#define SINGLE_TESTCASE 

#define FAST_IO 

#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11

// https://github.com/miscalculation53/library/tree/wip/template/template_all.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_all_but_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_types.hpp

#include <bits/stdc++.h>
using namespace std;

using ld = decltype(EPS);

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;

#define vc vector
template <class T>
using vvc = vc<vc<T>>;
template <class T>
using vvvc = vc<vc<vc<T>>>;

using vb = vc<bool>;
using vl = vc<ll>;
using vpll = vc<pll>;
using vtlll = vc<tlll>;
using vtllll = vc<tllll>;
using vstr = vc<string>;
using vvb = vvc<bool>;
using vvl = vvc<ll>;

template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;
template <class T>
using pqg = priority_queue<T>;

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (int i = neg; i < (int)s.size(); i++)
    res = 10 * res + s[i] - '0';
  if (neg)
    return -i128(res - 1) - 1;
  return i128(res);
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  u128 ux;
  if (x < 0)
    ux = u128(-(x + 1)) + 1, sign = "-";
  else
    ux = x;
  while (ux > 0)
  {
    res += '0' + ux % 10;
    ux /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  is >> s;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}

#define cauto const auto
// https://github.com/miscalculation53/library/tree/wip/template/template_rep.hpp

#define overload4(_1,_2,_3,_4,name,...) name
#define rep1(i,n) for (ll i = 0, nnnnn = ll(n); i < nnnnn; i++)
#define rep2(i,l,r) for (ll i = ll(l), rrrrr = ll(r); i < rrrrr; i++)
#define rep3(i,l,r,d) for (ll i = ll(l), rrrrr = ll(r), ddddd = ll(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i,n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i,l,r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi3(i,l,r,d) for (int i = int(l), rrrrr = int(r), ddddd = int(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
// https://github.com/miscalculation53/library/tree/wip/template/template_math.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/is_integral_ext.hpp

template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;

template <class T>
constexpr bool is_signed_ext = is_signed_v<T> || is_same_v<T, i128>;

template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;
// https://github.com/miscalculation53/library/tree/wip/utils/default_infty.hpp

namespace default_infty_detail
{
  template <class T, class = void>
  struct has_infty : false_type {};

  template <class T>
  struct has_infty<T, void_t<decltype(T::infty())>> : true_type {};

  template <class T>
  const T &custom_value()
  {
    static const T value = T::infty();
    return value;
  }

  template <class T>
  inline constexpr bool unsupported = false;
}

template <class T>
constexpr decltype(auto) default_infty()
{
  if constexpr (default_infty_detail::has_infty<T>::value)
    return default_infty_detail::custom_value<T>();
  else if constexpr (is_same_v<T, i128> || is_same_v<T, u128>)
    return T(INF) * T(INF);
  else if constexpr (is_integral_ext<T> && !is_same_v<T, bool>)
  {
    if constexpr (sizeof(T) >= sizeof(ll))
      return T(INF);
    else if constexpr (numeric_limits<T>::digits >= 31)
      return (T(1) << 30) - 1;
    else
      return T(numeric_limits<T>::max() / 2);
  }
  else if constexpr (numeric_limits<T>::has_infinity)
    return numeric_limits<T>::infinity();
  else
    static_assert(default_infty_detail::unsupported<T>, "No default infinity for this type; specify infty explicitly or define T::infty().");
}

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divround(U a, V b) { return divfloor<T>(2 * T(a) + T(b), 2 * T(b)); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;

  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;
  if (k == 2)
  {
    const T aa = T(a);
    T x = T(sqrtl((long double)a));
    while (x > aa / x)
      x--;
    while (x < numeric_limits<T>::max())
    {
      const T y = x + 1;
      if (y > aa / y)
        break;
      x = y;
    }
    return x;
  }

  auto isok = [&](T x) -> bool
  {
    if (x == 0)
      return true;
    T res = 1, k2 = k;
    while (true)
    {
      if (k2 & 1)
      {
        if (res > T(a) / x)
          return false;
        res *= x;
      }
      k2 >>= 1;
      if (k2 == 0)
        break;
      if (x > T(a) / x)
        return false;
      x *= x;
    }
    return res <= T(a);
  };

  T x = pow(a, 1.0 / k);
  bool up = true;
  while (!isok(x))
    up = false, x--;
  if (up)
  {
    while (x < numeric_limits<T>::max() && isok(x + 1))
      x++;
  }
  return x;
}

// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back

#define LMD(x,fx) ([&](const auto &x) { return fx; })
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_infty.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_value.hpp

template <class T, auto x, enable_if_t<!is_invocable_v<decltype(x)>, int> = 0>
constexpr T resolved_value()
{
  return T(x);
}

template <class T, auto x, enable_if_t<is_invocable_v<decltype(x)>, long> = 0>
const T &resolved_value()
{
  static const T value = T(x());
  return value;
}

template <class T, auto x = nullptr>
constexpr decltype(auto) resolved_infty()
{
  if constexpr (is_same_v<decltype(x), nullptr_t>)
    return default_infty<T>();
  else
    return resolved_value<T, x>();
}

template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}

template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};
template <class T, auto infty = nullptr>
struct MonoidMin
{
  using S = T;
  static constexpr decltype(auto) e() { return resolved_infty<T, infty>(); }
};
template <class T, auto infty = nullptr>
struct MonoidMax
{
  using S = T;
  static constexpr S e() { return -resolved_infty<T, infty>(); }
};

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

constexpr array<pll, 4> DRULgrid = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
constexpr array<pll, 4> DRULplane = {{{0, -1}, {1, 0}, {0, 1}, {-1, 0}}};
// https://github.com/miscalculation53/library/tree/wip/template/template_binsearch.hpp

template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;

#define DEFAULT_COMP ranges::less

namespace internal
{
};

// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp

template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }

inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }

inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }

inline constexpr ull lsb_pos(ull x) { assert(x != 0); return countr_zero(x); }
inline constexpr ull msb_pos(ull x) { assert(x != 0); return bit_width(x) - 1; }
inline constexpr ull lsb_mask(ull x) { assert(x != 0); return x & -x; }
inline constexpr ull msb_mask(ull x) { assert(x != 0); return bit_floor(x); }

inline constexpr bool btest(ull x, uint k) { return (x >> k) & 1; }
inline constexpr bool bsubset(ull x, ull y) { return (x & y) == x; }
inline constexpr bool bsupset(ull x, ull y) { return (x & y) == y; }
inline constexpr ull bsetminus(ull x, ull y) { return x & ~y; }
// https://github.com/miscalculation53/library/tree/wip/template/template_inout.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump_map.hpp

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) 
#define local(...) 
#define oj(...) __VA_ARGS__
#define local_oj(a,b) (b)

namespace fastio {
template <class T>
struct unsigned_integer
{
  using type = make_unsigned_t<T>;
};
template <>
struct unsigned_integer<i128>
{
  using type = u128;
};
template <>
struct unsigned_integer<u128>
{
  using type = u128;
};
template <class T>
using unsigned_integer_t = typename unsigned_integer<T>::type;

static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];

uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd1(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (c <= ' ');
}

void rd1(string &x) {
  x.clear();
  while (true) {
    if (pil == pir) load();
    while (pil < pir && ibuf[pil] <= ' ') ++pil;
    if (pil < pir) break;
  }
  while (true) {
    uint32_t p = pil;
    while (pil < pir && ibuf[pil] > ' ') ++pil;
    x.append(ibuf + p, pil - p);
    if (pil < pir) {
      ++pil;
      return;
    }
    load();
  }
}

template <typename T>
void rd1_real(T &x) {
  string s;
  rd1(s);

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = from_chars(s.data(), s.data() + s.size(), x);
    if (ec == errc{} && p == s.data() + s.size()) return;
  }

  if constexpr (is_same_v<T, long double>)
    x = stold(s);
  else
    x = stod(s);
}

template <bool check_buffer = true, typename T>
void rd1_integer(T &x) {
  using U = unsigned_integer_t<T>;
  bool minus = false;
  U val = 0;
  if constexpr (check_buffer)
    if (pil + 100 > pir) load();
  uint32_t p = pil;
  while (ibuf[p] < '-') ++p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (ibuf[p] == '-') minus = true, ++p;
  }
  while ('0' <= ibuf[p]) val = val * 10 + (ibuf[p++] & 15);
  pil = p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus)
    {
      const U min_abs = U(numeric_limits<T>::max()) + 1;
      x = val == min_abs ? numeric_limits<T>::lowest() : -T(val);
    }
    else x = T(val);
  }
  else
    x = T(val);
}

void rd1(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }

template <class... T>
void read(T &...x) {
  if constexpr (sizeof...(T) <= SIZ / 100 &&
                ((!is_same_v<T, char> &&
                  (is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>)) && ...)) {
    if (pil + 100 * sizeof...(T) > pir) load();
    (rd1_integer<false>(x), ...);
  }
  else
    (rd1(x), ...);
}

void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(string_view s) {
  while (!s.empty()) {
    if (por == SIZ) flush();
    size_t n = min<size_t>(s.size(), SIZ - por);
    memcpy(obuf + por, s.data(), n);
    por += n;
    s.remove_prefix(n);
  }
}

template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  using U = unsigned_integer_t<T>;
  U ux;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (x < 0)
      obuf[por++] = '-', ux = U(0) - U(x);
    else
      ux = U(x);
  }
  else
    ux = x;
  int outi;
  for (outi = 96; ux >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[ux % 10000], 4);
    ux /= 10000;
  }
  if (ux >= 1000) {
    memcpy(obuf + por, pre.num[ux], 4);
    por += 4;
  } else if (ux >= 100) {
    memcpy(obuf + por, pre.num[ux] + 1, 3);
    por += 3;
  } else if (ux >= 10) {
    int q = (ux * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (ux - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = ux | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt1_real(T x) {

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = to_chars(out, out + sizeof(out), x, chars_format::fixed, 15);
    if (ec == errc{}) {
      wt1(string_view(out, p));
      return;
    }
  }

  ostringstream oss;
  oss << fixed << setprecision(15) << x;
  wt1(oss.str());
}

void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }

template <class... T>
void write(T &&...x) {
  (wt1(std::forward<T>(x)), ...);
}

template <class... T>
void print(T &&...x) {
  if constexpr (sizeof...(T))
  {
    int i = 0;
    ((i++ ? wt1(' ') : void(), wt1(std::forward<T>(x))), ...);
  }
  wt1('\n');
}

} 

struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;

namespace internal
{

template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }

}; 

#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define IN(T,...) T __VA_ARGS__; READ(__VA_ARGS__)

#define CHAR(...) IN(char, __VA_ARGS__)
#define INT(...) IN(int, __VA_ARGS__)
#define LL(...) IN(ll, __VA_ARGS__)
#define STR(...) IN(string, __VA_ARGS__)
#define ARR(T,n,...) array<T, n> __VA_ARGS__; READ(__VA_ARGS__)

#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define READVEC2(...) internal::READVEC2nodump(__VA_ARGS__); dump(__VA_ARGS__)

#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define VEC2(T,n,m,...) vvc<T> __VA_ARGS__; READVEC2(n, m, __VA_ARGS__)

#define READJAG(...) internal::READJAGnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define JAG(T,n,...) vvc<T> __VA_ARGS__; READJAG(n, __VA_ARGS__)

#define ENDL '\n'

#define WRITE fastio::write
#define PRINT fastio::print

#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)

namespace internal
{

}; 

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = zip(tuple{__VA_ARGS__})

// https://github.com/miscalculation53/library/tree/wip/template/template_random.hpp

mt19937_64 mt;

template <class T = ll, class U1, class U2>
T randrange(U1 l, U2 r)
{
  assert(T(l) < T(r));
  return uniform_int_distribution<T>(T(l), T(r) - 1)(mt);
}

bool randbool(double p)
{
  assert(0 <= p && p <= 1);
  return bernoulli_distribution(p)(mt);
}

namespace internal
{
}; 

// https://github.com/miscalculation53/library/tree/wip/math/modint/template_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_static.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/larger_int.hpp

namespace larger_int_detail
{
}

template <class T>
struct larger_int
{
private:
  static constexpr bool check();
  static_assert(check());

public:
  using type = T;
};

#define LARGER_INT(T,U) template <> struct larger_int<T> { using type = U; };

LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)

LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)

#undef LARGER_INT

template <class T>
struct Rational;

template <class T>
struct larger_int<Rational<T>>
{
  using type = Rational<typename larger_int<T>::type>;
};

template <class T>
using larger_int_t = typename larger_int<T>::type;
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_isprime.hpp

namespace internal
{

template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}

template <auto n>
constexpr bool isprime = isprime_constexpr(n);

};

namespace internal
{

template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = larger_int_t<value_type>;

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr mod_type mod() { return M; }
  static constexpr value_type umod() { return M; }

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v) { return v; }
  static constexpr value_type mul(value_type a, value_type b)
  {
    return (value_type)((calc_type(a) * b) % M);
  }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_barrett32.hpp

namespace internal
{

struct barrett32
{
  uint m;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static void set_mod(mod_type m) { reducer = barrett32(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v) { return v; }
  static value_type mul(value_type a, value_type b) { return reducer.mul(a, b); }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_montgomery64.hpp

namespace internal
{

inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}

struct montgomery64odd
{
  ull m, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};

template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static void set_mod(mod_type m) { reducer = montgomery64odd(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{(1LL << 61) - 1};
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp

namespace internal
{

  template <class Policy>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;

  private:
    V _v;

  public:
    static constexpr M mod() { return Policy::mod(); }

    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m)) { return T::set_mod(m); }

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(Policy::umod());
        if (x < 0)
          x += Policy::umod();
        rem = x;
      }
      else
        rem = V(v % Policy::umod());
      _v = Policy::init(rem);
    };

    M val() const { return Policy::val(_v); }

    mint &operator+=(const mint &rhs)
    {
      _v += rhs._v;
      if (_v >= Policy::umod())
        _v -= Policy::umod();
      return *this;
    }
    mint &operator-=(const mint &rhs)
    {
      _v -= rhs._v;
      if (_v >= Policy::umod())
        _v += Policy::umod();
      return *this;
    }
    mint &operator*=(const mint &rhs)
    {
      _v = Policy::mul(_v, rhs._v);
      return *this;
    }

    template <class T>
    mint pow(T n) const
    {
      assert(n >= 0);
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }

    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; }
    friend M safe_hash_key(const mint &x) { return x.val(); }
  
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;
using modint1000000007 = static_modint32<1000000007>;
using modint = dynamic_modint32<-1>;
using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64 = dynamic_modint64<-1>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;

template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;

template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};

template <class mint>
struct modint_less
{
  bool operator()(const mint &a, const mint &b) const
  {
    if constexpr (is_modint_v<mint>)
      return a.val() < b.val();
    else
      return a < b;
  }
};

template <class mint>
struct modint_hash
{
  auto operator()(const mint &x) const
  {
    if constexpr (is_modint_v<mint>)
      return std::hash<decltype(x.val())>{}(x.val());
    else
      return std::hash<mint>{}(x);
  }
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

template <class mint>
struct PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;

public:

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp

template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
  
public:
  inline static vc<T> fac_, finv_, inv_;

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp

// https://github.com/miscalculation53/library/tree/wip/math/svp2d.hpp

template <class T = ll, class U = larger_int_t<T>>
pair<T, T> svp2d(const pair<T, T> &a, const pair<T, T> &b)
{
  assert((a != pair<T, T>{0, 0} && b != pair<T, T>{0, 0}));
  auto [a1, a2] = a;
  auto [b1, b2] = b;
  if ((U)a1 * a1 + (U)a2 * a2 < (U)b1 * b1 + (U)b2 * b2)
    swap(a1, b1), swap(a2, b2);
  while ((U)a1 * a1 + (U)a2 * a2 > (U)b1 * b1 + (U)b2 * b2)
  {
    swap(a1, b1), swap(a2, b2);
    T k = divround<U>((U)a1 * b1 + (U)a2 * b2, (U)a1 * a1 + (U)a2 * a2);
    b1 -= k * a1, b2 -= k * a2;
    if (b1 == 0 && b2 == 0)
      return {a1, a2};
  }
  return {a1, a2};
}

template <class mint>
pair<decltype(mint(0).val()), decltype(mint(0).val())>
mint_to_rat(const mint &x)
{
  auto [p, q] = svp2d({x.val(), 1}, {mint::mod(), 0});
  if (q < 0)
    p = -p, q = -q;
  return {p, q};
}

namespace cpp_dump
{
  template <class T>
  struct rat_value
  {
    T p, q;

    friend ostream &operator<<(ostream &os, const rat_value &x)
    {
      return os << x.p << '/' << x.q;
    }
  };

  struct mint_to_rat_fn
  {
    template <class T>
      requires requires(const T &x)
      {
        { T::mod() } -> std::convertible_to<ll>;
        { x.val() } -> std::convertible_to<ll>;
      }
    constexpr auto operator()(const T &x) const
        -> rat_value<typename decltype(mint_to_rat(x))::first_type>
    {
      auto [p, q] = mint_to_rat(x);
      return {p, q};
    }
  };

  template <class T>
  struct rat_object
  {
    T value;
  };

  template <class T>
  inline constexpr bool rat_is_map = false;
  template <class... Args>
  inline constexpr bool rat_is_map<std::map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::multimap<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_multimap<Args...>> = true;

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <class T>
    static constexpr bool can_convert()
    {
      if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;
        if constexpr (std::same_as<std::remove_cvref_t<T>, std::remove_cvref_t<Ref>>)
          return false;
        else
          return can_convert<Ref>();
      }
      else if constexpr (requires { std::tuple_size<std::remove_cvref_t<T>>::value; })
      {
        return []<size_t... I>(std::index_sequence<I...>)
        {
          if constexpr ((requires { std::get<I>(std::declval<T>()); } && ...))
            return (can_convert<decltype(std::get<I>(std::declval<T>()))>() || ...);
          else
            return false;
        }(std::make_index_sequence<std::tuple_size_v<std::remove_cvref_t<T>>>{});
      }
      else
        return std::invocable<mint_to_rat_fn, T>;
    }

    template <typename T>
      requires(!requires { std::views::all | std::declval<T>(); })
    constexpr decltype(auto) operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (can_convert<T>())
      {
        if constexpr (!std::ranges::range<T> || rat_is_map<std::remove_cvref_t<T>>)
          return rat_object<T>{std::forward<T>(t)};
        else
          return std::forward<T>(t) | std::views::transform(*this);
      }
      else
      {
        return std::forward<T>(t);
      }
    }
  };

  template <typename T>
    requires(!requires { std::views::all | std::declval<T>(); })

  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}

using mint = modint998244353;

using bi = Binomial<mint>;

void init()
{
  oj(mt.seed(random_device()()));
}

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/euler_product_prefix_sum.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/dirichlet_prefix_sum.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/dirichlet_convolution.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/enumerate_multiplicative.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_basic_ops.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_base.hpp

template <class S_, auto op_, auto e_>
struct Monoid
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
};

template <class S_, auto op_, auto e_, auto inv_>
struct Group
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
  static constexpr auto inv = inv_;
};

template <class S_, auto add_, auto e0_, auto mul_, auto e1_>
struct SemiRing
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_>
struct Ring
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_, auto inv_>
struct Field
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
  static constexpr auto inv = inv_;
};

template <class M>
struct OppositeMonoid
{
  using S = typename M::S;
  static constexpr auto e = M::e;
};
template <class G>
struct OppositeGroup
{
  using S = typename G::S;
  static constexpr auto e = G::e;
  static constexpr auto inv = G::inv;
};
template <class M>
struct NormalAndOppositeMonoid
{
  struct S
  {
    typename M::S normal;
    typename M::S opposite;
    S() {}
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename M::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args)
        : normal(std::forward<Args>(args)...), opposite(normal) {}
    S(const typename M::S &normal, const typename M::S &opposite) : normal(normal), opposite(opposite) {}
  };
  static constexpr S e() { return {M::e(), M::e()}; }
};
template <class G>
struct NormalAndOppositeGroup
{
  struct S
  {
    typename G::S normal;
    typename G::S opposite;
    S() {}
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename G::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args)
        : normal(std::forward<Args>(args)...), opposite(normal) {}
    S(const typename G::S &normal, const typename G::S &opposite) : normal(normal), opposite(opposite) {}
  };
  static constexpr S e() { return {G::e(), G::e()}; }
};

template <class SR>
using MonoidOfSemiRingAdd = Monoid<typename SR::S, SR::add, SR::e0>;
template <class SR>
using MonoidOfSemiRingMul = Monoid<typename SR::S, SR::mul, SR::e1>;
template <class R>
using GroupOfRingAdd = Group<typename R::S, R::add, R::e0, R::minus>;
template <class K>
using GroupOfFieldMul = Group<typename K::S, K::mul, K::e1, K::inv>;

template <class Madd, class Mmul>
struct SemiRingFromMonoidMonoid
{
  static_assert(is_same_v<typename Madd::S, typename Mmul::S>, "Madd::S and Mmul::S must be identical");
  using S = typename Madd::S;
  static constexpr auto add = Madd::op;
  static constexpr auto e0 = Madd::e;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

template <class Gadd, class Mmul>
struct RingFromGroupMonoid
{
  static_assert(is_same_v<typename Gadd::S, typename Mmul::S>, "Gadd::S and Mmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

template <class Gadd, class Gmul>
struct FieldFromGroupGroup
{
  static_assert(is_same_v<typename Gadd::S, typename Gmul::S>, "Gadd::S and Gmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Gmul::op;
  static constexpr auto e1 = Gmul::e;
  static constexpr auto inv = Gmul::inv;
};

template <class T, class = void>
struct has_e1 : false_type {};
template <class T>
struct has_e1<T, void_t<decltype(T::e1())>> : true_type {};
template <class T>
inline constexpr bool has_e1_v = has_e1<T>::value;

template <class T>
struct MonoidMul
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e()
  {
    if constexpr (has_e1_v<S>)
      return S::e1();
    else
      return 1;
  }
};

template <class T>
struct GroupAddSub
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return S{};
  }
  static constexpr S inv(S a) { return -a; }
  template <class I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k) { return a * k; }
};
template <class T>
struct GroupMulDiv
{
  using S = T;
  static constexpr S e()
  {
    if constexpr (has_e1_v<S>)
      return S::e1();
    else
      return S(1);
  }
};

template <class T, auto infty = nullptr>
using SemiRingMinPlus = SemiRingFromMonoidMonoid<MonoidMin<T, infty>, MonoidAdd<T>>;
template <class T, auto infty = nullptr>
using SemiRingMaxPlus = SemiRingFromMonoidMonoid<MonoidMax<T, infty>, MonoidAdd<T>>;
template <class T>
using RingAddSubMul = RingFromGroupMonoid<GroupAddSub<T>, MonoidMul<T>>;
template <class T>
using FieldAddSubMulDiv = FieldFromGroupGroup<GroupAddSub<T>, GroupMulDiv<T>>;
// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/linear_sieve.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/prime_power.hpp

template <class P>
struct PrimePower
{
  P p;
  int e;
  P pe;

  PrimePower() : p(-1), e(-1), pe(-1) {}
  PrimePower(P p, int e = 1) : p(p), e(e), pe(ipow(p, e)) {}
  PrimePower(P p, int e, P pe) : p(p), e(e), pe(pe) {}

};

tuple<int, ll, ll> ord_pow_div(ll n, ll m)
{
  assert(m >= 2);
  if (m == 2)
  {
    int e = countr_zero(n);
    return {e, 1LL << e, n >> e};
  }
  if (n % m != 0)
    return {0, 1, n};
  n /= m;
  if (n % m != 0)
    return {1, m, n};
  n /= m;
  ll m2 = m * m;
  auto [f, m2f, nn] = ord_pow_div(n, m2);
  int e = 2 + 2 * f;
  ll me = m2f * m2;
  if (nn % m == 0)
    e++, me *= m, nn /= m;
  return {e, me, nn};
}

template <class P>
vc<PrimePower<P>> factorized_mul
(const vc<PrimePower<P>> &fac1, const vc<PrimePower<P>> &fac2)
{
  const int n = fac1.size(), m = fac2.size();
  vc<PrimePower<P>> fac;
  fac.reserve(n + m);
  int i = 0, j = 0;
  while (i < n && j < m)
  {
    if (fac1[i].p < fac2[j].p)
      fac.emplace_back(fac1[i++]);
    else if (fac1[i].p > fac2[j].p)
      fac.emplace_back(fac2[j++]);
    else
    {
      using U = larger_int_t<P>;
      fac.emplace_back(fac1[i].p, fac1[i].e + fac2[j].e,
                       U(fac1[i].pe) * U(fac2[j].pe));
      i++, j++;
    }
  }
  fac.insert(fac.end(), fac1.begin() + i, fac1.end());
  fac.insert(fac.end(), fac2.begin() + j, fac2.end());
  return fac;
}

struct LinearSieve
{
public:
  static int n;
  static vc<PrimePower<int>> lpf_;
  static vc<int> primes;

  static void reserve(int n_)
  {
    if (n_ <= n)
      return;
    n = max(n_, 2 * n);
    lpf_.resize(n + 1);
    for (int d = 2; d <= n; d++)
    {
      if (lpf_[d].p == -1)
      {
        lpf_[d] = PrimePower<int>(d, 1, d);
        primes.eb(d);
      }
      fec(p : primes)
      {
        if (p > n / d || p > lpf_[d].p)
          break;
        if (lpf_[d].p == p)
          lpf_[p * d] = PrimePower<int>(p, lpf_[d].e + 1, lpf_[d].pe * p);
        else
          lpf_[p * d] = PrimePower<int>(p, 1, p);
      }
    }
  }

  template <class P = int>
  static PrimePower<P> lpf(int n)
  {
    assert(n >= 1);
    reserve(n);
    return lpf_[n];
  }

  static bool is_prime(int n)
  {
    if (n <= 1)
      return false;
    return lpf(n).p == n;
  }

  static int Omega(int n)
  {
    assert(1 <= n);
    static vc<unsigned char> table{0, 0};
    if (n >= int(table.size()))
    {
      reserve(n);
      const int first = int(table.size());
      table.resize(n + 1);
      for (int i = first; i <= n; i++)
        table[i] = table[i / lpf_[i].p] + 1;
    }
    return table[n];
  }

};
vc<PrimePower<int>> LinearSieve::lpf_{};
int LinearSieve::n{};
vc<int> LinearSieve::primes{};

namespace internal
{
  
  template <class M>
  typename M::S multiplicative_monoid_power(typename M::S a, ll k)
  {
    assert(k >= 0);
    if constexpr (HasMonoidPow<M, ll>::value)
      return M::pow(a, k);
    else
    {
      auto res = M::e();
      while (k > 0)
      {
        if (k & 1)
          res = M::op(res, a);
        k >>= 1;
        if (k > 0)
          a = M::op(a, a);
      }
      return res;
    }
  }

  template <class G>
  typename G::S multiplicative_integer_multiple(typename G::S a, ll n)
  {
    if (n < 0)
    {
      a = G::inv(a);
      return G::op(multiplicative_monoid_power<G>(a, -(n + 1)), a);
    }
    return multiplicative_monoid_power<G>(a, n);
  }

  template <class R>
  struct multiplicative_integer_embedding
  {
    static typename R::S get(ll n)
    { return multiplicative_integer_multiple<GroupOfRingAdd<R>>(R::e1(), n); }
  };
  
  template <class G, class M>
  struct multiplicative_integer_embedding<RingFromGroupMonoid<G, M>>
  {
    static typename G::S get(ll n)
    { return multiplicative_integer_multiple<G>(M::e(), n); }
  };
  template <class G, class H>
  struct multiplicative_integer_embedding<FieldFromGroupGroup<G, H>>
  {
    static typename G::S get(ll n)
    { return multiplicative_integer_multiple<G>(H::e(), n); }
  };
  template <class R>
  typename R::S multiplicative_from_integer(ll n)
  { return multiplicative_integer_embedding<R>::get(n); }

}

inline constexpr auto e_primepower = [](const auto &q, auto ring)
{ return q.e == 0 ? decltype(ring)::e1() : decltype(ring)::e0(); };

inline constexpr auto zeta_primepower = [](const auto &, auto ring)
{ return decltype(ring)::e1(); };

inline constexpr auto id_primepower = [](const auto &q, auto ring)
{ return internal::multiplicative_from_integer<decltype(ring)>(q.pe); };

inline constexpr auto pow_primepower = [](ll k)
{
  assert(k >= 0);
  return [k](const auto &q, auto ring)
  {
    using R = decltype(ring);
    return internal::multiplicative_monoid_power<MonoidOfSemiRingMul<R>>(
      internal::multiplicative_from_integer<R>(q.pe), k);
  };
};

inline constexpr auto pow_inv_primepower = [](ll k)
{
  assert(k >= 0);
  return [k](const auto &q, auto ring)
  {
    using R = decltype(ring);
    using S = typename R::S;
    if (k == 0 || q.e == 0)
      return R::e1();
    if (S::mod() % q.p == 0)
      return R::e0();
    return internal::multiplicative_monoid_power<MonoidOfSemiRingMul<R>>(
      R::inv(internal::multiplicative_from_integer<R>(q.pe)), k);
  };
};

inline constexpr auto mobius_primepower = [](const auto &q, auto ring)
{
  using R = decltype(ring);
  return q.e == 0 ? R::e1() : q.e == 1 ? R::minus(R::e1()) : R::e0();
};

inline constexpr auto divisor_count_primepower = [](const auto &q, auto ring)
{ return internal::multiplicative_from_integer<decltype(ring)>(ll(q.e) + 1); };

inline constexpr auto divisor_sum_primepower = [](const auto &q, auto ring)
{
  using R = decltype(ring);
  return R::add(internal::multiplicative_from_integer<R>(q.pe),
                internal::multiplicative_from_integer<R>((q.pe - 1) / (q.p - 1)));
};

inline constexpr auto divisor_k_primepower = [](ll k)
{
  assert(k >= 0);
  return [k](const auto &q, auto ring)
  {
    using R = decltype(ring);
    const auto pk = internal::multiplicative_monoid_power<MonoidOfSemiRingMul<R>>(
      internal::multiplicative_from_integer<R>(q.p), k);
    auto res = R::e1();
    for (int i = 0; i < q.e; i++)
      res = R::add(R::mul(res, pk), R::e1());
    return res;
  };
};

inline constexpr auto totient_primepower = [](const auto &q, auto ring)
{ return internal::multiplicative_from_integer<decltype(ring)>(q.pe - q.pe / q.p); };
// https://github.com/miscalculation53/library/tree/wip/math/modint/inv_many.hpp

namespace internal
{
  template <class R, class = void>
  struct dirichlet_has_inv : false_type {};
  template <class R>
  struct dirichlet_has_inv<R, void_t<decltype(R::inv(declval<const typename R::S &>()))>> : true_type {};

  template <class R>
  struct dirichlet_modint_field : false_type {};
  template <class S>
  struct dirichlet_modint_field<FieldAddSubMulDiv<S>> : is_modint<S> {};

  template <class R>
  struct dirichlet_divisor
  {
    using S = typename R::S;
    static constexpr bool exact_integer = is_integral_ext<S> && is_same_v<R, RingAddSubMul<S>>;
    S value;
    bool identity;
    S operator()(const S &a) const
    {
      if (identity)
        return a;
      if constexpr (dirichlet_has_inv<R>::value)
        return R::mul(a, value);
      else if constexpr (exact_integer)
        return a / value; 
      else
        return R::minus(a); 
    }
  };
}

template <class R>
struct DirichletSeries
{
  using S = typename R::S;

private:
  int n_;
  vc<S> f_;
  bool multiplicative_;

public:

  int n() const { return n_; }

  friend DirichletSeries operator*(const S &a, const DirichletSeries &f) { return f * a; }

private:
  
public:

};

inline constexpr auto e_prefix_sum = [](ll n, auto ring)
{ return n == 0 ? decltype(ring)::e0() : decltype(ring)::e1(); };
inline constexpr auto zeta_prefix_sum = [](ll n, auto ring)
{ return internal::multiplicative_from_integer<decltype(ring)>(n); };
inline constexpr auto id_prefix_sum = [](ll n, auto ring)
{
  using R = decltype(ring);
  const auto a = internal::multiplicative_from_integer<R>(n % 2 == 0 ? n / 2 : n);
  const auto b = n % 2 == 0
    ? R::add(internal::multiplicative_from_integer<R>(n), R::e1())
    : internal::multiplicative_from_integer<R>(n / 2 + 1);
  return R::mul(a, b);
};

namespace internal
{
  
  template <int D>
  struct dirichlet_root_prefix_sum_index
  {
    static_assert(D >= 1);
    ll n, maximum;
    int k, large;
    vc<ll> large_value;
    vc<int> raw_index;

  };

  template <>
  struct dirichlet_root_prefix_sum_index<1>
  {
    ll n;
    int k, large;
    explicit dirichlet_root_prefix_sum_index(ll n) : n(n)
    {
      assert(n >= 0);
      const ll root = iroot(n, 2), l = n / (root + 1);
      k = int(root), large = int(l);
    }
    int size() const { return k + large; }
    int index(ll v) const { return v <= k ? int(v) - 1 : size() - int(n / v); }
    ll value(int i) const { return i < k ? ll(i) + 1 : n / (size() - i); }
  };
  using dirichlet_prefix_sum_index = dirichlet_root_prefix_sum_index<1>;

  template <class R, class GetF>
  typename R::S eval_dirichlet_prefix(const GetF &getF, ll n)
  {
    if constexpr (is_invocable_v<const GetF &, ll>)
      return getF(n);
    else
      return getF(n, R{});
  }
}

template <class R, int D = 1>
struct DirichletPrefixSum
{
  using S = typename R::S;

private:
  ll n_;
  int k_, l_;
  vc<S> small_, large_;
  bool multiplicative_;
  internal::dirichlet_root_prefix_sum_index<D> coordinates_;

  struct same_shape {};
  
  int large_index(ll x) const
  {
    if constexpr (D == 1) return int(n_ / x);
    else return coordinates_.raw_index[size_t(n_ / ipow<ll>(x, D))];
  }

public:
  DirichletPrefixSum() : DirichletPrefixSum(0) {}

  explicit DirichletPrefixSum(ll n)
    : n_(n), multiplicative_(false), coordinates_(n)
  {
    k_ = coordinates_.k, l_ = coordinates_.large;
    small_.assign(k_ + 1, R::e0());
    large_.assign(l_ + 1, R::e0());
  }

  template <class GetF, enable_if_t<is_invocable_v<const GetF &, ll>
    || is_invocable_v<const GetF &, ll, R>, int> = 0>
  DirichletPrefixSum(ll n, const GetF &getF, bool multiplicative = false)
    : DirichletPrefixSum(n)
  {
    for (int x = 1; x <= k_; x++)
      small_[x] = internal::eval_dirichlet_prefix<R>(getF, x);
    for (int i = 1; i <= l_; i++)
      large_[i] = internal::eval_dirichlet_prefix<R>(getF, value(size() - i));
    multiplicative_ = multiplicative;
    assert(!multiplicative || n == 0 || small_[1] == R::e1());
  }

  ll n() const { return n_; }
  int size() const { return k_ + l_; }
  ll value(int i) const
  {
    assert(0 <= i && i < size());
    if constexpr (D == 1) return i < k_ ? ll(i) + 1 : n_ / (size() - i);
    else return i < k_ ? ll(i) + 1 : coordinates_.large_value[size() - i];
  }
  
  bool contains(ll x) const
  {
    if constexpr (D == 1)
      return 0 <= x && x <= n_ && (x <= k_ || n_ / (n_ / x) == x);
    else
      return 0 <= x && x <= coordinates_.maximum
        && (x <= k_ || coordinates_.large_value[large_index(x)] == x);
  }
  const S &F(ll x) const
  {
    assert(contains(x));
    return x <= k_ ? small_[size_t(x)] : large_[large_index(x)];
  }
  void setF(ll x, const S &value)
  {
    assert(x >= 1 && contains(x));
    (x <= k_ ? small_[size_t(x)] : large_[large_index(x)]) = value;
    multiplicative_ = false;
  }

private:

public:
  friend DirichletPrefixSum operator*(const S &a, const DirichletPrefixSum &f) { return f * a; }

private:
  
public:

};

template <class R, int D = 1>
typename R::S dirichlet_convolution_sum(const DirichletPrefixSum<R, D> &a, const DirichletPrefixSum<R, D> &b)
{
  assert(a.n() == b.n());
  const ll n = iroot(a.n(), D);
  const int m = int(iroot(n, 2));
  auto res = R::minus(R::mul(a.F(m), b.F(m)));
  for (int i = 1; i <= m; i++)
  {
    const auto ai = R::add(a.F(i), R::minus(a.F(i - 1)));
    const auto bi = R::add(b.F(i), R::minus(b.F(i - 1)));
    res = R::add(res, R::add(R::mul(ai, b.F(n / i)), R::mul(bi, a.F(n / i))));
  }
  return res;
}
// https://github.com/miscalculation53/library/tree/wip/ds/fenwick_tree/fenwick_tree.hpp

template <class G>
struct FenwickTree
{
  using S = typename G::S;

private:
  int n;
  vc<S> dat;

public:
  FenwickTree(int n) : n(n), dat(n + 1, G::e()) {}
  FenwickTree(const vc<S> &v) : FenwickTree(v.size())
  {
    repi(i, n) dat[i + 1] = v[i];
    repi(i, 1, n + 1)
    {
      int p = i + (i & -i);
      if (p <= n)
        dat[p] = G::op(dat[p], dat[i]);
    }
  }

  S sum(int r) const
  {
    assert(0 <= r && r <= n);
    S s = G::e();
    while (r > 0)
    {
      s = G::op(s, dat[r]);
      r -= r & -r;
    }
    return s;
  }
  
  S sum(int l, int r) const
  {
    assert(0 <= l && l <= r && r <= n);
    return G::op(G::inv(sum(l)), sum(r));
  }
  
  S get(int i) const
  {
    assert(0 <= i && i < n);
    return sum(i, i + 1);
  }

  void add(int i, S x)
  {
    assert(0 <= i && i < n);
    i++;
    while (i <= n)
    {
      dat[i] = G::op(dat[i], x);
      i += i & -i;
    }
  }
  
};

template <class R>
DirichletPrefixSum<R> euler_product_prefix_sum(const DirichletPrefixSum<R> &seed,
  const vvc<pair<ll, typename R::S>> &factors)
{
  using S = typename R::S;
  const ll n = seed.n();
  if (n == 0) return {};
  const internal::dirichlet_prefix_sum_index coordinates(n);
  const int k = coordinates.k;
  const int size = coordinates.size();
  vc<ll> x(size + 1, 0);
  vc<S> blocks(size, R::e0()), change(blocks);
  
  ll first = ll(k) + 1;
  for (int i = 0; i < size; i++)
  {
    x[i + 1] = coordinates.value(i);
    blocks[i] = R::add(seed.F(x[i + 1]), R::minus(seed.F(x[i])));
    if (1 <= i && i < k && blocks[i] != R::e0()) first = min(first, ll(i) + 1);
  }
  FenwickTree<GroupOfRingAdd<R>> bit(blocks);
  vc<int> order(factors.size());
  iota(order.begin(), order.end(), 0);
  vc<ll> minimum(factors.size(), LLONG_MAX);
  for (size_t i = 0; i < factors.size(); i++)
    for (const auto &[d, a] : factors[i])
    {
      assert(d >= 1);
      if (d >= 2 && d <= n && a != R::e0()) minimum[i] = min(minimum[i], d);
    }
  sort(order.begin(), order.end(), [&](int i, int j) { return minimum[i] > minimum[j]; });
  vc<int> touched;
  vc<bool> used(size, false);
  auto add_change = [&](int i, const S &a)
  {
    if (a == R::e0()) return;
    if (!used[i]) used[i] = true, touched.push_back(i);
    change[i] = R::add(change[i], a);
  };
  for (int fi : order)
  {
    for (const auto &[d, a] : factors[fi])
    {
      if (d > n || a == R::e0()) continue;
      add_change(coordinates.index(d), R::mul(a, blocks[0]));
      const ll limit = n / d;
      const ll b = min(limit, max(first - 1, ll(iroot(limit, 2))));
      
      for (ll v = first; v <= b; v++)
        if (blocks[v - 1] != R::e0())
          add_change(coordinates.index(v * d), R::mul(a, blocks[v - 1]));
      
      for (int i = size - 1; i >= 0 && x[i + 1] / d > b; i--)
      {
        const ll lo = max(b, x[i] / d), hi = x[i + 1] / d;
        const S value = bit.sum(coordinates.index(lo) + 1, coordinates.index(hi) + 1);
        add_change(i, R::mul(a, value));
      }
    }
    
    for (int i : touched)
    {
      if (change[i] != R::e0())
      {
        blocks[i] = R::add(blocks[i], change[i]);
        bit.add(i, change[i]);
      }
      change[i] = R::e0(); used[i] = false;
    }
    touched.clear();
    first = min(first, minimum[fi]);
  }
  DirichletPrefixSum<R> result(n);
  S prefix = R::e0();
  for (int i = 0; i < size; i++)
  {
    prefix = R::add(prefix, blocks[i]);
    result.setF(x[i + 1], prefix);
  }
  return result;
}

// https://github.com/miscalculation53/library/tree/wip/math/prime/sieve/multiplicative_prefix_sum.hpp

namespace internal
{

  template <class R>
  struct multiplicative_sieve_updates
  {
    ll n, bound;
    int root, large, top;
    dirichlet_prefix_sum_index coordinates;
    FenwickTree<GroupOfRingAdd<R>> bit;

  };

  inline constexpr ll multiplicative_prefix_sum_fenwick_threshold = 1'000'000;
} 

// https://github.com/miscalculation53/library/tree/wip/math/prime/large/factorize.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/large/primality_test.hpp

namespace internal
{

template <class mint, class Array>
bool is_prime_impl(ll n, const Array &bases)
{
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = (n - 1) >> countr_zero(n - 1);
  mint::set_mod(n);
  for (ll a : bases)
  {
    ll t = d;
    mint y = mint(a).pow(t);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y *= y;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}

}; 

bool is_prime(ll n)
{
  static constexpr array<ll, 3> bases32 = {2, 7, 61};
  static constexpr array<ll, 7> bases64 = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint32<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases32);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases64);
  }
}

namespace internal
{

template <class mint>
ll get_prime_factor_impl(ll n)
{
  mint::set_mod(n);
  int m = pow(n, .125);
  while (true)
  {
    int c = randrange(1, 100);
    mint x = 2, y = 2, prod = 1;
    ll g = 1;
    while (g == 1)
    {
      repi(i, m)
      {
        x = x * x + c;
        y = y * y + c, y = y * y + c;
        prod *= x - y;
      }
      g = gcd(prod.val(), n);
    }
    if (g == n)
      continue;
    if (is_prime(g))
      return g;
    else if (is_prime(n / g))
      return n / g;
    else
      return get_prime_factor_impl<mint>(g);
  }
}

ll get_prime_factor(ll n)
{
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint32<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
}

}; 

vc<PrimePower<ll>> factorize(ll n)
{
  vc<PrimePower<ll>> res;
  repi(p, 2, 100)
  {
    if (n % p == 0)
    {
      auto [e, pe, nn] = ord_pow_div(n, p);
      res.emplace_back(PrimePower<ll>(p, e, pe));
      n = nn;
    }
  }
  while (n > 1)
  {
    if (is_prime(n))
    {
      res.emplace_back(n);
      break;
    }
    ll p = internal::get_prime_factor(n);
    auto [e, pe, nn] = ord_pow_div(n, p);
    res.emplace_back(PrimePower<ll>(p, e, pe));
    n = nn;
  }
  sort(ALL(res), [&](const PrimePower<ll> &pp1, const PrimePower<ll> &pp2)
       { return pp1.p < pp2.p; });
  return res;
}

void main2()
{
  LL(N);

  const ll K = sqrtl(N) + 1;
  dump(K);

  DirichletPrefixSum<RingAddSubMul<ull>> zeta(N, zeta_prefix_sum);
  DirichletPrefixSum<RingAddSubMul<ull>> seed(N, LMD(n, n < K ? 1 : 1 + is_prime(n + 1)));
  dump(seed);
  vvc<pair<ll, ull>> factors;
  LinearSieve::reserve(K + 1);
  fe(p : LinearSieve::primes) if (p <= K)
  {
    vc<pair<ll, ull>> factor = {{p - 1, 1}, {p, -1}};
    factors.eb(factor);
  }
  dump(factors);
  auto res = euler_product_prefix_sum(seed, factors);
  ull ans = dirichlet_convolution_sum(res, zeta);
  PRINT(ans);
}

void test()
{

}

// https://github.com/miscalculation53/library/tree/wip/template/template_main.hpp

template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      
    };
  
    CERR("\n[FAST_IO]\n\n", "32");
    
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}