)) {}

	void shrink() {
		while(this->size() && this->back() == T(0)) this->pop_back();
	}

	P operator+(const P &r) const { return P(*this) += r; }

	P operator+(const T &v) const { return P(*this) += v; }

	P operator-(const P &r) const { return P(*this) -= r; }

	P operator-(const T &v) const { return P(*this) -= v; }

	P operator*(const P &r) const { return P(*this) *= r; }

	P operator*(const T &v) const { return P(*this) *= v; }

	P operator/(const P &r) const { return P(*this) /= r; }

	P operator%(const P &r) const { return P(*this) %= r; }

	P &operator+=(const P &r) {
		if(r.size() > this->size()) this->resize(r.size());
		for(int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
		return *this;
	}

	P &operator+=(const T &r) {
		if(this->empty()) this->resize(1);
		(*this)[0] += r;
		return *this;
	}

	P &operator-=(const P &r) {
		if(r.size() > this->size()) this->resize(r.size());
		for(int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
		shrink();
		return *this;
	}

	P &operator-=(const T &r) {
		if(this->empty()) this->resize(1);
		(*this)[0] -= r;
		shrink();
		return *this;
	}

	P &operator*=(const T &v) {
		const int n = (int) this->size();
		for(int k = 0; k < n; k++) (*this)[k] *= v;
		return *this;
	}

	P &operator*=(const P &r) {
		if(this->empty() || r.empty()) {
			this->clear();
			return *this;
		}
		assert(get_mult() != nullptr);
		return *this = get_mult()(*this, r);
	}

	P &operator%=(const P &r) {
		return *this -= *this / r * r;
	}

	P operator-() const {
		P ret(this->size());
		for(int i = 0; i < (int)(this->size()); i++) ret[i] = -(*this)[i];
		return ret;
	}

	P &operator/=(const P &r) {
		if(this->size() < r.size()) {
			this->clear();
			return *this;
		}
		int n = this->size() - r.size() + 1;
		return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
	}

	P pre(int sz) const {
		return P(begin(*this), begin(*this) + min((int) this->size(), sz));
	}

	P operator>>(int sz) const {
		if((int)(this->size()) <= sz) return {};
		P ret(*this);
		ret.erase(ret.begin(), ret.begin() + sz);
		return ret;
	}

	P operator<<(int sz) const {
		P ret(*this);
		ret.insert(ret.begin(), sz, T(0));
		return ret;
	}

	P rev(int deg = -1) const {
		P ret(*this);
		if(deg != -1) ret.resize(deg, T(0));
		reverse(begin(ret), end(ret));
		return ret;
	}

	P diff() const {
		const int n = (int) this->size();
		P ret(max(0, n - 1));
		for(int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
		return ret;
	}

	P integral() const {
		const int n = (int) this->size();
		P ret(n + 1);
		ret[0] = T(0);
		for(int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
		return ret;
	}

	// F(0) must not be 0
	P inv(int deg = -1) const {
		assert(((*this)[0]) != T(0));
		const int n = (int) this->size();
		if(deg == -1) deg = n;
		P ret({T(1) / (*this)[0]});
		for(int i = 1; i < deg; i <<= 1) {
			ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
		}
		return ret.pre(deg);
	}

	// F(0) must be 1
	P log(int deg = -1) const {
		assert((*this)[0] == T(1));
		const int n = (int) this->size();
		if(deg == -1) deg = n;
		return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
	}

	P sqrt(int deg = -1) const {
		const int n = (int) this->size();
		if(deg == -1) deg = n;

		if((*this)[0] == T(0)) {
			for(int i = 1; i < n; i++) {
				if((*this)[i] != T(0)) {
					if(i & 1) return {};
					if(deg - i / 2 <= 0) break;
					auto ret = (*this >> i).sqrt(deg - i / 2) << (i / 2);
					if(ret.size() < deg) ret.resize(deg, T(0));
					return ret;
				}
			}
			return P(deg, 0);
		}

		P ret({T(1)});
		T inv2 = T(1) / T(2);
		for(int i = 1; i < deg; i <<= 1) {
			ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
		}
		return ret.pre(deg);
	}

	// F(0) must be 0
	P exp(int deg = -1) const {
		assert((*this)[0] == T(0));
		const int n = (int) this->size();
		if(deg == -1) deg = n;
		P ret({T(1)});
		for(int i = 1; i < deg; i <<= 1) {
			ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
		}
		return ret.pre(deg);
	}

	P pow(int64_t k, int deg = -1) const {
		const int n = (int) this->size();
		if(deg == -1) deg = n;
		for(int i = 0; i < n; i++) {
			if((*this)[i] != T(0)) {
				T rev = T(1) / (*this)[i];
				P C(*this * rev);
				P D(n - i);
				for(int j = i; j < n; j++) D[j - i] = C[j];
				D = (D.log() * k).exp() * (*this)[i].pow(k);
				P E(deg);
				if(i * k > deg) return E;
				auto S = i * k;
				for(int j = 0; j + S < deg && j < D.size(); j++) E[j + S] = D[j];
				return E;
			}
		}
		return *this;
	}

	T eval(T x) const {
		T r = 0, w = 1;
		for(auto &v : *this) {
			r += w * v;
			w *= x;
		}
		return r;
	}

	//sparse
	P mul(vector<pair<ll, T>> g) {
		P ret(*this);
		int n = ret.size();
		auto [d, c] = g.front();
		if (d == 0) g.erase(g.begin());
		else c = 0;
		for(int i=n-1; i>=0; i--){
			ret[i] *= c;
			for (auto &[j, b] : g) {
				if (j > i) break;
				ret[i] += ret[i-j] * b;
			}
		}
		return ret;
	}
	
	//sparse, required: "g[0] == (0, c)" and "c != 0"
	P div(vector<pair<ll, T>> g) {
		P ret(*this);
		int n = ret.size();
		auto [d, c] = g.front();
		assert(d == 0 && c != T(0));
		g.erase(g.begin());
		for(int i=0; i<n; i++) {
			for (auto &[j, b] : g) {
				if (j > i) break;
				ret[i] -= ret[i-j] * b;
			}
			ret[i] /= c;
		}
		return ret;
	}
};


//partition(N): [0,N] の分割数を返す。
//分割数：自然数 n を 自然数の和で表す組み合わせの数 (順序は考慮しない）
//計算量：O(NlogN)
template< typename T >
FormalPowerSeries< T > partition(int N) {
	ArbitraryModConvolution< Modint > fft;
	using FPS = FormalPowerSeries< Modint >;
	auto mult = [&](const FPS::P &a, const FPS::P &b) { return fft.multiply(a, b); };
	FPS::set_fft(mult);

	FormalPowerSeries< T > po(N + 1);
	po[0] = 1;
	for(int k = 1; k <= N; k++) {
		if(1LL * k * (3 * k + 1) / 2 <= N) po[k * (3 * k + 1) / 2] += (k % 2 ? -1 : 1);
		if(1LL * k * (3 * k - 1) / 2 <= N) po[k * (3 * k - 1) / 2] += (k % 2 ? -1 : 1);
	}
	return po.inv();
}

/*-----8<-----library-----8<-----*/

//https://yukicoder.me/problems/no/3046
void yuki3046() {

	/*
	メモは形式的冪級数.cppを参照のこと
	*/

	ArbitraryModConvolution< Modint > fft;
	using FPS = FormalPowerSeries< Modint >;
	using SPARSE = vector<pair<ll,Modint>>;
	auto mult = [&](const FPS::P &a, const FPS::P &b) { return fft.multiply(a, b); };
	FPS::set_fft(mult);

	/*
	//任意mod畳み込みだけやりたいとき ModintのMODを変更しておくこと
	ArbitraryModConvolution< Modint > fft;
	vector<Modint> A(N + 1), B(N + 1);
	vector<Modint> C = fft.multiply(A, B);
	*/
	
	ll K, N;
	cin >> K >> N;

	//f(T)=(T^(進める歩数1) + T^(進める歩数2) + ... )とすると、
	//求めたいのは 1 + f(T) + f(T)^2 + ... = 1/(1-f(T))
	//まず X に 1-f(T) をつくる
	ll size = K+1;
	FormalPowerSeries<Modint> X(size);
	X[0] = 1;
	for(ll i = 0; i < N; i++) {
		ll t;
		cin >> t;
		if(t <= K) X[t] = -1;
	}

	//1/(1-f(T))
	FormalPowerSeries<Modint> v = X.inv(size);

	//x^Kの係数が解となる
	Modint ans = v[K];
	cout << ans << endl;

	//スパース(疎)な乗算、除算 O(NK) (K=係数が0でない項の数)
	//Y={xの次数, 係数}を詰めた配列
	vector<pair<ll,Modint>> Y{{0,1},{1,-1}};
	FormalPowerSeries<Modint> Z = X.mul(Y);
	FormalPowerSeries<Modint> U = X.div(Y);
}

void bunkatsusu(){
	FormalPowerSeries<Modint> partitionv = partition<Modint>(10);
	debug(partitionv);
}

signed main() {
	yuki3046();
	//bunkatsusu();

	return 0;
}
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