結果
| 問題 | No.8046 yukicoderの過去問 |
| ユーザー |
 minato
|
| 提出日時 | 2021-01-23 11:02:10 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 254 ms / 2,000 ms |
| コード長 | 33,676 bytes |
| コンパイル時間 | 3,503 ms |
| コンパイル使用メモリ | 251,048 KB |
| 実行使用メモリ | 11,800 KB |
| 最終ジャッジ日時 | 2024-06-09 15:32:40 |
| 合計ジャッジ時間 | 4,996 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge3 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 1 ms
5,248 KB |
| testcase_01 | AC | 1 ms
5,376 KB |
| testcase_02 | AC | 2 ms
5,376 KB |
| testcase_03 | AC | 85 ms
8,320 KB |
| testcase_04 | AC | 1 ms
5,376 KB |
| testcase_05 | AC | 162 ms
11,784 KB |
| testcase_06 | AC | 254 ms
11,800 KB |
| testcase_07 | AC | 249 ms
11,796 KB |
| testcase_08 | AC | 245 ms
11,800 KB |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ull = unsigned long long;
using i128 = __int128_t;
using pii = pair<int, int>;
using pll = pair<long long, long long>;
template<class T> using vec = vector<T>;
template<class T> using vvec = vector<vector<T>>;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define rrep(i, n) for (int i = int(n) - 1; i >= 0; i--)
#define all(x) begin(x), end(x)
constexpr char ln = '\n';
istream& operator>>(istream& is, __int128_t& x) {
x = 0;
string s;
is >> s;
int n = int(s.size()), it = 0;
if (s[0] == '-') it++;
for (; it < n; it++) x = (x * 10 + s[it] - '0');
if (s[0] == '-') x = -x;
return is;
}
ostream& operator<<(ostream& os, __int128_t x) {
if (x == 0) return os << 0;
if (x < 0) os << '-', x = -x;
deque<int> deq;
while (x) deq.emplace_front(x % 10), x /= 10;
for (int e : deq) os << e;
return os;
}
template<class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
return os << "(" << p.first << ", " << p.second << ")";
}
template<class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
for (int i = 0; i < int(v.size()); i++) {
if (i) os << ", ";
os << v[i];
}
return os << "}";
}
template<class Container> inline int SZ(const Container& v) { return int(v.size()); }
template<class T> inline void UNIQUE(vector<T>& v) { v.erase(unique(v.begin(), v.end()), v.end()); }
template<class T1, class T2> inline bool chmax(T1& a, T2 b) { if (a < b) { a = b; return true; } return false; }
template<class T1, class T2> inline bool chmin(T1& a, T2 b) { if (a > b) { a = b; return true; } return false; }
inline int topbit(ull x) { return x == 0 ? -1 : 63 - __builtin_clzll(x); }
inline int botbit(ull x) { return x == 0 ? 64 : __builtin_ctzll(x); }
inline int popcount(ull x) { return __builtin_popcountll(x); }
inline int kthbit(ull x, int k) { return (x >> k) & 1; }
inline constexpr long long TEN(int x) { return x == 0 ? 1 : TEN(x-1) * 10; }
template<class T> inline T ABS(T x) { return max(x, -x); }
const string YESNO[2] = {"NO", "YES"};
const string YesNo[2] = {"No", "Yes"};
const string yesno[2] = {"no", "yes"};
inline void YES(bool t = 1) { cout << YESNO[t] << "\n"; }
inline void Yes(bool t = 1) { cout << YesNo[t] << "\n"; }
inline void yes(bool t = 1) { cout << yesno[t] << "\n"; }
inline void print() { cout << "\n"; }
template<class T>
inline void print(const vector<T>& v) {
for (auto it = begin(v); it != end(v); ++it) {
if (it != begin(v)) cout << " ";
cout << *it;
}
print();
}
template<class T, class... Args>
inline void print(const T& x, const Args& ... args) {
cout << x << " ";
print(args...);
}
#ifdef MINATO_LOCAL
inline void debug_out() { cerr << endl; }
template <class T, class... Args>
inline void debug_out(const T& x, const Args& ... args) {
cerr << " " << x;
debug_out(args...);
}
#define debug(...) cerr << __LINE__ << " : [" << #__VA_ARGS__ << "] =", debug_out(__VA_ARGS__)
#else
#define debug(...) (void(0))
#endif
struct fast_ios { fast_ios() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(20); cerr << fixed << setprecision(7); }; } fast_ios_;
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
int bsf(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
} // namespace internal
} // namespace atcoder
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#include <utility>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
struct barrett {
unsigned int _m;
unsigned long long im;
barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
unsigned int umod() const { return _m; }
unsigned int mul(unsigned int a, unsigned int b) const {
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
} // namespace internal
} // namespace atcoder
#include <cassert>
#include <numeric>
#include <type_traits>
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = internal::barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
dynamic_modint(T v) {
long long x = (long long)(v % (long long)(mod()));
if (x < 0) x += mod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
dynamic_modint(T v) {
_v = (unsigned int)(v % mod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
auto eg = internal::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static internal::barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
static constexpr int g = internal::primitive_root<mint::mod()>;
int n = int(a.size());
int h = internal::ceil_pow2(n);
static bool first = true;
static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
if (first) {
first = false;
mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mint::mod() - 1);
mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
mint now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_e[i] = es[i] * now;
now *= ies[i];
}
}
for (int ph = 1; ph <= h; ph++) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint now = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * now;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
now *= sum_e[bsf(~(unsigned int)(s))];
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
static constexpr int g = internal::primitive_root<mint::mod()>;
int n = int(a.size());
int h = internal::ceil_pow2(n);
static bool first = true;
static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
if (first) {
first = false;
mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mint::mod() - 1);
mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
mint now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_ie[i] = ies[i] * now;
now *= es[i];
}
}
for (int ph = h; ph >= 1; ph--) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint inow = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(mint::mod() + l.val() - r.val()) *
inow.val();
}
inow *= sum_ie[bsf(~(unsigned int)(s))];
}
}
}
} // namespace internal
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 60) {
if (n < m) {
std::swap(n, m);
std::swap(a, b);
}
std::vector<mint> ans(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
ans[i + j] += a[i] * b[j];
}
}
return ans;
}
int z = 1 << internal::ceil_pow2(n + m - 1);
a.resize(z);
internal::butterfly(a);
b.resize(z);
internal::butterfly(b);
for (int i = 0; i < z; i++) {
a[i] *= b[i];
}
internal::butterfly_inv(a);
a.resize(n + m - 1);
mint iz = mint(z).inv();
for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
return a;
}
template <unsigned int mod = 998244353,
class T,
std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
using mint = static_modint<mod>;
std::vector<mint> a2(n), b2(m);
for (int i = 0; i < n; i++) {
a2[i] = mint(a[i]);
}
for (int i = 0; i < m; i++) {
b2[i] = mint(b[i]);
}
auto c2 = convolution(move(a2), move(b2));
std::vector<T> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
c[i] = c2[i].val();
}
return c;
}
std::vector<long long> convolution_ll(const std::vector<long long>& a,
const std::vector<long long>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
static constexpr unsigned long long MOD1 = 754974721; // 2^24
static constexpr unsigned long long MOD2 = 167772161; // 2^25
static constexpr unsigned long long MOD3 = 469762049; // 2^26
static constexpr unsigned long long M2M3 = MOD2 * MOD3;
static constexpr unsigned long long M1M3 = MOD1 * MOD3;
static constexpr unsigned long long M1M2 = MOD1 * MOD2;
static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
static constexpr unsigned long long i1 =
internal::inv_gcd(MOD2 * MOD3, MOD1).second;
static constexpr unsigned long long i2 =
internal::inv_gcd(MOD1 * MOD3, MOD2).second;
static constexpr unsigned long long i3 =
internal::inv_gcd(MOD1 * MOD2, MOD3).second;
auto c1 = convolution<MOD1>(a, b);
auto c2 = convolution<MOD2>(a, b);
auto c3 = convolution<MOD3>(a, b);
std::vector<long long> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
unsigned long long x = 0;
x += (c1[i] * i1) % MOD1 * M2M3;
x += (c2[i] * i2) % MOD2 * M1M3;
x += (c3[i] * i3) % MOD3 * M1M2;
long long diff =
c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
if (diff < 0) diff += MOD1;
static constexpr unsigned long long offset[5] = {
0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
x -= offset[diff % 5];
c[i] = x;
}
return c;
}
} // namespace atcoder
using namespace atcoder;
//using mint = modint998244353;
using mint = modint1000000007;
//using mint = modint;
istream& operator>>(istream& is, modint998244353& M) { long long x; is >> x; M = x; return is; }
ostream& operator<<(ostream& os, const modint998244353& M) { return os << M.val(); }
istream& operator>>(istream& is, modint1000000007& M) { long long x; is >> x; M = x; return is; }
ostream& operator<<(ostream& os, const modint1000000007& M) { return os << M.val(); }
template<int m> istream& operator>>(istream& is, static_modint<m>& M) { long long x; is >> x; M = x; return is; }
template<int m> ostream& operator<<(ostream& os, const static_modint<m>& M) { return os << M.val(); }
istream& operator>>(istream& is, modint& M) { long long x; is >> x; M = x; return is; }
ostream& operator<<(ostream& os, const modint& M) { return os << M.val(); }
vector<mint> multiply(vector<mint> a, vector<mint> b) { return convolution(a,b); }
vector<mint> multiply_naive(vector<mint> a, vector<mint> b) {
int n = int(a.size()), m = int(b.size());
if (!n or !m) return {};
vector<mint> ret(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
ret[i + j] += a[i] * b[j];
}
}
return ret;
}
//59501818244292734739283969=5.95*10^25 までの値を正しく計算
//最終的な列の大きさが 2^24 までなら動く
//最終的な列の大きさが 2^20 以下のときは，下の 3 つの素数を使ったほうが速い
namespace arbitrary_convolution {
constexpr int m0 = 167772161;
constexpr int m1 = 469762049;
constexpr int m2 = 754974721;
constexpr int r01 = 104391568; // mint1(m0).inv()
constexpr int r02 = 323560596; // mint2(m0).inv()
constexpr int r12 = 399692502; // mint2(m1).inv()
// constexpr int m0 = 1045430273;
// constexpr int m1 = 1051721729;
// constexpr int m2 = 1053818881;
// constexpr int r01 = 175287122; // mint1(m0).inv()
// constexpr int r02 = 395182206; // mint2(m0).inv()
// constexpr int r12 = 526909943; // mint2(m1).inv()
constexpr int r02r12 = (long long)(r02) * r12 % m2;
constexpr long long w1 = m0;
constexpr long long w2 = (long long)(m0) * m1;
template<class T>
vector<int> multiply(const vector<T>& a, const vector<T>& b, int mod) {
int n = int(a.size()), m = int(b.size());
vector<T> v0 = convolution<m0>(a, b);
vector<T> v1 = convolution<m1>(a, b);
vector<T> v2 = convolution<m2>(a, b);
vector<int> ret(n + m - 1);
const int W1 = w1 % mod;
const int W2 = w2 % mod;
for (int i = 0; i < n + m - 1; i++) {
int n1 = v1[i], n2 = v2[i], x = v0[i];
int y = (long long)(n1 + m1 - x) * r01 % m1;
int z = ((long long)(n2 + m2 - x) * r02r12 + (long long)(m2 - y) * r12) % m2;
ret[i] = ((long long)(x) + (long long)(y) * W1 + (long long)(z) * W2) % mod;
}
return ret;
}
template<class D>
vector<D> multiply(vector<D> a, vector<D> b) {
int n = int(a.size()), m = int(b.size());
if (!n or !m) return {};
if (min(n,m) < 128) {
if (n < m) {
swap(n, m);
swap(a, b);
}
vector<D> ret(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
ret[i + j] += a[i] * b[j];
}
}
return ret;
}
vector<int> a_(n), b_(m);
for (int i = 0; i < n; i++) a_[i] = a[i].val();
for (int i = 0; i < m; i++) b_[i] = b[i].val();
vector<int> c = multiply<int>(a_, b_, D::mod());
vector<D> ret(n + m - 1);
for (int i = 0; i < n + m - 1; i++) ret[i] = D::raw(c[i]);
return ret;
}
}
template<class D, vector<D> (*op)(vector<D>, vector<D>)>
struct FormalPowerSeries {
using Poly = FormalPowerSeries;
vector<D> v;
FormalPowerSeries(const vector<D>& v_ = {}) : v(v_) { shrink(); }
void shrink() {
while (v.size() && v.back() == 0) v.pop_back();
}
int size() const { return int(v.size()); }
D freq(int p) const { return (p < size()) ? v[p] : D(0); }
Poly operator+(const Poly& r) const {
int n = max(size(), r.size());
vector<D> res(n);
for (int i = 0; i < n; i++) res[i] = freq(i) + r.freq(i);
return res;
}
Poly operator-(const Poly& r) const {
int n = max(size(), r.size());
vector<D> res(n);
for (int i = 0; i < n; i++) res[i] = freq(i) - r.freq(i);
return res;
}
Poly operator*(const Poly& r) const { return op(v, r.v); }
Poly operator*(const D& r) const {
int n = size();
vector<D> res(n);
for (int i = 0; i < n; i++) res[i] = v[i] * r;
return res;
}
Poly operator*(const vector<pair<int, D>>& r) const {
assert(!r.empty());
int n = size();
int m = r.back().first;
vector<D> res(n + m);
for (int i = 0; i < n; i++) {
for (auto e : r) {
res[i + e.first] += v[i] * e.second;
}
}
return res;
}
Poly operator/(const D& r) const {
return *this * r.inv();
}
Poly operator/(const Poly& r) const {
if (size() < r.size()) return {{}};
int n = size() - r.size() + 1;
return (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
}
Poly operator/(const vector<pair<int, D>>& r) const {
assert(!r.empty());
int n = size();
auto [d, c] = r.front();
assert(d == 0 && c != D(0));
D ic = D(1) / c;
vector<D> res(n);
for (int i = 0; i < n; i++) {
for (auto& e : r) {
if (e.first and e.first <= i) {
res[i] -= res[i - e.first] * e.second;
}
}
res[i] += v[i];
res[i] *= ic;
}
return res;
}
Poly operator%(const Poly& r) const { return *this - *this / r * r; }
Poly operator<<(int s) const {
vector<D> res(size() + s);
for (int i = 0; i < size(); i++) res[i + s] = v[i];
return res;
}
Poly operator>>(int s) const {
if (size() <= s) return Poly();
vector<D> res(size() - s);
for (int i = 0; i < size() - s; i++) res[i] = v[i + s];
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator*=(const vector<pair<int, D>>& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D &r) {return *this = *this/r;}
Poly& operator/=(const vector<pair<int, D>>& r) { return *this = *this / r; }
Poly& operator%=(const Poly& r) { return *this = *this % r; }
Poly& operator<<=(const size_t& n) { return *this = *this << n; }
Poly& operator>>=(const size_t& n) { return *this = *this >> n; }
Poly mul(int d, const D& c) const {
vector<D> res(size() + d);
for (int i = 0; i < size(); i++) {
res[i] += v[i];
res[i+d] += v[i] * c;
}
return res;
}
Poly div(int d, const D& c) const {
vector<D> res(size());
for (int i = 0; i < size(); i++) {
res[i] = v[i];
if (i >= d) res[i] -= res[i - d] * c;
}
return res;
}
Poly pre(int le) const {
return {{v.begin(), v.begin() + min(size(), le)}};
}
Poly rev(int n = -1) const {
vector<D> res = v;
if (n != -1) res.resize(n);
reverse(res.begin(), res.end());
return res;
}
Poly diff() const {
vector<D> res(max(0, size() - 1));
for (int i = 1; i < size(); i++) res[i - 1] = freq(i) * i;
return res;
}
Poly inte() const {
vector<D> res(size() + 1);
for (int i = 0; i < size(); i++) res[i + 1] = freq(i) / (i + 1);
return res;
}
// f * f.inv() = 1 + g(x)x^m
Poly inv(int m) const {
Poly res = Poly({D(1) / freq(0)});
for (int i = 1; i < m; i *= 2) {
res = (res * D(2) - res * res * pre(2 * i)).pre(2 * i);
}
return res.pre(m);
}
Poly exp(int n) const {
assert(freq(0) == 0);
Poly f({1}), g({1});
for (int i = 1; i < n; i *= 2) {
g = (g * 2 - f * g * g).pre(i);
Poly q = diff().pre(i - 1);
Poly w = (q + g * (f.diff() - f * q)).pre(2 * i - 1);
f = (f + f * (*this - w.inte()).pre(2 * i)).pre(2 * i);
}
return f.pre(n);
}
Poly log(int n) const {
assert(freq(0) == 1);
auto f = pre(n);
return (f.diff() * f.inv(n - 1)).pre(n - 1).inte();
}
Poly sqrt(int n) const {
assert(freq(0) == 1);
Poly f = pre(n + 1);
Poly g({1});
for (int i = 1; i < n; i *= 2) {
g = (g + f.pre(2 * i) * g.inv(2 * i)) / 2;
}
return g.pre(n + 1);
}
Poly pow_mod(ll n, const Poly& mod) {
Poly x = *this, r = {{1}};
while (n) {
if (n & 1) r = r * x % mod;
x = x * x % mod;
n >>= 1;
}
return r;
}
friend ostream& operator<<(ostream& os, const Poly& p) {
if (p.size() == 0) return os << "0";
for (auto i = 0; i < p.size(); i++) {
if (p.v[i] != 0) {
os << p.v[i] << " x^" << i;
if (i != p.size() - 1) os << " + ";
}
}
return os;
}
};
using Poly = FormalPowerSeries<mint, arbitrary_convolution::multiply>;
int main() {
int K,N; cin >> K >> N;
Poly f;
f.v.assign(K+1,0);
f.v[0] = 1;
rep(i,N) {
int x; cin >> x;
if (x <= K) f.v[x]--;
}
f = f.inv(K+1);
cout << f.freq(K) << ln;
}