結果

問題 No.8046 yukicoderの過去問
ユーザー HEXAebmrHEXAebmr
提出日時 2022-05-02 13:52:20
言語 C++14
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 138 ms / 2,000 ms
コード長 20,293 bytes
コンパイル時間 4,110 ms
コンパイル使用メモリ 209,936 KB
実行使用メモリ 21,280 KB
最終ジャッジ日時 2024-07-01 16:49:27
合計ジャッジ時間 5,562 ms
ジャッジサーバーID
(参考情報)
judge1 / judge5
このコードへのチャレンジ
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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 2 ms
6,812 KB
testcase_01 AC 2 ms
6,940 KB
testcase_02 AC 2 ms
6,940 KB
testcase_03 AC 130 ms
20,524 KB
testcase_04 AC 2 ms
6,948 KB
testcase_05 AC 128 ms
20,332 KB
testcase_06 AC 136 ms
21,280 KB
testcase_07 AC 133 ms
20,932 KB
testcase_08 AC 138 ms
21,268 KB
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ソースコード

diff #

#include <algorithm>
#include <atcoder/all>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <cstdint>
#include <fstream>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <queue>
#include <random>
#include <regex>
#include <set>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using namespace atcoder;
using ll = long long;
using ld = long double;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using vll = vector<ll>;
using vvll = vector<vector<ll>>;
using vvvll = vector<vector<vector<ll>>>;
using vii = vector<int>;
using vvii = vector<vector<int>>;
using vvvii = vector<vector<vector<int>>>;
using vdd = vector<ld>;
using vvdd = vector<vector<ld>>;
using vvvdd = vector<vector<vector<ld>>>;
using vbb = vector<bool>;
using vvbb = vector<vector<bool>>;
using vvvbb = vector<vector<vector<bool>>>;
using vpll = vector<pll>;
using vvpll = vector<vector<pll>>;
using vvvpll = vector<vector<vector<pll>>>;
using vmm1 = vector<modint1000000007>;
using vvmm1 = vector<vector<modint1000000007>>;
using vvvmm1 = vector<vector<vector<modint1000000007>>>;
using vmm2 = vector<modint998244353>;
using vvmm2 = vector<vector<modint998244353>>;
using vvvmm2 = vector<vector<vector<modint998244353>>>;
#define pb push_back
#define mp make_pair
#define sc second
#define fr first
#define endl '\n'
#define stpr std::fixed << setprecision
#define cYES cout << "YES" << endl
#define cNO cout << "NO" << endl
#define cYes cout << "Yes" << endl
#define cNo cout << "No" << endl
#define cerr cout << -1 << endl
#define rep(i, n) for (ll i = 0; i < (n); ++i)
#define drep(i, a, b, d) for (ll i = (a); i <= (b); i += d)
#define Rep(i, a, b) for (ll i = (a); i < (b); ++i)
#define rrep(i, n) for (ll i = n - 1; i >= 0; i--)
#define drrep(i, a, b, d) for (ll i = (a); i >= (b); i -= d)
#define rRep(i, a, b) for (ll i = a; i >= b; i--)
#define crep(i) for (char i = 'a'; i <= 'z'; ++i)
#define Crep(i) for (char i = 'A'; i <= 'Z'; ++i)
#define ALL(x) (x).begin(), (x).end()
#define rALL(x) (x).rbegin(), (x).rend()
#define sort2(A, N) \
  sort(A, A + N,    \
       [](const pii &a, const pii &b) { return a.second < b.second; });
#define debug(v)      \
  cout << #v << ":";  \
  for (auto x : v) {  \
    cout << x << ' '; \
  }                   \
  cout << endl;
int ctoi(const char c) {
  if ('0' <= c && c <= '9') return (c - '0');
  return -1;
}
ll gcd(ll a, ll b) { return (b == 0 ? a : gcd(b, a % b)); }
ll lcm(ll a, ll b) { return a * b / gcd(a, b); }
ll rup(ll a, ll b) { return a + (b - a % b) % b; }
ll rdc(ll a, ll b) { return a / b * b; }
constexpr ll INF = 1000000011;
constexpr ll MOD = 1000000007;
constexpr ll MOD2 = 998244353;
constexpr ll LINF = 1001002003004005006ll;
constexpr ld EPS = 10e-15;
// using mint = modint1000000007;
// using mint2 = modint998244353;
template <class T, class U>
inline bool chmax(T &lhs, const U &rhs) {
  if (lhs < rhs) {
    lhs = rhs;
    return 1;
  }
  return 0;
}
template <class T, class U>
inline bool chmin(T &lhs, const U &rhs) {
  if (lhs > rhs) {
    lhs = rhs;
    return 1;
  }
  return 0;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (auto &&x : v) is >> x;
  return is;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
  is >> p.first;
  is >> p.second;
  return is;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
  os << p.first << ' ' << p.second;
  return os;
}
template <class T>
ostream &operator<<(ostream &os, vector<T> &v) {
  for (auto i = begin(v); i != end(v); ++i) {
    if (i != begin(v)) os << ' ';
    os << *i;
  }
  return os;
}

using int64 = long long;
// const int mod = 1e9 + 7;
const int mod = 998244353;

const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;

template <typename T1, typename T2>
inline bool chmax(T1 &a, T2 b) {
  return a < b && (a = b, true);
}

template <typename T1, typename T2>
inline bool chmin(T1 &a, T2 b) {
  return a > b && (a = b, true);
}

template <typename T = int64>
vector<T> make_v(size_t a) {
  return vector<T>(a);
}

template <typename T, typename... Ts>
auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}

template <typename T, typename V>
typename enable_if<is_class<T>::value == 0>::type fill_v(T &t, const V &v) {
  t = v;
}

template <typename T, typename V>
typename enable_if<is_class<T>::value != 0>::type fill_v(T &t, const V &v) {
  for (auto &e : t) fill_v(e, v);
}

template <typename F>
struct FixPoint : F {
  FixPoint(F &&f) : F(forward<F>(f)) {}

  template <typename... Args>
  decltype(auto) operator()(Args &&...args) const {
    return F::operator()(*this, forward<Args>(args)...);
  }
};

template <typename F>
inline decltype(auto) MFP(F &&f) {
  return FixPoint<F>{forward<F>(f)};
}

#line 1 "math/fft/fast-fourier-transform.cpp"
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); }
};

const real PI = acosl(-1);
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 = 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;
      }
    }
  }
}

vector<int64_t> multiply(const vector<int> &a, const vector<int> &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++) {
    int x = (i < (int)a.size() ? a[i] : 0);
    int 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<int64_t> ret(need);
  for (int i = 0; i < need; i++) {
    ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
  }
  return ret;
}
};  // namespace FastFourierTransform
#line 2 "math/fft/arbitrary-mod-convolution.cpp"

/*
 * @brief Arbitrary-Mod-Convolution(任意mod畳み込み)
 */
template <typename T>
struct ArbitraryModConvolution {
  using real = FastFourierTransform::real;
  using C = FastFourierTransform::C;

  ArbitraryModConvolution() = default;

  static 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 < a.size(); i++) {
      fa[i] = C(a[i].x & ((1 << 15) - 1), a[i].x >> 15);
    }
    fft(fa, sz);
    vector<C> fb(sz);
    if (a == b) {
      fb = fa;
    } else {
      for (int i = 0; i < b.size(); i++) {
        fb[i] = C(b[i].x & ((1 << 15) - 1), b[i].x >> 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++) {
      int64_t aa = llround(fa[i].x);
      int64_t bb = llround(fb[i].x);
      int64_t cc = llround(fa[i].y);
      aa = T(aa).x, bb = T(bb).x, cc = T(cc).x;
      ret[i] = aa + (bb << 15) + (cc << 30);
    }
    return ret;
  }
};

#line 2 "math/fps/formal-power-series.cpp"

/**
 * @brief Formal-Power-Series(形式的冪級数)
 */
template <typename T>
struct FormalPowerSeries : vector<T> {
  using vector<T>::vector;
  using P = FormalPowerSeries;
  using Conv = ArbitraryModConvolution<T>;

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

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

  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 < r.size(); i++) (*this)[i] += r[i];
    return *this;
  }

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

  // https://judge.yosupo.jp/problem/convolution_mod
  P &operator*=(const P &r) {
    if (this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    auto ret = Conv::multiply(*this, r);
    return *this = {begin(ret), end(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 &operator%=(const P &r) { return *this -= *this / r * r; }

  // https://judge.yosupo.jp/problem/division_of_polynomials
  pair<P, P> div_mod(const P &r) {
    P q = *this / r;
    return make_pair(q, *this - q * 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 T &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

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

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

  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 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;
  }

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

  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;
  }

  // https://judge.yosupo.jp/problem/inv_of_formal_power_series
  // 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);
  }

  // https://judge.yosupo.jp/problem/log_of_formal_power_series
  // 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();
  }

  // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
  P sqrt(
      int deg = -1,
      const function<T(T)> &get_sqrt = [](T) { return T(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, get_sqrt);
          if (ret.empty()) return {};
          ret = ret << (i / 2);
          if (ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }
    auto sqr = T(get_sqrt((*this)[0]));
    if (sqr * sqr != (*this)[0]) return {};
    P ret{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);
  }

  P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
    return sqrt(deg, get_sqrt);
  }

  // https://judge.yosupo.jp/problem/exp_of_formal_power_series
  // F(0) must be 0
  P exp(int deg = -1) const {
    if (deg == -1) deg = this->size();
    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);
  }

  // https://judge.yosupo.jp/problem/pow_of_formal_power_series
  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() * ((*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;
  }

  P 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;
        ret.shrink();
      }
      x *= x;
      x -= get_div(x) * g;
      x.shrink();
      k >>= 1;
    }
    return ret;
  }

  // https://judge.yosupo.jp/problem/polynomial_taylor_shift
  P 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;
  }
};

/**
 * @brief Linear-Recursion-Formula
 */
template <template <typename> class FPS, typename Mint>
Mint linear_recursion_formula(FPS<Mint> P, FPS<Mint> Q, int64_t k) {
  // compute the coefficient [x^k] P/Q of rational power series
  Mint ret = 0;
  if (P.size() >= Q.size()) {
    auto R = P / Q;
    P -= R * Q;
    P.shrink();
    if (k < (int)R.size()) ret += R[k];
  }
  if (P.empty()) return ret;
  P.resize((int)Q.size() - 1);
  while (k > 0) {
    auto Q2 = Q;
    for (int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
    auto S = P * Q2;
    auto T = Q * Q2;
    if (k & 1) {
      for (int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
      for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
    } else {
      for (int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
      for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
    }
    k >>= 1;
  }
  return ret + P[0];
}

template <typename Mint>
using FPS = FormalPowerSeries<Mint>;

template <int mod>
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if ((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if ((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int)(1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }

  ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }

  ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }

  ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x; }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt<mod>(t);
    return (is);
  }

  static int get_mod() { return mod; }
};

using modint = ModInt<mod>;
using mint = ModInt<MOD>;

mint Bostan_Mori(FPS<mint> p, FPS<mint> q, ll k) {
  if (k < 0) return 0;
  p.resize(max(p.size(), q.size()));
  q.resize(max(p.size(), q.size()));
  while (k) {
    FPS<mint> _q = q;
    for (int i = 1; i < _q.size(); i += 2) _q[i] = -_q[i];
    FPS<mint> v = q, u = p;
    v *= _q;
    u *= _q;
    for (int i = k % 2; i < u.size(); i += 2) p[i / 2] = u[i];
    for (int i = 0; i < v.size(); i += 2) q[i / 2] = v[i];
    k /= 2;
  }
  return p[0] / q[0];
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);
  ll K, N;
  cin >> K >> N;
  vll A(N);
  cin >> A;
  FPS<mint> X(K + 1);
  X[0] = 1;
  rep(i, N) { X[A[i]] -= 1; }
  X = X.inv();
  cout << X[K] << endl;
}
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