#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <limits>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

using namespace std;

using Int = long long;

template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }

////////////////////////////////////////////////////////////////////////////////
template <unsigned M_> struct ModInt {
  static constexpr unsigned M = M_;
  unsigned x;
  constexpr ModInt() : x(0U) {}
  constexpr ModInt(unsigned x_) : x(x_ % M) {}
  constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
  constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
  constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
  ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
  ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
  ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
  ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
  ModInt pow(long long e) const {
    if (e < 0) return inv().pow(-e);
    ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
  }
  ModInt inv() const {
    unsigned a = M, b = x; int y = 0, z = 1;
    for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
    assert(a == 1U); return ModInt(y);
  }
  ModInt operator+() const { return *this; }
  ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
  ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
  ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
  ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
  ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
  template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
  template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
  template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
  template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
  explicit operator bool() const { return x; }
  bool operator==(const ModInt &a) const { return (x == a.x); }
  bool operator!=(const ModInt &a) const { return (x != a.x); }
  friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
constexpr unsigned MO = 998244353U;
constexpr unsigned MO2 = 2U * MO;
constexpr int FFT_MAX = 23;
using Mint = ModInt<MO>;
constexpr Mint FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 911660635U, 372528824U, 929031873U, 452798380U, 922799308U, 781712469U, 476477967U, 166035806U, 258648936U, 584193783U, 63912897U, 350007156U, 666702199U, 968855178U, 629671588U, 24514907U, 996173970U, 363395222U, 565042129U, 733596141U, 267099868U, 15311432U};
constexpr Mint INV_FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 86583718U, 509520358U, 337190230U, 87557064U, 609441965U, 135236158U, 304459705U, 685443576U, 381598368U, 335559352U, 129292727U, 358024708U, 814576206U, 708402881U, 283043518U, 3707709U, 121392023U, 704923114U, 950391366U, 428961804U, 382752275U, 469870224U};
constexpr Mint FFT_RATIOS[FFT_MAX] = {911660635U, 509520358U, 369330050U, 332049552U, 983190778U, 123842337U, 238493703U, 975955924U, 603855026U, 856644456U, 131300601U, 842657263U, 730768835U, 942482514U, 806263778U, 151565301U, 510815449U, 503497456U, 743006876U, 741047443U, 56250497U, 867605899U};
constexpr Mint INV_FFT_RATIOS[FFT_MAX] = {86583718U, 372528824U, 373294451U, 645684063U, 112220581U, 692852209U, 155456985U, 797128860U, 90816748U, 860285882U, 927414960U, 354738543U, 109331171U, 293255632U, 535113200U, 308540755U, 121186627U, 608385704U, 438932459U, 359477183U, 824071951U, 103369235U};

// as[rev(i)] <- \sum_j \zeta^(ij) as[j]
void fft(Mint *as, int n) {
  assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
  int m = n;
  if (m >>= 1) {
    for (int i = 0; i < m; ++i) {
      const unsigned x = as[i + m].x;  // < MO
      as[i + m].x = as[i].x + MO - x;  // < 2 MO
      as[i].x += x;  // < 2 MO
    }
  }
  if (m >>= 1) {
    Mint prod = 1U;
    for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
      for (int i = i0; i < i0 + m; ++i) {
        const unsigned x = (prod * as[i + m]).x;  // < MO
        as[i + m].x = as[i].x + MO - x;  // < 3 MO
        as[i].x += x;  // < 3 MO
      }
      prod *= FFT_RATIOS[__builtin_ctz(++h)];
    }
  }
  for (; m; ) {
    if (m >>= 1) {
      Mint prod = 1U;
      for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
        for (int i = i0; i < i0 + m; ++i) {
          const unsigned x = (prod * as[i + m]).x;  // < MO
          as[i + m].x = as[i].x + MO - x;  // < 4 MO
          as[i].x += x;  // < 4 MO
        }
        prod *= FFT_RATIOS[__builtin_ctz(++h)];
      }
    }
    if (m >>= 1) {
      Mint prod = 1U;
      for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
        for (int i = i0; i < i0 + m; ++i) {
          const unsigned x = (prod * as[i + m]).x;  // < MO
          as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x;  // < 2 MO
          as[i + m].x = as[i].x + MO - x;  // < 3 MO
          as[i].x += x;  // < 3 MO
        }
        prod *= FFT_RATIOS[__builtin_ctz(++h)];
      }
    }
  }
  for (int i = 0; i < n; ++i) {
    as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x;  // < 2 MO
    as[i].x = (as[i].x >= MO) ? (as[i].x - MO) : as[i].x;  // < MO
  }
}

// as[i] <- (1/n) \sum_j \zeta^(-ij) as[rev(j)]
void invFft(Mint *as, int n) {
  assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
  int m = 1;
  if (m < n >> 1) {
    Mint prod = 1U;
    for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
      for (int i = i0; i < i0 + m; ++i) {
        const unsigned long long y = as[i].x + MO - as[i + m].x;  // < 2 MO
        as[i].x += as[i + m].x;  // < 2 MO
        as[i + m].x = (prod.x * y) % MO;  // < MO
      }
      prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
    }
    m <<= 1;
  }
  for (; m < n >> 1; m <<= 1) {
    Mint prod = 1U;
    for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
      for (int i = i0; i < i0 + (m >> 1); ++i) {
        const unsigned long long y = as[i].x + MO2 - as[i + m].x;  // < 4 MO
        as[i].x += as[i + m].x;  // < 4 MO
        as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x;  // < 2 MO
        as[i + m].x = (prod.x * y) % MO;  // < MO
      }
      for (int i = i0 + (m >> 1); i < i0 + m; ++i) {
        const unsigned long long y = as[i].x + MO - as[i + m].x;  // < 2 MO
        as[i].x += as[i + m].x;  // < 2 MO
        as[i + m].x = (prod.x * y) % MO;  // < MO
      }
      prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
    }
  }
  if (m < n) {
    for (int i = 0; i < m; ++i) {
      const unsigned y = as[i].x + MO2 - as[i + m].x;  // < 4 MO
      as[i].x += as[i + m].x;  // < 4 MO
      as[i + m].x = y;  // < 4 MO
    }
  }
  const Mint invN = Mint(n).inv();
  for (int i = 0; i < n; ++i) {
    as[i] *= invN;
  }
}

void fft(vector<Mint> &as) {
  fft(as.data(), as.size());
}
void invFft(vector<Mint> &as) {
  invFft(as.data(), as.size());
}

vector<Mint> convolve(vector<Mint> as, vector<Mint> bs) {
  if (as.empty() || bs.empty()) return {};
  const int len = as.size() + bs.size() - 1;
  int n = 1;
  for (; n < len; n <<= 1) {}
  as.resize(n); fft(as);
  bs.resize(n); fft(bs);
  for (int i = 0; i < n; ++i) as[i] *= bs[i];
  invFft(as);
  as.resize(len);
  return as;
}
vector<Mint> square(vector<Mint> as) {
  if (as.empty()) return {};
  const int len = as.size() + as.size() - 1;
  int n = 1;
  for (; n < len; n <<= 1) {}
  as.resize(n); fft(as);
  for (int i = 0; i < n; ++i) as[i] *= as[i];
  invFft(as);
  as.resize(len);
  return as;
}
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
// inv: log, exp, pow
// fac: shift
// invFac: shift
constexpr int LIM_INV = 1 << 20;  // @
Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV];
struct ModIntPreparator {
  ModIntPreparator() {
    inv[1] = 1;
    for (int i = 2; i < LIM_INV; ++i) inv[i] = -((Mint::M / i) * inv[Mint::M % i]);
    fac[0] = 1;
    for (int i = 1; i < LIM_INV; ++i) fac[i] = fac[i - 1] * i;
    invFac[0] = 1;
    for (int i = 1; i < LIM_INV; ++i) invFac[i] = invFac[i - 1] * inv[i];
  }
} preparator;

// polyWork0: *, inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork1: inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork2: divAt, exp, pow, sqrt
// polyWork3: exp, pow, sqrt
static constexpr int LIM_POLY = 1 << 20;  // @
static_assert(LIM_POLY <= 1 << FFT_MAX, "Poly: LIM_POLY <= 1 << FFT_MAX must hold.");
static Mint polyWork0[LIM_POLY], polyWork1[LIM_POLY], polyWork2[LIM_POLY], polyWork3[LIM_POLY];

struct Poly : public vector<Mint> {
  Poly() {}
  explicit Poly(int n) : vector<Mint>(n) {}
  Poly(const vector<Mint> &vec) : vector<Mint>(vec) {}
  Poly(std::initializer_list<Mint> il) : vector<Mint>(il) {}
  int size() const { return vector<Mint>::size(); }
  Mint at(long long k) const { return (0 <= k && k < size()) ? (*this)[k] : 0U; }
  int ord() const { for (int i = 0; i < size(); ++i) if ((*this)[i]) return i; return -1; }
  int deg() const { for (int i = size(); --i >= 0; ) if ((*this)[i]) return i; return -1; }
  Poly mod(int n) const { return Poly(vector<Mint>(data(), data() + min(n, size()))); }
  friend std::ostream &operator<<(std::ostream &os, const Poly &fs) {
    os << "[";
    for (int i = 0; i < fs.size(); ++i) { if (i > 0) os << ", "; os << fs[i]; }
    return os << "]";
  }

  Poly &operator+=(const Poly &fs) {
    if (size() < fs.size()) resize(fs.size());
    for (int i = 0; i < fs.size(); ++i) (*this)[i] += fs[i];
    return *this;
  }
  Poly &operator-=(const Poly &fs) {
    if (size() < fs.size()) resize(fs.size());
    for (int i = 0; i < fs.size(); ++i) (*this)[i] -= fs[i];
    return *this;
  }
  // 3 E(|t| + |f|)
  Poly &operator*=(const Poly &fs) {
    if (empty() || fs.empty()) return *this = {};
    const int nt = size(), nf = fs.size();
    int n = 1;
    for (; n < nt + nf - 1; n <<= 1) {}
    assert(n <= LIM_POLY);
    resize(n);
    fft(data(), n);  // 1 E(n)
    memcpy(polyWork0, fs.data(), nf * sizeof(Mint));
    memset(polyWork0 + nf, 0, (n - nf) * sizeof(Mint));
    fft(polyWork0, n);  // 1 E(n)
    for (int i = 0; i < n; ++i) (*this)[i] *= polyWork0[i];
    invFft(data(), n);  // 1 E(n)
    resize(nt + nf - 1);
    return *this;
  }
  // 13 E(deg(t) - deg(f) + 1)
  // rev(t) = rev(f) rev(q) + x^(deg(t)-deg(f)+1) rev(r)
  Poly &operator/=(const Poly &fs) {
    const int m = deg(), n = fs.deg();
    assert(n != -1);
    if (m < n) return *this = {};
    Poly tsRev(m - n + 1), fsRev(min(m - n, n) + 1);
    for (int i = 0; i <= m - n; ++i) tsRev[i] = (*this)[m - i];
    for (int i = 0, i0 = min(m - n, n); i <= i0; ++i) fsRev[i] = fs[n - i];
    const Poly qsRev = tsRev.div(fsRev, m - n + 1);  // 13 E(m - n + 1)
    resize(m - n + 1);
    for (int i = 0; i <= m - n; ++i) (*this)[i] = qsRev[m - n - i];
    return *this;
  }
  // 13 E(deg(t) - deg(f) + 1) + 3 E(|t|)
  Poly &operator%=(const Poly &fs) {
    const Poly qs = *this / fs;  // 13 E(deg(t) - deg(f) + 1)
    *this -= fs * qs;  // 3 E(|t|)
    resize(deg() + 1);
    return *this;
  }
  Poly &operator*=(const Mint &a) {
    for (int i = 0; i < size(); ++i) (*this)[i] *= a;
    return *this;
  }
  Poly &operator/=(const Mint &a) {
    const Mint b = a.inv();
    for (int i = 0; i < size(); ++i) (*this)[i] *= b;
    return *this;
  }
  Poly operator+() const { return *this; }
  Poly operator-() const {
    Poly fs(size());
    for (int i = 0; i < size(); ++i) fs[i] = -(*this)[i];
    return fs;
  }
  Poly operator+(const Poly &fs) const { return (Poly(*this) += fs); }
  Poly operator-(const Poly &fs) const { return (Poly(*this) -= fs); }
  Poly operator*(const Poly &fs) const { return (Poly(*this) *= fs); }
  Poly operator/(const Poly &fs) const { return (Poly(*this) /= fs); }
  Poly operator%(const Poly &fs) const { return (Poly(*this) %= fs); }
  Poly operator*(const Mint &a) const { return (Poly(*this) *= a); }
  Poly operator/(const Mint &a) const { return (Poly(*this) /= a); }
  friend Poly operator*(const Mint &a, const Poly &fs) { return fs * a; }

  // 10 E(n)
  // f <- f - (t f - 1) f
  Poly inv(int n) const {
    assert(!empty()); assert((*this)[0]); assert(1 <= n);
    assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY);
    Poly fs(n);
    fs[0] = (*this)[0].inv();
    for (int m = 1; m < n; m <<= 1) {
      memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint));
      memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint));
      fft(polyWork0, m << 1);  // 2 E(n)
      memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint));
      memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint));
      fft(polyWork1, m << 1);  // 2 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
      invFft(polyWork0, m << 1); // 2 E(n)
      memset(polyWork0, 0, m * sizeof(Mint));
      fft(polyWork0, m << 1); // 2 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
      invFft(polyWork0, m << 1); // 2 E(n)
      for (int i = m, i0 = min(m << 1, n); i < i0; ++i) fs[i] = -polyWork0[i];
    }
    return fs;
  }
  // 9 E(n)
  // Need (4 m)-th roots of unity to lift from (mod x^m) to (mod x^(2m)).
  // f <- f - (t f - 1) f
  // (t f^2) mod ((x^(2m) - 1) (x^m - 1^(1/4)))
  /*
  Poly inv(int n) const {
    assert(!empty()); assert((*this)[0]); assert(1 <= n);
    assert(n == 1 || 3 << (31 - __builtin_clz(n - 1)) <= LIM_POLY);
    assert(n <= 1 << (FFT_MAX - 1));
    Poly fs(n);
    fs[0] = (*this)[0].inv();
    for (int h = 2, m = 1; m < n; ++h, m <<= 1) {
      const Mint a = FFT_ROOTS[h], b = INV_FFT_ROOTS[h];
      memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint));
      memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint));
      {
        Mint aa = 1;
        for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] = aa * polyWork0[i]; aa *= a; }
        for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] += aa * polyWork0[m + i]; aa *= a; }
      }
      fft(polyWork0, m << 1);  // 2 E(n)
      fft(polyWork0 + (m << 1), m);  // 1 E(n)
      memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint));
      memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint));
      {
        Mint aa = 1;
        for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] = aa * polyWork1[i]; aa *= a; }
        for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] += aa * polyWork1[m + i]; aa *= a; }
      }
      fft(polyWork1, m << 1);  // 2 E(n)
      fft(polyWork1 + (m << 1), m);  // 1 E(n)
      for (int i = 0; i < (m << 1) + m; ++i) polyWork0[i] *= polyWork1[i] * polyWork1[i];
      invFft(polyWork0, m << 1);  // 2 E(n)
      invFft(polyWork0 + (m << 1), m);  // 1 E(n)
      // 2 f0 + (-f2), (-f1) + (-f3), 1^(1/4) (-f1) - (-f2) - 1^(1/4) (-f3)
      {
        Mint bb = 1;
        for (int i = 0, i0 = min(m, n - m); i < i0; ++i) {
          unsigned x = polyWork0[i].x + (bb * polyWork0[(m << 1) + i]).x + MO2 - (fs[i].x << 1);  // < 4 MO
          fs[m + i] = Mint(static_cast<unsigned long long>(FFT_ROOTS[2].x) * x) - polyWork0[m + i];
          fs[m + i].x = ((fs[m + i].x & 1) ? (fs[m + i].x + MO) : fs[m + i].x) >> 1;
          bb *= b;
        }
      }
    }
    return fs;
  }
  */
  // 13 E(n)
  // g = (1 / f) mod x^m
  // h <- h - (f h - t) g
  Poly div(const Poly &fs, int n) const {
    assert(!fs.empty()); assert(fs[0]); assert(1 <= n);
    if (n == 1) return {at(0) / fs[0]};
    // m < n <= 2 m
    const int m = 1 << (31 - __builtin_clz(n - 1));
    assert(m << 1 <= LIM_POLY);
    Poly gs = fs.inv(m);  // 5 E(n)
    gs.resize(m << 1);
    fft(gs.data(), m << 1);  // 1 E(n)
    memcpy(polyWork0, data(), min(m, size()) * sizeof(Mint));
    memset(polyWork0 + min(m, size()), 0, ((m << 1) - min(m, size())) * sizeof(Mint));
    fft(polyWork0, m << 1);  // 1 E(n)
    for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
    invFft(polyWork0, m << 1);  // 1 E(n)
    Poly hs(n);
    memcpy(hs.data(), polyWork0, m * sizeof(Mint));
    memset(polyWork0 + m, 0, m * sizeof(Mint));
    fft(polyWork0, m << 1);  // 1 E(n)
    memcpy(polyWork1, fs.data(), min(m << 1, fs.size()) * sizeof(Mint));
    memset(polyWork1 + min(m << 1, fs.size()), 0, ((m << 1) - min(m << 1, fs.size())) * sizeof(Mint));
    fft(polyWork1, m << 1);  // 1 E(n)
    for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
    invFft(polyWork0, m << 1);  // 1 E(n)
    memset(polyWork0, 0, m * sizeof(Mint));
    for (int i = m, i0 = min(m << 1, size()); i < i0; ++i) polyWork0[i] -= (*this)[i];
    fft(polyWork0, m << 1);  // 1 E(n)
    for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
    invFft(polyWork0, m << 1);  // 1 E(n)
    for (int i = m; i < n; ++i) hs[i] = -polyWork0[i];
    return hs;
  }
  // (4 (floor(log_2 k) - ceil(log_2 |f|)) + 16) E(|f|)  for  |t| < |f|
  // [x^k] (t(x) / f(x)) = [x^k] ((t(x) f(-x)) / (f(x) f(-x))
  // polyWork0: half of (2 m)-th roots of unity, inversed, bit-reversed
  Mint divAt(const Poly &fs, long long k) const {
    assert(k >= 0);
    if (size() >= fs.size()) {
      const Poly qs = *this / fs;  // 13 E(deg(t) - deg(f) + 1)
      Poly rs = *this - fs * qs;  // 3 E(|t|)
      rs.resize(rs.deg() + 1);
      return qs.at(k) + rs.divAt(fs, k);
    }
    int h = 0, m = 1;
    for (; m < fs.size(); ++h, m <<= 1) {}
    if (k < m) {
      const Poly gs = fs.inv(k + 1);  // 10 E(|f|)
      Mint sum;
      for (int i = 0, i0 = min<int>(k + 1, size()); i < i0; ++i) sum += (*this)[i] * gs[k - i];
      return sum;
    }
    assert(m << 1 <= LIM_POLY);
    polyWork0[0] = Mint(2U).inv();
    for (int hh = 0; hh < h; ++hh) for (int i = 0; i < 1 << hh; ++i) polyWork0[1 << hh | i] = polyWork0[i] * INV_FFT_ROOTS[hh + 2];
    const Mint a = FFT_ROOTS[h + 1];
    memcpy(polyWork2, data(), size() * sizeof(Mint));
    memset(polyWork2 + size(), 0, ((m << 1) - size()) * sizeof(Mint));
    fft(polyWork2, m << 1);  // 2 E(|f|)
    memcpy(polyWork1, fs.data(), fs.size() * sizeof(Mint));
    memset(polyWork1 + fs.size(), 0, ((m << 1) - fs.size()) * sizeof(Mint));
    fft(polyWork1, m << 1);  // 2 E(|f|)
    for (; ; ) {
      if (k & 1) {
        for (int i = 0; i < m; ++i) polyWork2[i] = polyWork0[i] * (polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] - polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0]);
      } else {
        for (int i = 0; i < m; ++i) {
          polyWork2[i] = polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] + polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0];
          polyWork2[i].x = ((polyWork2[i].x & 1) ? (polyWork2[i].x + MO) : polyWork2[i].x) >> 1;
        }
      }
      for (int i = 0; i < m; ++i) polyWork1[i] = polyWork1[i << 1 | 0] * polyWork1[i << 1 | 1];
      if ((k >>= 1) < m) {
        invFft(polyWork2, m);  // 1 E(|f|)
        invFft(polyWork1, m);  // 1 E(|f|)
        // Poly::inv does not use polyWork2
        const Poly gs = Poly(vector<Mint>(polyWork1, polyWork1 + k + 1)).inv(k + 1);  // 10 E(|f|)
        Mint sum;
        for (int i = 0; i <= k; ++i) sum += polyWork2[i] * gs[k - i];
        return sum;
      }
      memcpy(polyWork2 + m, polyWork2, m * sizeof(Mint));
      invFft(polyWork2 + m, m);  // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
      memcpy(polyWork1 + m, polyWork1, m * sizeof(Mint));
      invFft(polyWork1 + m, m);  // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
      Mint aa = 1;
      for (int i = m; i < m << 1; ++i) { polyWork2[i] *= aa; polyWork1[i] *= aa; aa *= a; }
      fft(polyWork2 + m, m);  // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
      fft(polyWork1 + m, m);  // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
    }
  }
  // 13 E(n)
  // D log(t) = (D t) / t
  Poly log(int n) const {
    assert(!empty()); assert((*this)[0].x == 1U); assert(n <= LIM_INV);
    Poly fs = mod(n);
    for (int i = 0; i < fs.size(); ++i) fs[i] *= i;
    fs = fs.div(*this, n);
    for (int i = 1; i < n; ++i) fs[i] *= ::inv[i];
    return fs;
  }
  // (16 + 1/2) E(n)
  // f = exp(t) mod x^m  ==>  (D f) / f == D t  (mod x^m)
  // g = (1 / exp(t)) mod x^m
  // f <- f - (log f - t) / (1 / f)
  //   =  f - (I ((D f) / f) - t) f
  //   == f - (I ((D f) / f + (f g - 1) ((D f) / f - D (t mod x^m))) - t) f  (mod x^(2m))
  //   =  f - (I (g (D f - f D (t mod x^m)) + D (t mod x^m)) - t) f
  // g <- g - (f g - 1) g
  // polyWork1: DFT(f, 2 m), polyWork2: g, polyWork3: DFT(g, 2 m)
  Poly exp(int n) const {
    assert(!empty()); assert(!(*this)[0]); assert(1 <= n);
    assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= min(LIM_INV, LIM_POLY));
    if (n == 1) return {1U};
    if (n == 2) return {1U, at(1)};
    Poly fs(n);
    fs[0].x = polyWork1[0].x = polyWork1[1].x = polyWork2[0].x = 1U;
    int m;
    for (m = 1; m << 1 < n; m <<= 1) {
      for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i];
      memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint));
      fft(polyWork0, m);  // (1/2) E(n)
      for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i];
      invFft(polyWork0, m);  // (1/2) E(n)
      for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i];
      memset(polyWork0 + m, 0, m * sizeof(Mint));
      fft(polyWork0, m << 1);  // 1 E(n)
      memcpy(polyWork3, polyWork2, m * sizeof(Mint));
      memset(polyWork3 + m, 0, m * sizeof(Mint));
      fft(polyWork3, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i];
      for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i];
      memset(polyWork0 + m, 0, m * sizeof(Mint));
      fft(polyWork0, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      memcpy(fs.data() + m, polyWork0, m * sizeof(Mint));
      memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint));
      memset(polyWork1 + (m << 1), 0, (m << 1) * sizeof(Mint));
      fft(polyWork1, m << 2);  // 2 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      memset(polyWork0, 0, m * sizeof(Mint));
      fft(polyWork0, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i];
    }
    for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i];
    memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint));
    fft(polyWork0, m);  // (1/2) E(n)
    for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i];
    invFft(polyWork0, m);  // (1/2) E(n)
    for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i];
    memcpy(polyWork0 + m, polyWork0 + (m >> 1), (m >> 1) * sizeof(Mint));
    memset(polyWork0 + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    memset(polyWork0 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    fft(polyWork0, m);  // (1/2) E(n)
    fft(polyWork0 + m, m);  // (1/2) E(n)
    memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint));
    memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    fft(polyWork3 + m, m);  // (1/2) E(n)
    for (int i = 0; i < m; ++i) polyWork0[m + i] = polyWork0[i] * polyWork3[m + i] + polyWork0[m + i] * polyWork3[i];
    for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork3[i];
    invFft(polyWork0, m);  // (1/2) E(n)
    invFft(polyWork0 + m, m);  // (1/2) E(n)
    for (int i = 0; i < m >> 1; ++i) polyWork0[(m >> 1) + i] += polyWork0[m + i];
    for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i];
    for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i];
    memset(polyWork0 + m, 0, m * sizeof(Mint));
    fft(polyWork0, m << 1);  // 1 E(n)
    for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
    invFft(polyWork0, m << 1);  // 1 E(n)
    memcpy(fs.data() + m, polyWork0, (n - m) * sizeof(Mint));
    return fs;
  }
  // (29 + 1/2) E(n)
  // g <- g - (log g - a log t) g
  Poly pow(Mint a, int n) const {
    assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n);
    return (a * log(n)).exp(n);  // 13 E(n) + (16 + 1/2) E(n)
  }
  // (29 + 1/2) E(n - a ord(t))
  Poly pow(long long a, int n) const {
    assert(a >= 0); assert(1 <= n);
    if (a == 0) { Poly gs(n); gs[0].x = 1U; return gs; }
    const int o = ord();
    if (o == -1 || o > (n - 1) / a) return Poly(n);
    const Mint b = (*this)[o].inv(), c = (*this)[o].pow(a);
    const int ntt = min<int>(n - a * o, size() - o);
    Poly tts(ntt);
    for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i];
    tts = tts.pow(Mint(a), n - a * o);  // (29 + 1/2) E(n - a ord(t))
    Poly gs(n);
    for (int i = 0; i < n - a * o; ++i) gs[a * o + i] = c * tts[i];
    return gs;
  }
  // (10 + 1/2) E(n)
  // f = t^(1/2) mod x^m,  g = 1 / t^(1/2) mod x^m
  // f <- f - (f^2 - h) g / 2
  // g <- g - (f g - 1) g
  // polyWork1: DFT(f, m), polyWork2: g, polyWork3: DFT(g, 2 m)
  Poly sqrt(int n) const {
    assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n);
    assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY);
    if (n == 1) return {1U};
    if (n == 2) return {1U, at(1) / 2};
    Poly fs(n);
    fs[0].x = polyWork1[0].x = polyWork2[0].x = 1U;
    int m;
    for (m = 1; m << 1 < n; m <<= 1) {
      for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i];
      invFft(polyWork1, m);  // (1/2) E(n)
      for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i];
      for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i];
      memset(polyWork1 + m, 0, m * sizeof(Mint));
      fft(polyWork1, m << 1);  // 1 E(n)
      memcpy(polyWork3, polyWork2, m * sizeof(Mint));
      memset(polyWork3 + m, 0, m * sizeof(Mint));
      fft(polyWork3, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i];
      invFft(polyWork1, m << 1);  // 1 E(n)
      for (int i = 0; i < m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; }
      memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint));
      fft(polyWork1, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      memset(polyWork0, 0, m * sizeof(Mint));
      fft(polyWork0, m << 1);  // 1 E(n)
      for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
      invFft(polyWork0, m << 1);  // 1 E(n)
      for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i];
    }
    for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i];
    invFft(polyWork1, m);  // (1/2) E(n)
    for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i];
    for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i];
    memcpy(polyWork1 + m, polyWork1 + (m >> 1), (m >> 1) * sizeof(Mint));
    memset(polyWork1 + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    memset(polyWork1 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    fft(polyWork1, m);  // (1/2) E(n)
    fft(polyWork1 + m, m);  // (1/2) E(n)
    memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint));
    memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
    fft(polyWork3 + m, m);  // (1/2) E(n)
    // for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i];
    for (int i = 0; i < m; ++i) polyWork1[m + i] = polyWork1[i] * polyWork3[m + i] + polyWork1[m + i] * polyWork3[i];
    for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork3[i];
    invFft(polyWork1, m);  // (1/2) E(n)
    invFft(polyWork1 + m, m);  // (1/2) E(n)
    for (int i = 0; i < m >> 1; ++i) polyWork1[(m >> 1) + i] += polyWork1[m + i];
    for (int i = 0; i < n - m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; }
    return fs;
  }
  // (10 + 1/2) E(n)
  // modSqrt must return a quadratic residue if exists, or anything otherwise.
  // Return {} if *this does not have a square root.
  template <class F> Poly sqrt(int n, F modSqrt) const {
    assert(1 <= n);
    const int o = ord();
    if (o == -1) return Poly(n);
    if (o & 1) return {};
    const Mint c = modSqrt((*this)[o]);
    if (c * c != (*this)[o]) return {};
    if (o >> 1 >= n) return Poly(n);
    const Mint b = (*this)[o].inv();
    const int ntt = min(n - (o >> 1), size() - o);
    Poly tts(ntt);
    for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i];
    tts = tts.sqrt(n - (o >> 1));  // (10 + 1/2) E(n)
    Poly gs(n);
    for (int i = 0; i < n - (o >> 1); ++i) gs[(o >> 1) + i] = c * tts[i];
    return gs;
  }
  // 6 E(|t|)
  // x -> x + a
  Poly shift(const Mint &a) const {
    if (empty()) return {};
    const int n = size();
    int m = 1;
    for (; m < n; m <<= 1) {}
    for (int i = 0; i < n; ++i) polyWork0[i] = fac[i] * (*this)[i];
    memset(polyWork0 + n, 0, ((m << 1) - n) * sizeof(Mint));
    fft(polyWork0, m << 1);  // 2 E(|t|)
    {
      Mint aa = 1;
      for (int i = 0; i < n; ++i) { polyWork1[n - 1 - i] = invFac[i] * aa; aa *= a; }
    }
    memset(polyWork1 + n, 0, ((m << 1) - n) * sizeof(Mint));
    fft(polyWork1, m << 1);  // 2 E(|t|)
    for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
    invFft(polyWork0, m << 1);  // 2 E(|t|)
    Poly fs(n);
    for (int i = 0; i < n; ++i) fs[i] = invFac[i] * polyWork0[n - 1 + i];
    return fs;
  }
};

Mint linearRecurrenceAt(const vector<Mint> &as, const vector<Mint> &cs, long long k) {
  assert(!cs.empty()); assert(cs[0]);
  const int d = cs.size() - 1;
  assert(as.size() >= static_cast<size_t>(d));
  return (Poly(vector<Mint>(as.begin(), as.begin() + d)) * cs).mod(d).divAt(cs, k);
}

struct SubproductTree {
  int logN, n, nn;
  vector<Mint> xs;
  // [DFT_4((X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3]))] [(X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3])mod X^4]
  // [         DFT_4((X-xs[0])(X-xs[1]))         ] [         DFT_4((X-xs[2])(X-xs[3]))         ]
  // [   DFT_2(X-xs[0])   ] [   DFT_2(X-xs[1])   ] [   DFT_2(X-xs[2])   ] [   DFT_2(X-xs[3])   ]
  vector<Mint> buf;
  vector<Mint *> gss;
  // (1 - xs[0] X) ... (1 - xs[nn-1] X)
  Poly all;
  // (ceil(log_2 n) + O(1)) E(n)
  SubproductTree(const vector<Mint> &xs_) {
    n = xs_.size();
    for (logN = 0, nn = 1; nn < n; ++logN, nn <<= 1) {}
    xs.assign(nn, 0U);
    memcpy(xs.data(), xs_.data(), n * sizeof(Mint));
    buf.assign((logN + 1) * (nn << 1), 0U);
    gss.assign(nn << 1, nullptr);
    for (int h = 0; h <= logN; ++h) for (int u = 1 << h; u < 1 << (h + 1); ++u) {
      gss[u] = buf.data() + (h * (nn << 1) + ((u - (1 << h)) << (logN - h + 1)));
    }
    for (int i = 0; i < nn; ++i) {
      gss[nn + i][0] = -xs[i] + 1;
      gss[nn + i][1] = -xs[i] - 1;
    }
    if (nn == 1) gss[1][1] += 2;
    for (int h = logN; --h >= 0; ) {
      const int m = 1 << (logN - h);
      for (int u = 1 << (h + 1); --u >= 1 << h; ) {
        for (int i = 0; i < m; ++i) gss[u][i] = gss[u << 1][i] * gss[u << 1 | 1][i];
        memcpy(gss[u] + m, gss[u], m * sizeof(Mint));
        invFft(gss[u] + m, m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        if (h > 0) {
          gss[u][m] -= 2;
          const Mint a = FFT_ROOTS[logN - h + 1];
          Mint aa = 1;
          for (int i = m; i < m << 1; ++i) { gss[u][i] *= aa; aa *= a; };
          fft(gss[u] + m, m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        }
      }
    }
    all.resize(nn + 1);
    all[0] = 1;
    for (int i = 1; i < nn; ++i) all[i] = gss[1][nn + nn - i];
    all[nn] = gss[1][nn] - 1;
  }
  // ((3/2) ceil(log_2 n) + O(1)) E(n) + 10 E(|f|) + 3 E(|f| + 2^(ceil(log_2 n)))
  vector<Mint> multiEval(const Poly &fs) const {
    vector<Mint> work0(nn), work1(nn), work2(nn);
    {
      const int m = max(fs.size(), 1);
      auto invAll = all.inv(m);  // 10 E(|f|)
      std::reverse(invAll.begin(), invAll.end());
      int mm;
      for (mm = 1; mm < m - 1 + nn; mm <<= 1) {}
      invAll.resize(mm, 0U);
      fft(invAll);  // E(|f| + 2^(ceil(log_2 n)))
      vector<Mint> ffs(mm, 0U);
      memcpy(ffs.data(), fs.data(), fs.size() * sizeof(Mint));
      fft(ffs);  // E(|f| + 2^(ceil(log_2 n)))
      for (int i = 0; i < mm; ++i) ffs[i] *= invAll[i];
      invFft(ffs);  // E(|f| + 2^(ceil(log_2 n)))
      memcpy(((logN & 1) ? work1 : work0).data(), ffs.data() + m - 1, nn * sizeof(Mint));
    }
    for (int h = 0; h < logN; ++h) {
      const int m = 1 << (logN - h);
      for (int u = 1 << h; u < 1 << (h + 1); ++u) {
        Mint *hs = (((logN - h) & 1) ? work1 : work0).data() + ((u - (1 << h)) << (logN - h));
        Mint *hs0 = (((logN - h) & 1) ? work0 : work1).data() + ((u - (1 << h)) << (logN - h));
        Mint *hs1 = hs0 + (m >> 1);
        fft(hs, m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        for (int i = 0; i < m; ++i) work2[i] = gss[u << 1 | 1][i] * hs[i];
        invFft(work2.data(), m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        memcpy(hs0, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
        for (int i = 0; i < m; ++i) work2[i] = gss[u << 1][i] * hs[i];
        invFft(work2.data(), m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        memcpy(hs1, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
      }
    }
    work0.resize(n);
    return work0;
  }
  // ((5/2) ceil(log_2 n) + O(1)) E(n)
  Poly interpolate(const vector<Mint> &ys) const {
    assert(static_cast<int>(ys.size()) == n);
    Poly gs(n);
    for (int i = 0; i < n; ++i) gs[i] = (i + 1) * all[n - (i + 1)];
    const vector<Mint> denoms = multiEval(gs);  // ((3/2) ceil(log_2 n) + O(1)) E(n)
    vector<Mint> work(nn << 1, 0U);
    for (int i = 0; i < n; ++i) {
      // xs[0], ..., xs[n - 1] are not distinct
      assert(denoms[i]);
      work[i << 1] = work[i << 1 | 1] = ys[i] / denoms[i];
    }
    for (int h = logN; --h >= 0; ) {
      const int m = 1 << (logN - h);
      for (int u = 1 << (h + 1); --u >= 1 << h; ) {
        Mint *hs = work.data() + ((u - (1 << h)) << (logN - h + 1));
        for (int i = 0; i < m; ++i) hs[i] = gss[u << 1 | 1][i] * hs[i] + gss[u << 1][i] * hs[m + i];
        if (h > 0) {
          memcpy(hs + m, hs, m * sizeof(Mint));
          invFft(hs + m, m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
          const Mint a = FFT_ROOTS[logN - h + 1];
          Mint aa = 1;
          for (int i = m; i < m << 1; ++i) { hs[i] *= aa; aa *= a; };
          fft(hs + m, m);  // ((1/2) ceil(log_2 n) + O(1)) E(n)
        }
      }
    }
    invFft(work.data(), nn);  // E(n)
    return Poly(vector<Mint>(work.data() + nn - n, work.data() + nn));
  }
};
////////////////////////////////////////////////////////////////////////////////


vector<Mint> getSurjs(int S, int L) {
  vector<Mint> xs(L);
  for (int i = 0; i < L; ++i) {
    xs[i] = i + 1;
  }
  const SubproductTree st(xs);
  const auto res = st.all.inv(S - L + 1);
  vector<Mint> ret(S + 1, 0);
  for (int n = L; n <= S; ++n) {
    ret[n] = fac[L] * res[n - L];
  }
// cerr<<"getSurjs "<<S<<" "<<L<<" = "<<ret<<endl;
  return ret;
}

Mint solve(const vector<Mint> &A, int L) {
// cerr<<"solve "<<A<<" "<<L<<endl;
  const int S = A.size();
  const auto surj = getSurjs(S, L);
  vector<Mint> as(S);
  for (int i = 0; i < S; ++i) {
    as[i] = -A[i];
  }
  const SubproductTree st(as);
  Mint ans = 0;
  for (int k = 0; k <= S; ++k) {
    const Mint way = st.all[S - k];
// cerr<<k<<": "<<way<<endl;
    ans += way * surj[k];
  }
// cerr<<"solve "<<A<<" "<<L<<" = "<<ans<<endl;
  return ans;
}

int main() {
  int N;
  vector<int> A;
  for (; ~scanf("%d", &N); ) {
    A.resize(N);
    for (int i = 0; i < N; ++i) {
      scanf("%d", &A[i]);
    }
    sort(A.begin(), A.end(), greater<Int>());
    
    vector<int> B(N, 0);
    for (int i = 0; i < N; ++i) {
      ++B[A[i] - 1];
    }
    for (int j = N; --j > 0; ) {
      B[j - 1] += B[j];
    }
    
    int S = 0;
    for (int i = 0; i < N; ++i) if (i < A[i]) {
      ++S;
    }
// cerr<<"A = "<<A<<", B = "<<B<<", S = "<<S<<endl;
    
    Mint ans = 0;
    {
      int need = 0;
      for (int j = 0; j < S; ++j) if (B[j] > S) {
        ++need;
      }
      vector<Mint> cs(S);
      for (int i = 0; i < S; ++i) {
        cs[i] = A[i] - need;
      }
      ans += solve(cs, need);
    }
    {
      int need = 0;
      for (int i = 0; i < S; ++i) if (A[i] > S) {
        ++need;
      }
      vector<Mint> cs(S);
      for (int j = 0; j < S; ++j) {
        cs[j] = B[j] - need;
      }
      ans += solve(cs, need);
    }
    ans -= fac[S];
    printf("%d %u\n", S, ans.x);
  }
  return 0;
}