#602865 (C++17) No.8046 yukicoderの過去問

• 問題: No.8046 yukicoderの過去問
• ユーザー: kyon2326
• 提出日時: 2021-01-13 13:55:20
• 言語: C++17 (gcc 13.3.0 + boost 1.89.0)
• 結果: AC
• 実行時間: 1,560 ms / 2,000 ms
• コード長: 38,423 bytes
• コンパイル時間: 3,197 ms
• コンパイル使用メモリ: 234,632 KB
• 最終ジャッジ日時: 2025-01-17 17:15:43
• ジャッジサーバーID （参考情報）: judge5 / judge4

ファイルパターン	結果
other	AC * 9

ソースコード

#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;
*/
#if defined(LOCAL_TEST) || defined(LOCAL_DEV)
	#define BOOST_STACKTRACE_USE_ADDR2LINE
	#define BOOST_STACKTRACE_ADDR2LINE_LOCATION /usr/local/opt/binutils/bin/addr2line
	#define _GNU_SOURCE 1
	#include <boost/stacktrace.hpp>
#endif
#ifdef LOCAL_TEST
	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)
	constexpr inline long long prodlocal([[maybe_unused]] long long prod, [[maybe_unused]] long long local) { return local; }
#else
	#define debug(...) do {} while (false)
	constexpr inline long long prodlocal([[maybe_unused]] long long prod, [[maybe_unused]] long long local) { return prod; }
#endif
//#define int long long
using ll = long long;
//INT_MAX = (1<<31)-1 = 2147483647, INT64_MAX = (1LL<<63)-1 = 9223372036854775807
constexpr ll INF = numeric_limits<ll>::max() == INT_MAX ? (ll)1e9 + 7 : (ll)1e18;
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 : inv(); if(x<0) x = -x; 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]; }

////
/*
template< typename T >
struct FormalPowerSeries : vector< T > {
	using vector< T >::vector;
	using P = FormalPowerSeries;

	using MULT = function< vector< T >(P, P) >;
	using FFT = function< void(P &) >;
	using SQRT = function< T(T) >;

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

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

	static FFT &get_fft() {
		static FFT fft = nullptr;
		return fft;
	}

	static FFT &get_ifft() {
		static FFT ifft = nullptr;
		return ifft;
	}

	static void set_fft(FFT f, FFT g) {
		get_fft() = f;
		get_ifft() = g;
	}

	static SQRT &get_sqrt() {
		static SQRT sqr = nullptr;
		return sqr;
	}

	static void set_sqrt(SQRT sqr) {
		get_sqrt() = sqr;
	}

	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);
		auto ret = get_mult()(*this, r);
		return *this = P(begin(ret), end(ret));
	}

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

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

	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;
		if(get_fft() != nullptr) {
			P ret(*this);
			ret.resize(deg, T(0));
			return ret.inv_fast();
		}
		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] == 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);
					if(ret.empty()) return {};
					ret = ret << (i / 2);
					if(ret.size() < deg) ret.resize(deg, T(0));
					return ret;
				}
			}
			return P(deg, 0);
		}

		P ret;
		if(get_sqrt() == nullptr) {
			assert((*this)[0] == T(1));
			ret = {T(1)};
		} else {
			auto sqr = get_sqrt()((*this)[0]);
			if(sqr * sqr != (*this)[0]) return {};
			ret = {T(sqr)};
		}

		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;
		if(get_fft() != nullptr) {
			P ret(*this);
			ret.resize(deg, T(0));
			return ret.exp_rec();
		}
		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 online_convolution_exp(const P &conv_coeff) const {
		const int n = (int) conv_coeff.size();
		assert((n & (n - 1)) == 0);
		vector< P > conv_ntt_coeff;
		auto& fft = get_fft();
		auto& ifft = get_ifft();
		for(int i = n; i >= 1; i >>= 1) {
			P g(conv_coeff.pre(i));
			fft(g);
			conv_ntt_coeff.emplace_back(g);
		}
		P conv_arg(n), conv_ret(n);
		auto rec = [&](auto rec, int l, int r, int d) -> void {
			if(r - l <= 16) {
				for(int i = l; i < r; i++) {
					T sum = 0;
					for(int j = l; j < i; j++) sum += conv_arg[j] * conv_coeff[i - j];
					conv_ret[i] += sum;
					conv_arg[i] = i == 0 ? T(1) : conv_ret[i] / i;
				}
			} else {
				int m = (l + r) / 2;
				rec(rec, l, m, d + 1);
				P pre(r - l);
				for(int i = 0; i < m - l; i++) pre[i] = conv_arg[l + i];
				fft(pre);
				for(int i = 0; i < r - l; i++) pre[i] *= conv_ntt_coeff[d][i];
				ifft(pre);
				for(int i = 0; i < r - m; i++) conv_ret[m + i] += pre[m + i - l];
				rec(rec, m, r, d + 1);
			}
		};
		rec(rec, 0, n, 0);
		return conv_arg;
	}

	P exp_rec() const {
		assert((*this)[0] == T(0));
		const int n = (int) this->size();
		int m = 1;
		while(m < n) m *= 2;
		P conv_coeff(m);
		for(int i = 1; i < n; i++) conv_coeff[i] = (*this)[i] * i;
		return online_convolution_exp(conv_coeff).pre(n);
	}


	P inv_fast() const {
		assert(((*this)[0]) != T(0));

		const int n = (int) this->size();
		P res{T(1) / (*this)[0]};

		for(int d = 1; d < n; d <<= 1) {
			P f(2 * d), g(2 * d);
			for(int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
			for(int j = 0; j < d; j++) g[j] = res[j];
			get_fft()(f);
			get_fft()(g);
			for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
			get_ifft()(f);
			for(int j = 0; j < d; j++) {
				f[j] = 0;
				f[j + d] = -f[j + d];
			}
			get_fft()(f);
			for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
			get_ifft()(f);
			for(int j = 0; j < d; j++) f[j] = res[j];
			res = f;
		}
		return res.pre(n);
	}

	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 ret = (((*this * rev) >> i).log() * k).exp() * (T((*this)[i]).pow(k));
				if(i * k > deg) return P(deg, T(0));
				ret = (ret << (i * k)).pre(deg);
				if((int)ret.size() < deg) ret.resize(deg, T(0));
				return ret;
			}
		}
		return *this;
	}

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

	P pow_mod(int64_t n, P mod) const {
		P modinv = mod.rev().inv();
		auto get_div = [&](P base) {
			if(base.size() < mod.size()) {
				base.clear();
				return base;
			}
			int n = base.size() - mod.size() + 1;
			return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
		};
		P x(*this), ret{1};
		while(n > 0) {
			if(n & 1) {
				ret *= x;
				ret -= get_div(ret) * mod;
			}
			x *= x;
			x -= get_div(x) * mod;
			n >>= 1;
		}
		return ret;
	}

	
};
*/

template< typename T >
struct FormalPowerSeries : vector< T > {
	using vector< T >::vector;
	using P = FormalPowerSeries;

	using MULT = function< vector< T >(P, P) >;
	using FFT = function< void(P &) >;
	using SQRT = function< T(T) >;

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

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

	static FFT &get_fft() {
		static FFT fft = nullptr;
		return fft;
	}

	static FFT &get_ifft() {
		static FFT ifft = nullptr;
		return ifft;
	}

	static void set_fft(FFT f, FFT g) {
		get_fft() = f;
		get_ifft() = g;
		if(get_mult() == nullptr) {
			auto mult = [&](P a, P b) {
				int need = a.size() + b.size() - 1;
				int nbase = 1;
				while((1 << nbase) < need) nbase++;
				int sz = 1 << nbase;
				a.resize(sz, T(0));
				b.resize(sz, T(0));
				get_fft()(a);
				get_fft()(b);
				for(int i = 0; i < sz; i++) a[i] *= b[i];
				get_ifft()(a);
				a.resize(need);
				return a;
			};
			set_mult(mult);
		}
	}

	static SQRT &get_sqrt() {
		static SQRT sqr = nullptr;
		return sqr;
	}

	static void set_sqrt(SQRT sqr) {
		get_sqrt() = sqr;
	}

	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);
		auto ret = get_mult()(*this, r);
		return *this = P(begin(ret), end(ret));
	}

	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 dot(P r) const {
		P ret(min(this->size(), r.size()));
		for(int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
		return ret;
	}

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

	P operator>>(int sz) const {
		if(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;
	}

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

	// https://opt-cp.com/fps-implementation/
	// multiply and divide (1 + cz^d)
	P mul(const ll d, const T c) {
		P ret(*this);
		int n = ret.size();
		if (c == T(1)) for(int i=n-d-1; i>=0; --i) ret[i+d] += ret[i];
		else if (c == T(-1)) for(int i=n-d-1; i>=0; --i) ret[i+d] -= ret[i];
		else for(int i=n-d-1; i>=0; --i) ret[i+d] += ret[i] * c;
		return ret;
	}
	P div(const ll d, const T c) {
		P ret(*this);
		int n = ret.size();
		if (c == T(1)) for(int i=0; i<n-d; ++i) ret[i+d] -= ret[i];
		else if (c == T(-1)) for(int i=0; i<n-d; ++i) ret[i+d] += ret[i];
		else for(int i=0; i<n-d; ++i) ret[i+d] -= ret[i] * c;
		return ret;
	}
	// sparse
	P mul(vector<pair<ll, T>> g) {
		if ((int)g.size() == 2 && g[0] == pair<ll, T>(0, 1))
			return mul(g[1].first, g[1].second);
		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) {
		if ((int)g.size() == 2 && g[0] == pair<ll, T>(0, 1))
			return div(g[1].first, g[1].second);
		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;
	}

	P diff() const;

	P integral() const;

	// F(0) must not be 0
	P inv_fast() const;

	P inv(int deg = -1) const;

	// F(0) must be 1
	P log(int deg = -1) const;

	P sqrt(int deg = -1) const;

	// F(0) must be 0
	P exp_fast(int deg = -1) const;

	P exp(int deg = -1) const;

	P pow(int64_t k, int deg = -1) const;

	P mod_pow(int64_t k, P g) const;

	P taylor_shift(T c) const;
};
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::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;
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::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;
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::inv_fast() const {
	assert(((*this)[0]) != T(0));

	const int n = (int) this->size();
	P res{T(1) / (*this)[0]};

	for(int d = 1; d < n; d <<= 1) {
		P f(2 * d), g(2 * d);
		for(int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
		for(int j = 0; j < d; j++) g[j] = res[j];
		get_fft()(f);
		get_fft()(g);
		for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
		get_ifft()(f);
		for(int j = 0; j < d; j++) {
			f[j] = 0;
			f[j + d] = -f[j + d];
		}
		get_fft()(f);
		for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
		get_ifft()(f);
		for(int j = 0; j < d; j++) f[j] = res[j];
		res = f;
	}
	return res.pre(n);
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::inv(int deg) const {
	assert(((*this)[0]) != T(0));
	const int n = (int) this->size();
	if(deg == -1) deg = n;
	if(get_fft() != nullptr) {
		P ret(*this);
		ret.resize(deg, T(0));
		return ret.inv_fast();
	}
	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);
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::log(int deg) const {
	assert((*this)[0] == 1);
	const int n = (int) this->size();
	if(deg == -1) deg = n;
	return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::sqrt(int deg) 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);
				if(ret.empty()) return {};
				ret = ret << (i / 2);
				if(ret.size() < deg) ret.resize(deg, T(0));
				return ret;
			}
		}
		return P(deg, 0);
	}

	P ret;
	if(get_sqrt() == nullptr) {
		assert((*this)[0] == T(1));
		ret = {T(1)};
	} else {
		auto sqr = get_sqrt()((*this)[0]);
		if(sqr * sqr != (*this)[0]) return {};
		ret = {T(sqr)};
	}

	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);
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::exp_fast(int deg) const {
	if(deg == -1) deg = this->size();
	assert((*this)[0] == T(0));

	P inv;
	inv.reserve(deg + 1);
	inv.push_back(T(0));
	inv.push_back(T(1));

	auto inplace_integral = [&](P &F) -> void {
		const int n = (int) F.size();
		auto mod = T::get_mod();
		while((int) inv.size() <= n) {
			int i = inv.size();
			inv.push_back((-inv[mod % i]) * (mod / i));
		}
		F.insert(begin(F), T(0));
		for(int i = 1; i <= n; i++) F[i] *= inv[i];
	};

	auto inplace_diff = [](P &F) -> void {
		if(F.empty()) return;
		F.erase(begin(F));
		T coeff = 1, one = 1;
		for(int i = 0; i < (int) F.size(); i++) {
			F[i] *= coeff;
			coeff += one;
		}
	};

	P b{1, 1 < (int) this->size() ? (*this)[1] : T(0)}, c{1}, z1, z2{1, 1};
	for(int m = 2; m < deg; m *= 2) {
		auto y = b;
		y.resize(2 * m);
		get_fft()(y);
		z1 = z2;
		P z(m);
		for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
		get_ifft()(z);
		fill(begin(z), begin(z) + m / 2, T(0));
		get_fft()(z);
		for(int i = 0; i < m; ++i) z[i] *= -z1[i];
		get_ifft()(z);
		c.insert(end(c), begin(z) + m / 2, end(z));
		z2 = c;
		z2.resize(2 * m);
		get_fft()(z2);
		P x(begin(*this), begin(*this) + min< int >(this->size(), m));
		inplace_diff(x);
		x.push_back(T(0));
		get_fft()(x);
		for(int i = 0; i < m; ++i) x[i] *= y[i];
		get_ifft()(x);
		x -= b.diff();
		x.resize(2 * m);
		for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
		get_fft()(x);
		for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
		get_ifft()(x);
		x.pop_back();
		inplace_integral(x);
		for(int i = m; i < min< int >(this->size(), 2 * m); ++i) x[i] += (*this)[i];
		fill(begin(x), begin(x) + m, T(0));
		get_fft()(x);
		for(int i = 0; i < 2 * m; ++i) x[i] *= y[i];
		get_ifft()(x);
		b.insert(end(b), begin(x) + m, end(x));
	}
	return P(begin(b), begin(b) + deg);
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::exp(int deg) const {
	assert((*this)[0] == T(0));
	const int n = (int) this->size();
	if(deg == -1) deg = n;
	if(get_fft() != nullptr) {
		P ret(*this);
		ret.resize(deg, T(0));
		return ret.exp_fast(deg);
	}
	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);
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::pow(int64_t k, int deg) 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 ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
			if(i * k > deg) return P(deg, T(0));
			ret = (ret << (i * k)).pre(deg);
			if(ret.size() < deg) ret.resize(deg, T(0));
			return ret;
		}
	}
	return *this;
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::mod_pow(int64_t k, P g) const {
	P modinv = g.rev().inv();
	auto get_div = [&](P base) {
		if(base.size() < g.size()) {
			base.clear();
			return base;
		}
		int n = base.size() - g.size() + 1;
		return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
	};
	P x(*this), ret{1};
	while(k > 0) {
		if(k & 1) {
			ret *= x;
			ret -= get_div(ret) * g;
		}
		x *= x;
		x -= get_div(x) * g;
		k >>= 1;
	}
	return ret;
}
template< typename T >
typename FormalPowerSeries< T >::P FormalPowerSeries< T >::taylor_shift(T c) const {
	int n = (int) this->size();
	vector< T > fact(n), rfact(n);
	fact[0] = rfact[0] = T(1);
	for(int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
	rfact[n - 1] = T(1) / fact[n - 1];
	for(int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
	P p(*this);
	for(int i = 0; i < n; i++) p[i] *= fact[i];
	p = p.rev();
	P bs(n, T(1));
	for(int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
	p = (p * bs).pre(n);
	p = p.rev();
	for(int i = 0; i < n; i++) p[i] *= rfact[i];
	return p;
}

////

template< typename Mint >
struct NumberTheoreticTransformFriendlyModInt {

	vector< Mint > dw, idw;
	int max_base;
	Mint root;

	NumberTheoreticTransformFriendlyModInt() {
		const unsigned mod = Mint::get_mod();
		assert(mod >= 3 && mod % 2 == 1);
		auto tmp = mod - 1;
		max_base = 0;
		while(tmp % 2 == 0) tmp >>= 1, max_base++;
		root = 2;
		while(root.pow((mod - 1) >> 1) == 1) root += 1;
		assert(root.pow(mod - 1) == 1);
		dw.resize(max_base);
		idw.resize(max_base);
		for(int i = 0; i < max_base; i++) {
			dw[i] = -root.pow((mod - 1) >> (i + 2));
			idw[i] = Mint(1) / dw[i];
		}
	}

	void ntt(vector< Mint > &a) {
		const int n = (int) a.size();
		assert((n & (n - 1)) == 0);
		assert(__builtin_ctz(n) <= max_base);
		for(int m = n; m >>= 1;) {
			Mint w = 1;
			for(int s = 0, k = 0; s < n; s += 2 * m) {
				for(int i = s, j = s + m; i < s + m; ++i, ++j) {
					auto x = a[i], y = a[j] * w;
					a[i] = x + y, a[j] = x - y;
				}
				w *= dw[__builtin_ctz(++k)];
			}
		}
	}

	void intt(vector< Mint > &a, bool f = true) {
		const int n = (int) a.size();
		assert((n & (n - 1)) == 0);
		assert(__builtin_ctz(n) <= max_base);
		for(int m = 1; m < n; m *= 2) {
			Mint w = 1;
			for(int s = 0, k = 0; s < n; s += 2 * m) {
				for(int i = s, j = s + m; i < s + m; ++i, ++j) {
					auto x = a[i], y = a[j];
					a[i] = x + y, a[j] = (x - y) * w;
				}
				w *= idw[__builtin_ctz(++k)];
			}
		}
		if(f) {
			Mint inv_sz = Mint(1) / n;
			for(int i = 0; i < n; i++) a[i] *= inv_sz;
		}
	}

	vector< Mint > multiply(vector< Mint > a, vector< Mint > b) {
		int need = a.size() + b.size() - 1;
		int nbase = 1;
		while((1 << nbase) < need) nbase++;
		int sz = 1 << nbase;
		a.resize(sz, 0);
		b.resize(sz, 0);
		ntt(a);
		ntt(b);
		Mint inv_sz = Mint(1) / sz;
		for(int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
		intt(a, false);
		a.resize(need);
		return a;
	}
};

/////////////////////



template<class T>
T extgcd(T a, T b, T& x, T& y) {
	for (T u=y=1,v=x=0; a; ) {
		T q = b / a;
		swap(x -= q*u, u);
		swap(y -= q*v, v);
		swap(b -= q*a, a);
	}
	return b;
}

inline long long mod_pow(long long x, long long e, long long mod) {
 	long long v = 1LL;
	for ( ; e; e>>=1) {
		if (e & 1) v = (v * x) % mod;
		x = (x * x) % mod;
	}
	return v;
}

inline long long mod_inv(long long a, long long mod) {
	// return mod_pow(a, mod-2, mod); // slower
	long long x, y;
	extgcd(a, mod, x, y);
	return (mod + x % mod) % mod;
	// ax + MODy = 1
	// aとmodが互いに素である限り解が存在する
	// ax = 1 - MODy
	// ax % MOD = 1
	// x = 1/a % MOD
}

long long _garner(vector<long long>& xs, vector<long long>& mods) {
	int M = xs.size();

	vector<long long> coeffs(M, 1), constants(M, 0);

	for (int i=0; i<M-1; ++i) {
		long long mod_i = mods[i];
		// coffs[i] * v + constants[i] == mr[i].val (mod mr[i].first) を解く
		long long v = (xs[i] - constants[i] + mod_i) % mod_i;
		v = (v * mod_inv(coeffs[i], mod_i)) % mod_i;

		for (int j=i+1; j<M; j++) {
			long long mod_j = mods[j];
			constants[j] = (constants[j] + coeffs[j] * v) % mod_j;
			coeffs[j] = (coeffs[j] * mod_i) % mod_j;
		}
	}

	return constants.back();
}

template<typename T>
inline void bit_reverse(vector<T>& a) {
	int n = a.size();
	int i = 0;
	for (int j=1; j<n-1; ++j) {
		for (int k = n >> 1; k >(i ^= k); k >>= 1);
		if (j < i) swap(a[i], a[j]);
	}
}


template<long long mod, long long primitive_root>
class NTT {
public:
	long long get_mod() { return mod; }

	void _ntt(vector<long long>& a, int sign) {
		const int n = a.size();
		assert((n ^ (n&-n)) == 0); //n = 2^k

		const long long g = primitive_root; // g is primitive root of mod

		long long tmp = (mod - 1) * mod_inv(n, mod) % mod; // -1/n
		long long h = mod_pow(g, tmp, mod); // ^n√g
		if (sign == -1) h = mod_inv(h, mod);

		bit_reverse(a);

		for (int m=1; m<n; m<<=1) {
			const int m2 = 2 * m;
			// long long _base = mod_pow(h.val, n/m2, mod);
			long long _base = mod_pow(h, n/m2, mod);
			long long _w = 1;
			for (int x=0; x<m; ++x) {
				for (int s=x; s<n; s+=m2) {
					long long u = a[s];
					long long d = (a[s + m] * _w) % mod;
					a[s] = (u+d) % mod;
					a[s+m] = (u-d+mod) % mod;
				}
				_w = (_w * _base) % mod;
			}
		}
	}
	void ntt(vector<long long>& input) {
		_ntt(input, 1);
	}
	void intt(vector<long long>& input) {
		_ntt(input, -1);

		const long long n_inv = mod_inv(input.size(), mod);
		for (auto &x : input) x = (x * n_inv) % mod;
	}

	// 畳み込み演算を行う
	vector<long long> convolution(const vector<long long>& a, const vector<long long>& b){
		int result_size = a.size() + b.size() - 1;
		int n = 1; while (n < result_size) n <<= 1;

		vector<long long> _a = a, _b = b;
		_a.resize(n, 0);
		_b.resize(n, 0);

		ntt(_a);
		ntt(_b);
		for (int i=0; i<n; ++i) _a[i] = (_a[i] * _b[i]) % mod;
		intt(_a);

		_a.resize(result_size);
		return _a;
	}
};

template <typename T>
vector<T> convolution_ntt(const vector<T>& x, const vector<T>& y) {
	vector<ll> a(x.size()), b(y.size());
	for (int i = 0; i < (int)x.size(); i++) a[i] = x[i];
	for (int i = 0; i < (int)y.size(); i++) b[i] = y[i];
	// ll maxval = max(a.size(), b.size()) * *max_element(a.begin(), a.end()) * *max_element(b.begin(), b.end());
	// if (maxval < 1224736769) {
	// 	NTT<1224736769, 3> ntt3;
	//	 return ntt3.convolution(a, b);
	// }

	NTT<167772161, 3> ntt1;
	NTT<469762049, 3> ntt2;
	NTT<1224736769, 3> ntt3;

	vector<long long> x1 = ntt1.convolution(a, b);
	vector<long long> x2 = ntt2.convolution(a, b);
	vector<long long> x3 = ntt3.convolution(a, b);

	vector<T> ret(x1.size());
	vector<long long> mods { 167772161, 469762049, 1224736769, T::get_mod() };
	for (int i=0; i<x1.size(); ++i) {
		vector<long long> xs { x1[i], x2[i], x3[i], 0 };
		ret[i] = _garner(xs, mods);
	}

	return ret;
}

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

void solve() {
	/*
	NumberTheoreticTransformFriendlyModInt< Modint > ntt;
	using FPS = FormalPowerSeries< Modint >;
	using SPARSE = vector<pair<ll,Modint>>;
	auto mult = [&](const FPS::P &a, const FPS::P &b) {
		auto ret = ntt.multiply(a, b);
		return FPS::P(ret.begin(), ret.end());
	};
	FPS::set_mult(mult);
	FPS::set_fft([&](FPS::P &a) { ntt.ntt(a); }, [&](FPS::P &a) { ntt.intt(a); });
	*/
	using FPS = FormalPowerSeries< Modint >;
	using SPARSE = vector<pair<ll,Modint>>;
	auto mult = [&](const FPS::P& a, const FPS::P& b) { return convolution_ntt(a, b); };
	FPS::set_mult(mult);	

	ll K, N;
	cin >> K >> N;
	
	//T=(x^(進める歩数1) + x^(進める歩数2) + ... )とすると、
	//求めたいのは 1 + T + T^2 + ... = 1/(1-T)
	//まず X に 1-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-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);
}

signed main() {
	std::cin.tie(nullptr);
	std::ios::sync_with_stdio(false);
	//ll Q; cin >> Q; while(Q--)solve();
	solve();
	return 0;
}