#define SINGLE_TESTCASE
#define FAST_CIO
#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11
#include <bits/stdc++.h>
using namespace std;
namespace {
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>>;
using vpll = vc<pll>;
using vstr = vc<string>;
#ifdef __SIZEOF_INT128__
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;
}
#endif
#define cauto const auto
#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__)
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>;
template <class T>
constexpr T monoid_default_infty()
{
  if constexpr (numeric_limits<T>::is_specialized && numeric_limits<T>::is_integer)
    return numeric_limits<T>::max() / T(2);
  else
    return T(INF);
}
template <class T>
constexpr T monoid_default_identity()
{
  if constexpr (numeric_limits<T>::is_specialized && numeric_limits<T>::is_integer)
    return numeric_limits<T>::max();
  else
    return T(INF) + T(1);
}
template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T, class U>
inline bool chmax(T &a, U b) { return a < b ? a = b, true : false; }
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 divceil(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 B, class M>
T mul_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (b == 0)
    return 0;
  return T(a) > T(m) / T(b) ? T(m) : T(a) * T(b);
}
template <class T = ll, class A, class B>
T mul_limited(A a, B b) { return mul_limited<T>(a, b, INF); }
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (a <= 1 || b == 0)
    return min(ipow<T>(a, b), T(m));
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
    {
      if (res > T(m) / tmp)
        return m;
      res *= tmp;
    }
    b >>= 1;
    if (b == 0)
      break;
    if (tmp > T(m) / tmp)
      return m;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B>
T pow_limited(A a, B b) { return pow_limited<T>(a, b, INF); }
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;
}
template <class T = ll, class A, class K>
constexpr T iroot_ceil(A a, K k)
{
  T x = iroot<T>(a, k);
  return ipow<T>(x, k) == a ? x : x + 1;
}
template <class D = decltype(EPS), class A>
int SGN(A a, D eps = EPS) { return int(a > eps) - int(a < -eps); }
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base)
{
  assert(val >= 0);
  assert(base >= 2);
  if (val == 0)
    return {0};
  vc<T> a;
  while (val > 0)
  {
    a.emplace_back(val % base);
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base, int n)
{
  assert(val >= 0);
  assert(base >= 2);
  assert(n >= 0);
  vc<T> a(n);
  repi(i, n)
  {
    a[i] = val % base;
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  auto a = base_repr(val, base);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base, int n)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  assert(n >= 0);
  auto a = base_repr(val, base, n);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
#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) ([&](auto x) { return fx; })
template <class F>
auto gen_vec(int n, const F &f)
{
  vc<decltype(f(0))> res(n);
  repi(i, n) res[i] = f(i);
  return res;
}
#define GEN_VEC(n, i, fi) (gen_vec(n, LMD(i, fi)))
template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}
template <class T = ll>
T ctol(const char &c, const string &s)
{
  repi(i, SZ<int>(s)) if (s[i] == c) return i;
  return -1;
}
template <class T = ll>
vc<T> stov(const string &s, char first)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return s[i] - first; });
}
template <class T = ll>
vc<T> stov(const string &s, const string &t)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return ctol(s[i], t); });
}
template <class T>
string vtos(const vc<T> &v, char first)
{
  string res = "";
  fe(vi : v) res += vi + first;
  return res;
}
template <class T>
string vtos(const vc<T> &v, const string &t)
{
  string res = "";
  fe(vi : v) res += t[vi];
  return res;
}
template <class T>
vc<T> concat(const vvc<T> &vs)
{
  vc<T> res;
  for (cauto &v : vs)
    res.insert(res.end(), ALL(v));
  return res;
}
template <class T>
vc<T> concat(const vc<T> &v) { return v; }
template <class T, class... Ts>
vc<T> concat(vc<T> v, const vc<Ts> &...vs)
{
  (v.insert(v.end(), ALL(vs)), ...);
  return v;
}
template <class T>
vc<T> merged(const vc<T> &a, const vc<T> &b)
{
  vc<T> res;
  merge(ALL(a), ALL(b), back_inserter(res));
  return res;
}
template <class T, class I>
T vecget(const vc<T> &v, I i, const T &dflt_negative = -monoid_default_infty<T>(), const T &dflt_positive = monoid_default_infty<T>())
{
  if (i < 0)
    return dflt_negative;
  if (i >= SZ<int>(v))
    return dflt_positive;
  return v[i];
}
template <class V>
auto SUM(const V &v)
{
  typename V::value_type s{};
  fec(vi : v) s += vi;
  return s;
}
template <class T, class V>
T SUM(const V &v)
{
  T s{};
  fec(vi : v) s += vi;
  return s;
}
template <class V>
auto MAX(const V &v) { return *max_element(ALL(v)); }
template <class V>
auto MIN(const V &v) { return *min_element(ALL(v)); }
template <class I = ll, class V>
I ARGMAX(const V &v) { return max_element(ALL(v)) - v.begin(); }
template <class I = ll, class V>
I ARGMIN(const V &v) { return min_element(ALL(v)) - v.begin(); }
template<class T = ll, class V>
T mex(const V &a)
{
  int n = a.size();
  vector<bool> exists(n, false);
  repi(i, n) if (0 <= a[i] && a[i] < n) exists[a[i]] = true;
  repi(x, n) if (!exists[x]) return x;
  return n;
}
template <class I>
bool is_permutation(const vc<I> &p)
{
  const int n = p.size();
  vc<bool> b(n, false);
  repi(i, n)
  {
    if (!(0 <= p[i] && p[i] < n))
      return false;
    b[p[i]] = true;
  }
  return all_of(ALL(b), [](bool bi)
                { return bi; });
}
template <class T = ll>
vc<T> permid(const int &n, const int &base_index = 0)
{
  vc<T> p(n);
  repi(i, n) p[i] = i + base_index;
  return p;
}
template <class T>
vc<T> perminv(const vc<T> &p)
{
  if (p.empty())
    return {};
  const int n = p.size();
  vc<T> q(MAX(p) + 1);
  repi(i, n) if (p[i] >= 0) q[p[i]] = i;
  return q;
}
template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p)
{
  const int n = p.size();
  vc<T> res(n);
  repi(i, n)
  {
    assert(0 <= p[i] && p[i] < U(a.size()));
    res[i] = a[p[i]];
  }
  return res;
}
template <class T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs)
{
  return permuted(permuted(p, q), rs...);
}
template <class V>
V reversed(const V &v) { return V(v.rbegin(), v.rend()); }
#if __cplusplus < 202002L
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  sort(ALL(v), forward<Args>(args)...);
  return v;
}
#else
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  ranges::sort(v, forward<Args>(args)...);
  return v;
}
#endif
template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V>
V uniqued(V v) { unique(v); return v; }
template <class V>
void sortunique(V &v)
{
  sort(ALL(v));
  unique(v);
}
template <class V>
V sortuniqued(V v) { sortunique(v); return v; }
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  if (n == 0)
    return;
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
template <class V, class U>
V rotated(V v, U k) { rotate(v, k); return v; }
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 op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};
template <class T, const T infty = monoid_default_infty<T>()>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr S e() { return infty; }
};
template <class T, const T infty = monoid_default_infty<T>()>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b) { return max(a, b); }
  static constexpr S e() { return -infty; }
};
template <class M>
typename M::S pow_monoid(typename M::S a, ll k)
{
  typename M::S c = M::e();
  for (; k; k >>= 1)
  {
    if (k & 1)
      c = M::op(c, a);
    a = M::op(a, a);
  }
  return c;
}
template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0)
{
  const int n = v.size();
  vc<typename M::S> res(n + 1);
  res[0] = M::e();
  repi(i, n) res[i + 1] = M::op(res[i], v[i]);
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}
template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0)
{ return reversed(cuml<M>(reversed(v), right_index)); }
template <class T>
vc<T> cumlsum(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidAdd<T>>(v, left_index); }
template <class T>
vc<T> cumrsum(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidAdd<T>>(v, right_index); }
template <class T>
vc<T> cumlmin(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMin<T>>(v, left_index); }
template <class T>
vc<T> cumrmin(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMin<T>>(v, right_index); }
template <class T>
vc<T> cumlmax(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMax<T>>(v, left_index); }
template <class T>
vc<T> cumrmax(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMax<T>>(v, right_index); }
template <class T>
vc<T> adjd(const vc<T> &v, int left_index = 0, int right_index = 0)
{
  int n = v.size();
  assert(0 <= left_index && 0 <= right_index && left_index + right_index <= n + 1);
  vc<T> res(n + 1);
  if (n == 0)
  {
    res[0] = T{};
    res.erase(res.end() - right_index, res.end());
    res.erase(res.begin(), res.begin() + left_index);
    return res;
  }
  res[0] = v[0];
  repi(i, 1, n) res[i] = v[i] - v[i - 1];
  res[n] = -v[n - 1];
  res.erase(res.end() - right_index, res.end());
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}
const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};
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;
#if __cplusplus < 202002L
struct identity
{
  template <class T>
  constexpr T &&operator()(T &&t) const noexcept
  { return forward<T>(t); }
};
namespace internal
{
  template <class T = ll, class V, class Judge>
  inline T bound_helper(const V &v, Judge judge)
  {
    int l = -1, r = v.size();
    while (r - l > 1)
    {
      int m = (l + r) / 2;
      if (judge(m))
        l = m;
      else
        r = m;
    }
    return r;
  }
};
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return comp(proj(*(v.begin() + i)), val); });
}
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return !comp(val, proj(*(v.begin() + i))); });
}
#define DEFAULT_COMP less<>
#else
template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::lower_bound(v, val, comp, proj) - v.begin(); }
template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::upper_bound(v, val, comp, proj) - v.begin(); }
#define DEFAULT_COMP ranges::less
#endif
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj) - 1; }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj) - 1; }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj); }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj); }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj); }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj); }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - UB<T>(v, val, comp, proj); }
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - LB<T>(v, val, comp, proj); }
template <class T = ll, class V, class L, class R, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto in_cnt(const V &v, L l, R r, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{
  if (l > r)
    return 0;
  return lt_cnt<T>(v, r, comp, proj) - lt_cnt<T>(v, l, comp, proj);
}
template <class V, class Value>
inline auto lt_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.lower_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}
template <class V, class Value>
inline auto leq_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.upper_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}
template <class V, class Value>
inline auto gt_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.upper_bound(val); }
template <class V, class Value>
inline auto geq_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.lower_bound(val); }
namespace internal
{
template <class T>
bool binsearch_adjacent(T a, T b)
{
  if (a < b)
    return a + 1 == b;
  if (b < a)
    return b + 1 == a;
  return false;
}
};
template <class T = ll, class Judge, class InitOk, class InitNg>
pair<T, T> binsearch(const Judge &judge, InitOk init_ok, InitNg init_ng, bool check_ok = true, bool check_ng = true)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  while (!internal::binsearch_adjacent(ok, ng))
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}
template <class T = ld, class Judge, class InitOk, class InitNg>
T binsearch_real(const Judge &judge, InitOk init_ok, InitNg init_ng, int iteration_count = 100, bool check_ok = true, bool check_ng = true)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  repi(_, iteration_count)
  {
    T mid = (ok + ng) / 2;
    (judge(mid) ? ok : ng) = mid;
  }
  return ok;
}
template <class T = ll, class Judge, class InitVal>
pair<T, T> expsearch(const Judge &judge, InitVal init_val, bool positive = true)
{
  T cur(init_val), step = 1;
  const bool cur_ok = judge(cur);
  while (true)
  {
    T nxt;
    if (positive)
      nxt = cur > numeric_limits<T>::max() - step ? numeric_limits<T>::max() : cur + step;
    else
      nxt = cur < numeric_limits<T>::lowest() + step ? numeric_limits<T>::lowest() : cur - step;
    assert(nxt != cur && "the boundary must exist in the searched direction");
    if (nxt == cur || judge(nxt) != cur_ok)
    {
      T ok = cur_ok ? cur : nxt;
      T ng = cur_ok ? nxt : cur;
      return binsearch<T>(judge, ok, ng, false, false);
    }
    cur = nxt;
    if (step > numeric_limits<T>::max() / 2)
      step = numeric_limits<T>::max();
    else
      step *= 2;
  }
}
template <class T>
inline constexpr ull pow2(T k) { return 1ULL << k; }
template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }
#if __cplusplus < 202002L
inline constexpr ull bit_width(ull x) { return x == 0 ? 0 : 64 - __builtin_clzll(x); }
inline constexpr ull bit_floor(ull x) { return x == 0 ? 0ULL : 1ULL << (bit_width(x) - 1); }
inline constexpr ull bit_ceil(ull x) { return x == 0 ? 1ULL : 1ULL << bit_width(x - 1); }
inline constexpr ull countr_zero(ull x) { assert(x != 0); return __builtin_ctzll(x); }
inline constexpr ull popcount(ull x) { return __builtin_popcountll(x); }
inline constexpr bool has_single_bit(ull x) { return popcount(x) == 1; }
#else
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); }
#endif
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; }
template <class T>
inline void bset(T &x, uint k, bool b = 1) { b ? x |= (1ULL << k) : x &= ~(1ULL << k); }
template <class T>
inline void bflip(T &x, uint k) { x ^= (1ULL << k); }
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; }
#define dump(...)
#define local(...)
#define oj(...) __VA_ARGS__
#define local_oj(a, b) (b)
template <class T, class Sequence>
vc<T> content(queue<T, Sequence> que)
{
  vc<T> res;
  while (!que.empty())
  {
    res.eb(que.front());
    que.pop();
  }
  return res;
}
template <class T, class Sequence, class Compare>
vc<T> content(priority_queue<T, Sequence, Compare> pque)
{
  vc<T> res;
  while (!pque.empty())
  {
    res.eb(pque.top());
    pque.pop();
  }
  return res;
}
template <class T>
auto content(const T &obj) { return obj.content(); }
template <class T, class U>
istream &operator>>(istream &is, pair<T, U> &p)
{
  is >> p.first >> p.second;
  return is;
}
template <class... Ts>
istream &operator>>(istream &is, tuple<Ts...> &t)
{
  apply([&](auto &...a)
        { (is >> ... >> a); }, t);
  return is;
}
template <class T, size_t n>
istream &operator>>(istream &is, array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
template <class T>
istream &operator>>(istream &is, vc<T> &a)
{
  const size_t n = a.size();
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
namespace internal
{
template <class... Ts>
void CIN(Ts &...a) { (cin >> ... >> a); }
template <class... Ts>
void READnodump(Ts &...a) { CIN(a...); }
template <class T>
void READVECnodump(int n, vc<T> &v)
{
  v.resize(n);
  READnodump(v);
}
template <class T, class... Ts>
void READVECnodump(int n, vc<T> &v, vc<Ts> &...vs)
{ READVECnodump(n, v), READVECnodump(n, vs...); }
template <class T>
void READVEC2nodump(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  READnodump(v);
}
template <class T, class... Ts>
void READVEC2nodump(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ READVEC2nodump(n, m, v), READVEC2nodump(n, m, vs...); }
template <class T>
void READJAGnodump(int n, vvc<T> &v)
{
  v.resize(n);
  repi(i, n)
  {
    int k;
    READnodump(k);
    READVECnodump(k, v[i]);
  }
}
template <class T, class... Ts>
void READJAGnodump(int n, vvc<T> &v, vvc<Ts> &...vs)
{ READJAGnodump(n, v), READJAGnodump(n, vs...); }
}; // namespace internal
#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'
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p)
{
  os << p.first << ' ' << p.second;
  return os;
}
namespace internal
{
template <size_t N = 0, typename T>
void cout_tuple(ostream &os, const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { os << ' '; }
    const auto x = std::get<N>(t);
    os << x;
    cout_tuple<N + 1>(os, t);
  }
}
}; // namespace internal
template <class... Ts>
ostream &operator<<(ostream &os, const tuple<Ts...> &t)
{
  internal::cout_tuple(os, t);
  return os;
}
template <class T, size_t n>
ostream &operator<<(ostream &os, const array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << a[i];
  }
  return os;
}
template <class T>
ostream &operator<<(ostream &os, const vc<T> &v)
{
  const size_t n = v.size();
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << v[i];
  }
  return os;
}
namespace internal
{
template <class T>
void COUTW() {}
template <class... Ts>
void COUTW(const Ts &...a) { (cout << ... << a); }
inline void COUTP() { cout << ENDL; }
template <class T>
void COUTP(const T &a) { cout << a << ENDL; }
template <class T, class... Ts>
void COUTP(const T &a, const Ts &...b)
{
  cout << a;
  (cout << ... << (cout << ' ', b));
  cout << ENDL;
}
}; // namespace internal
#define WRITE internal::COUTW
#define PRINT internal::COUTP
#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)
template <class T>
void PRINTV(const vc<T> &v) { for (auto &vi : v) PRINT(vi); }
#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)
template <class T, class U, class P>
pair<T, U> &operator+=(pair<T, U> &a, const P &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> a, const P &b) { return a += b; }
template <class T, size_t n, class A>
array<T, n> &operator+=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> a, const A &b) { return a += b; }
namespace internal
{
template <size_t... I, class A, class B>
auto &tuple_add_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}
}; // namespace internal
template <class... Ts, class Tp>
tuple<Ts...> &operator+=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> a, const Tp &b) { return a += b; }
template <class T, class Add>
void offset(vc<T> &v, const Add &add) { for (auto &vi : v) vi += add; }
template <class T, class Add>
void offset(vvc<T> &v, const Add &add) { for (auto &vi : v) for (auto &vij : vi) vij += add; }
template <class T, const size_t m>
array<vc<T>, m> unzip(const vc<array<T, m>> &vt)
{
  const size_t n = vt.size();
  array<vc<T>, m> tv;
  tv.fill(vc<T>(n));
  for (size_t i = 0; i < n; i++)
    for (size_t j = 0; j < m; j++)
      tv[j][i] = vt[i][j];
  return tv;
}
template <class T, const size_t m>
vc<array<T, m>> zip(const array<vc<T>, m> &tv)
{
  if (tv.empty()) return {};
  const size_t n = tv[0].size();
  vc<array<T, m>> vt(n);
  for (size_t j = 0; j < m; j++)
  {
    assert(tv[j].size() == n);
    for (size_t i = 0; i < n; i++)
      vt[i][j] = tv[j][i];
  }
  return vt;
}
template <class T, class U>
pair<vc<T>, vc<U>> unzip(const vc<pair<T, U>> &vt)
{
  const size_t n = vt.size();
  pair<vc<T>, vc<U>> tv;
  tv.first.resize(n), tv.second.resize(n);
  for (size_t i = 0; i < n; i++)
    tie(tv.first[i], tv.second[i]) = vt[i];
  return tv;
}
template <class T, class U>
vc<pair<T, U>> zip(const pair<vc<T>, vc<U>> &tv)
{
  const size_t n = tv.first.size();
  assert(n == tv.second.size());
  vc<pair<T, U>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = make_pair(tv.first[i], tv.second[i]);
  return vt;
}
namespace internal
{
template <size_t... I, class V, class Tp>
auto vt_to_tv_impl(V &tv, const Tp &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }
template <size_t... I, class Tp>
auto tv_to_vt_impl(const Tp &tv, index_sequence<I...>, size_t index)
{ return make_tuple(get<I>(tv)[index]...); }
};
template <class... Ts>
auto unzip(const vc<tuple<Ts...>> &vt)
{
  const size_t n = vt.size();
  tuple<vc<Ts>...> tv;
  apply([&](auto &...v)
        { ((v.resize(n)), ...); }, tv);
  for (size_t i = 0; i < n; i++)
    internal::vt_to_tv_impl(tv, vt[i], make_index_sequence<tuple_size_v<decltype(tv)>>{}, i);
  return tv;
}
template <class... Ts>
auto zip(const tuple<vc<Ts>...> &tv)
{
  size_t n = get<0>(tv).size();
  apply([&](auto &...v)
        { ((void(v), assert(v.size() == n)), ...); }, tv);
  vc<tuple<Ts...>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = internal::tv_to_vt_impl(tv, index_sequence_for<Ts...>{}, i);
  return vt;
}
#define UNZIP(vt, ...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt, ...) auto vt = zip(tuple{__VA_ARGS__})
mt19937_64 mt;
template <class T = ll, class U1, class U2>
T randint(U1 l, U2 r)
{
  assert(T(l) <= T(r));
  return T(l) + mt() % (T(r) - T(l) + 1);
}
template <class T = ll, class U1, class U2>
T randrange(U1 l, U2 r)
{
  assert(T(l) < T(r));
  return T(l) + mt() % (T(r) - T(l));
}
template <int k, bool does_sort, class T = ll, class U1, class U2>
array<T, k> random_sample_range_array(U1 l, U2 r)
{
  assert(T(r) - T(l) >= T(k));
  array<T, k> res;
  repi(i, k) res[i] = randint<T>(T(l), T(r) - T(k));
  sort(ALL(res));
  repi(i, k) res[i] += i;
  if (!does_sort)
    shuffle(ALL(res), mt);
  return res;
}
template <bool does_sort, class T = ll, class U1, class U2>
vc<T> random_sample_range_vector(U1 l, U2 r, int k)
{
  assert(T(r) - T(l) >= T(k));
  vc<T> res(k);
  repi(i, k) res[i] = randint<T>(T(l), T(r) - T(k));
  sort(ALL(res));
  repi(i, k) res[i] += i;
  if (!does_sort)
    shuffle(ALL(res), mt);
  return res;
}
template <class T>
struct larger_int
{
  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>
using larger_int_t = typename larger_int<T>::type;
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 <class T>
constexpr bool isprime_constexpr(T n)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_constexpr<int>(n);
  }
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;
  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };
  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}
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);
  }
};
};
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); }
};
};
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 void set_mod(mod_type m) { reducer = montgomery64(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 <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    T q = z1 / z2;
    tie(x1, x2) = make_pair(x2, x1 - q * x2);
    tie(y1, y2) = make_pair(y2, y1 - q * y2);
    tie(z1, z2) = make_pair(z2, z1 - q * z2);
  }
  if (z1 < 0)
    x1 = -x1, y1 = -y1, z1 = -z1;
  return {z1, x1, y1};
}
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); }
    static mint raw(V v)
    {
      mint x;
      x._v = v;
      return x;
    }
    modint_impl() : _v(0) {}
    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;
    }
    mint &operator/=(const mint &rhs)
    {
      return *this *= rhs.inv();
    }
    mint &operator++()
    {
      _v++;
      if (_v == Policy::umod())
        _v = 0;
      return *this;
    }
    mint &operator--()
    {
      if (_v == 0)
        _v = Policy::umod();
      _v--;
      return *this;
    }
    mint operator++(int)
    {
      mint res = *this;
      ++(*this);
      return res;
    }
    mint operator--(int)
    {
      mint res = *this;
      --(*this);
      return res;
    }
    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *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;
    }
    mint inv() const
    {
      if constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(x);
      }
    }
    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 std::istream &operator>>(std::istream &is, mint &x)
    {
      long long a;
      is >> a;
      x = a;
      return is;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint &x)
    {
      os << x.val();
      return os;
    }
  };
};
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>;
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 PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;
public:
  PowerTable() {}
  PowerTable(const mint &base) : mod(mint::mod()), base(base), pw(1, 1) {}
  void reserve(int n)
  {
    if (mod != mint::mod())
    {
      mod = mint::mod();
      pw = {1};
    }
    int i = pw.size();
    if (n < i)
      return;
    pw.resize(n + 1);
    for (; i <= n; i++)
      pw[i] = pw[i - 1] * base;
  }
  mint pow(int n)
  {
    reserve(n);
    return pw[n];
  }
};
template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
public:
  inline static vc<T> fac_, finv_, inv_;
  static void reserve(int n)
  {
    if constexpr (is_dynamic_modint_v<T>)
    {
      if (mod != T::mod())
      {
        mod = T::mod();
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
      }
    }
    else
    {
      if (fac_.empty())
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    if (n < SZ(fac_))
      return;
    chmin(n, T::mod() - 1);
    int si = fac_.size();
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    repi(i, si, n + 1)
    {
      fac_[i] = fac_[i - 1] * T::raw(i);
      inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  static T fac(int n)
  {
    assert(n >= 0);
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n];
  }
  static T finv(int n)
  {
    assert(n < T::mod());
    if (n < 0)
      return 0;
    reserve(n);
    return finv_[n];
  }
  static T inv(int n)
  {
    n %= T::mod();
    if (n < 0)
      n += T::mod();
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }
  static T P(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n] * finv_[n - k];
  }
  static T C(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n] * finv_[k] * finv_[n - k];
  }
  static T H(int n, int k)
  {
    if (n == 0 && k == 0)
      return 1;
    return C(n + k - 1, k);
  }
};
template <class mint>
mint stom(string s)
{
  mint res = 0;
  fec(c : s)
  {
    res *= 10;
    res += c - '0';
  }
  return res;
}
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};
}
#if __cplusplus >= 202302L
namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };
  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;
        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };
  template <typename T>
    requires(!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
  constexpr auto operator|(T &&t, const rat_closure &c)
  {
    return c(std::forward<T>(t));
  }
  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}
#endif
using mint = modint998244353;
using bi = Binomial<mint>;
void init()
{
  oj(mt.seed(random_device()()));
}
template <class T = ll, class R0, class R1, class M0, class M1>
constexpr tuple<bool, T, T> crt2(R0 r0_, R1 r1_, M0 m0_, M1 m1_)
{
  T m0 = m0_, m1 = m1_;
  assert(m0 >= 1 && m1 >= 1);
  T r0 = safemod(r0_, m0), r1 = safemod(r1_, m1);
  if (m0 < m1)
    swap(r0, r1), swap(m0, m1);
  if (m0 % m1 == 0)
  {
    if (r0 % m1 != r1)
      return {false, 0, 0};
    return {true, r0, m0};
  }
  auto [g, im, _] = extgcd<T>(m0, m1);
  T u1 = m1 / g;
  if ((r1 - r0) % g)
    return {false, 0, 0};
  T x = (r1 - r0) / g % u1 * im % u1;
  r0 += x * m0;
  m0 *= u1;
  if (r0 < 0)
    r0 += m0;
  return {true, r0, m0};
}
template <class T = ll, class V1, class V2>
constexpr tuple<bool, T, T> crt(const V1 &rs, const V2 &ms)
{
  assert(rs.size() == ms.size());
  const int n = rs.size();
  T r = 0, m = 1;
  repi(i, n)
  {
    auto [ok, nr, nm] = crt2<T>(r, rs[i], m, ms[i]);
    if (!ok)
      return {false, 0, 0};
    r = nr, m = nm;
  }
  return {true, r, m};
}
template <class mint, class V1, class V2>
pair<mint, mint> crt_mod(const V1 &rs, const V2 &ms)
{
  using T = decay_t<decltype(ms[0])>;
  assert(rs.size() == ms.size());
  const int n = rs.size();
  mint r = 0, m = 1;
  if constexpr (sizeof(T) <= 4)
  {
    vc<internal::barrett32> ba;
    ba.reserve(n);
    repi(i, n) ba.eb(ms[i]);
    vc<uint> rr(n, 0), mm(n, 1);
    repi(i, n)
    {
      assert(ms[i] >= 1);
      auto [g, im, _] = extgcd<ll>(mm[i], ms[i]);
      assert(g == 1);
      if (im < 0)
        im += ms[i];
      ll diff = safemod(ll(rs[i]) - rr[i], ms[i]);
      uint t = ba[i].mul(diff, im);
      r += t * m, m *= ms[i];
      repi(j, i + 1, n)
      {
        rr[j] += ba[j].mul(t, mm[j]);
        if (rr[j] >= (uint)ms[j])
          rr[j] -= ms[j];
        mm[j] = ba[j].mul(mm[j], ms[i]);
      }
    }
  }
  else
  {
    vc<ull> rr(n, 0), mm(n, 1);
    repi(i, n)
    {
      assert(ms[i] >= 1);
      auto [g, im, _] = extgcd<ll>(mm[i], ms[i]);
      assert(g == 1);
      if (im < 0)
        im += ms[i];
      i128 diff = safemod((i128)rs[i] - rr[i], ms[i]);
      ull t = (ull)((u128)diff * im % ms[i]);
      r += t * m, m *= ms[i];
      repi(j, i + 1, n)
      {
        rr[j] += (ull)((u128)t * mm[j] % ms[j]);
        if (rr[j] >= (ull)ms[j]) rr[j] -= ms[j];
        mm[j] = (ull)((u128)mm[j] * ms[i] % ms[j]);
      }
    }
  }
  return {r, m};
}
template <class mint, class U1, class U2, size_t n>
constexpr pair<mint, mint> crt_mod_constexpr(const array<U1, n> &rs, const array<U2, n> &ms)
{
  using T = larger_int_t<U2>;
  assert(rs.size() == ms.size());
  mint r = 0, m = 1;
  array<T, n> rr{}, mm;
  fill(ALL(mm), 1);
  repi(i, n)
  {
    assert(ms[i] >= U2(1));
    assert(U1(0) <= rs[i] && U2(rs[i]) < ms[i]);
    auto [g, im, _] = extgcd<T>(mm[i], ms[i]);
    assert(g == 1);
    T t = safemod((rs[i] - rr[i]) * im, ms[i]);
    r += t * m, m *= ms[i];
    repi(j, i + 1, n)
    {
      rr[j] += t * mm[j] % ms[j];
      if (rr[j] >= ms[j])
        rr[j] -= ms[j];
      mm[j] *= ms[i], mm[j] %= ms[j];
    }
  }
  return {r, m};
}
template <class V1, class V2>
bool pre_crt(const V1 &rs, V2 &ms)
{
  using T = typename V2::value_type;
  assert(rs.size() == ms.size());
  const int n = rs.size();
  repi(i, n) repi(j, i + 1, n)
  {
    T g = gcd(ms[i], ms[j]);
    if ((rs[i] - rs[j]) % g)
      return false;
    ms[i] /= g, ms[j] /= g;
    T gi = gcd(ms[i], g), gj = g / gi;
    do
    {
      g = gcd(gi, gj);
      gi *= g, gj /= g;
    } while (g > 1);
    ms[i] *= gi, ms[j] *= gj;
  }
  return true;
}
template <class T>
T convolution_point_get(const vc<T> &a, const vc<T> &b, int p)
{
  const int n = a.size(), m = b.size();
  T res = 0;
  repi(i, max(0, p - m + 1), min(n, p + 1)) res += a[i] * b[p - i];
  return res;
}
namespace internal
{
constexpr int primitive_root_constexpr(int m)
{
  if (m == 2)
    return 1;
  if (m == 167772161)
    return 3;
  if (m == 469762049)
    return 3;
  if (m == 754974721)
    return 11;
  if (m == 998244353)
    return 3;
  if (m == 1107296257)
    return 10;
  if (m == 1711276033)
    return 29;
  if (m == 1811939329)
    return 13;
  if (m == 2013265921)
    return 31;
  if (m == 2113929217)
    return 5;
  int divs[20] = {};
  divs[0] = 2;
  int cnt = 1;
  int x = (m - 1) / 2;
  while (x % 2 == 0)
    x /= 2;
  for (int i = 3; (long long)(i)*i <= x; i += 2)
  {
    if (x % i == 0)
    {
      divs[cnt++] = i;
      while (x % i == 0)
      {
        x /= i;
      }
    }
  }
  if (x > 1)
  {
    divs[cnt++] = x;
  }
  for (int g = 2;; g++)
  {
    bool ok = true;
    for (int i = 0; i < cnt; i++)
    {
      if (powmod_constexpr(g, (m - 1) / divs[i], m) == 1)
      {
        ok = false;
        break;
      }
    }
    if (ok)
      return g;
  }
}
template <int m>
constexpr int primitive_root_for_convolution = primitive_root_constexpr(m);
template <class mint, int g = internal::primitive_root_for_convolution<mint::mod()>>
struct fft_info
{
  static constexpr int rank2 = countr_zero(mint::mod() - 1);
  std::array<mint, std::max(3, rank2 + 1)> root;  // root[i]^(2^i) == 1 (i <= rank2)
  std::array<mint, std::max(3, rank2 + 1)> iroot; // root[i] * iroot[i] == 1 (i <= rank2)
  std::array<mint, std::max(1, rank2 - 2 + 1)> rate2;
  std::array<mint, std::max(1, rank2 - 2 + 1)> irate2;
  std::array<mint, std::max(1, rank2 - 3 + 1)> rate3;
  std::array<mint, std::max(1, rank2 - 3 + 1)> irate3;
  fft_info()
  {
    root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
    iroot[rank2] = root[rank2].inv();
    for (int i = rank2 - 1; i >= 0; i--)
    {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 2; i++)
      {
        rate2[i] = root[i + 2] * prod;
        irate2[i] = iroot[i + 2] * iprod;
        prod *= iroot[i + 2];
        iprod *= root[i + 2];
      }
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 3; i++)
      {
        rate3[i] = root[i + 3] * prod;
        irate3[i] = iroot[i + 3] * iprod;
        prod *= iroot[i + 3];
        iprod *= root[i + 3];
      }
    }
  }
};
}  // namespace internal
template <class mint>
bool ntt_ok(int n)
{
  if (n <= 0)
    return false;
  if constexpr (is_static_modint_v<mint>)
  {
    if constexpr (!internal::isprime<mint::mod()>)
      return false;
    static constexpr int rank2 = countr_zero(mint::mod() - 1);
    return n <= (1 << rank2);
  }
  else
    return false;
}
template <int id>
void ntt(vc<dynamic_modint32<id>> &) { assert(false); }
template <int id>
void intt(vc<dynamic_modint32<id>> &) { assert(false); }
template <auto mod>
void ntt(vc<internal::modint_impl<internal::policy_static<mod>>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));
  static const internal::fft_info<mint> info;
  int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
  while (len < h)
  {
    if (h - len == 1)
    {
      int p = 1 << (h - len - 1);
      mint rot = 1;
      for (int s = 0; s < (1 << len); s++)
      {
        int offset = s << (h - len);
        for (int i = 0; i < p; i++)
        {
          auto l = a[i + offset];
          auto r = a[i + offset + p] * rot;
          a[i + offset] = l + r;
          a[i + offset + p] = l - r;
        }
        if (s + 1 != (1 << len))
          rot *= info.rate2[countr_zero(~(unsigned int)(s))];
      }
      len++;
    }
    else
    {
      int p = 1 << (h - len - 2);
      mint rot = 1, imag = info.root[2];
      for (int s = 0; s < (1 << len); s++)
      {
        mint rot2 = rot * rot;
        mint rot3 = rot2 * rot;
        int offset = s << (h - len);
        for (int i = 0; i < p; i++)
        {
          auto mod2 = 1ULL * mint::mod() * mint::mod();
          auto a0 = 1ULL * a[i + offset].val();
          auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
          auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
          auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
          auto a1na3imag =
              1ULL * mint(a1 + mod2 - a3).val() * imag.val();
          auto na2 = mod2 - a2;
          a[i + offset] = a0 + a2 + a1 + a3;
          a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
          a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
          a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
        }
        if (s + 1 != (1 << len))
          rot *= info.rate3[countr_zero(~(unsigned int)(s))];
      }
      len += 2;
    }
  }
}
template <auto mod>
void intt(vc<internal::modint_impl<internal::policy_static<mod>>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));
  static const internal::fft_info<mint> info;
  int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
  while (len)
  {
    if (len == 1)
    {
      int p = 1 << (h - len);
      mint irot = 1;
      for (int s = 0; s < (1 << (len - 1)); s++)
      {
        int offset = s << (h - len + 1);
        for (int i = 0; i < p; i++)
        {
          auto l = a[i + offset];
          auto r = a[i + offset + p];
          a[i + offset] = l + r;
          a[i + offset + p] =
              ((unsigned long long)mint::mod() + l.val() - (uint)r.val()) *
              irot.val();
          ;
        }
        if (s + 1 != (1 << (len - 1)))
          irot *= info.irate2[countr_zero(~(unsigned int)(s))];
      }
      len--;
    }
    else
    {
      int p = 1 << (h - len);
      mint irot = 1, iimag = info.iroot[2];
      for (int s = 0; s < (1 << (len - 2)); s++)
      {
        mint irot2 = irot * irot;
        mint irot3 = irot2 * irot;
        int offset = s << (h - len + 2);
        for (int i = 0; i < p; i++)
        {
          auto a0 = 1ULL * a[i + offset + 0 * p].val();
          auto a1 = 1ULL * a[i + offset + 1 * p].val();
          auto a2 = 1ULL * a[i + offset + 2 * p].val();
          auto a3 = 1ULL * a[i + offset + 3 * p].val();
          auto a2na3iimag =
              1ULL *
              mint((mint::mod() + a2 - a3) * iimag.val()).val();
          a[i + offset] = a0 + a1 + a2 + a3;
          a[i + offset + 1 * p] =
              (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
          a[i + offset + 2 * p] =
              (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
              irot2.val();
          a[i + offset + 3 * p] =
              (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
              irot3.val();
        }
        if (s + 1 != (1 << (len - 2)))
          irot *= info.irate3[countr_zero(~(unsigned int)(s))];
      }
      len -= 2;
    }
  }
}
namespace internal
{
template <class mint>
vc<mint> convolution_naive(const vc<mint> &a, const vc<mint> &b)
{
  const int n = a.size(), m = b.size();
  const int cnta = n - count(ALL(a), 0), cntb = m - count(ALL(b), 0);
  vc<mint> c(n + m - 1);
  if ((ll)m * cnta > (ll)n * cntb)
  {
    repi(j, m)
    {
      if (b[j] == 0)
        continue;
      repi(i, n) c[i + j] += a[i] * b[j];
    }
  }
  else
  {
    repi(i, n)
    {
      if (a[i] == 0)
        continue;
      repi(j, m) c[i + j] += a[i] * b[j];
    }
  }
  return c;
}
template <class mint>
vc<mint> convolution_ntt(vc<mint> a, vc<mint> b)
{
  const int n = a.size(), m = b.size();
  const int z = bit_ceil(n + m - 1);
  if (a == b)
  {
    a.resize(z);
    ntt(a);
    repi(i, z) a[i] *= a[i];
  }
  else
  {
    a.resize(z), b.resize(z);
    ntt(a), ntt(b);
    repi(i, z) a[i] *= b[i];
  }
  intt(a);
  mint iz = mint(z).inv();
  fem(ai : a) ai *= iz;
  a.resize(n + m - 1);
  return a;
}
template <size_t j, int mod, class T, size_t k>
void convolution_crt_helper(const vc<T> &a, const vc<T> &b, vc<array<T, k>> &cs)
{
  using mint = static_modint32<mod>;
  const int n = a.size(), m = b.size();
  auto c = convolution_ntt(vc<mint>(ALL(a)), vc<mint>(ALL(b)));
  repi(i, n + m - 1) cs[i][j] = c[i].val();
}
template <int ...ms, class T>
vc<T> convolution_crt(const vc<T> &a, const vc<T> &b)
{
  const int n = a.size(), m = b.size();
  constexpr size_t k = sizeof...(ms);
  vc<array<T, k>> cs(n + m - 1);
  constexpr array<int, k> ms_arr = {ms...};
  [&]<size_t... Is>(index_sequence<Is...>)
  {
    (convolution_crt_helper<Is, ms_arr[Is], T, k>(a, b, cs), ...);
  }(make_index_sequence<k>{});
  vc<T> c(n + m - 1);
  repi(i, n + m - 1) c[i] = get<1>(crt(cs[i], ms_arr));
  return c;
}
template <class mint, int... ms, class T>
vc<mint> convolution_crt_mod(const vc<T> &a, const vc<T> &b)
{
  const int n = a.size(), m = b.size();
  constexpr size_t k = sizeof...(ms);
  vc<array<T, k>> cs(n + m - 1);
  constexpr array<int, k> ms_arr = {ms...};
  [&]<size_t... Is>(index_sequence<Is...>)
  {
    (convolution_crt_helper<Is, ms_arr[Is], T, k>(a, b, cs), ...);
  }(make_index_sequence<k>{});
  vc<mint> c(n + m - 1);
  repi(i, n + m - 1) c[i] = crt_mod_constexpr<mint>(cs[i], ms_arr).first;
  return c;
}
}  // namespace internal
template <class mint, typename = std::enable_if_t<!std::is_integral<mint>::value>>
vc<mint> convolution(const vc<mint> &a, const vc<mint> &b)
{
  const int n = a.size(), m = b.size();
  const int cnta = n - count(ALL(a), 0), cntb = m - count(ALL(b), 0);
  if (n == 0 || m == 0)
    return {};
  if (ntt_ok<mint>(n + m - 1))
  {
    if (min(cnta, cntb) <= 60)
      return internal::convolution_naive(a, b);
    return internal::convolution_ntt(a, b);
  }
  else
  {
    if (min(cnta, cntb) <= 300)
      return internal::convolution_naive(a, b);
    assert(ntt_ok<static_modint32<469762049>>(n + m - 1) && "|a| + |b| - 1 <= 2^26");
    vc<ll> a_(n), b_(m);
    repi(i, n) a_[i] = a[i].val();
    repi(j, m) b_[j] = b[j].val();
    return internal::convolution_crt_mod<mint, 469762049, 1811939329, 2013265921>(a_, b_);
  }
}
template <int mod = 998244353, class T, typename = enable_if_t<is_integral<T>::value>>
vc<T> convolution(const vc<T> &a, const vc<T> &b)
{
  using mint = static_modint32<mod>;
  auto c = convolution(vc<mint>(ALL(a)), vc<mint>(ALL(b)));
  vc<T> c_(c.size());
  repi(i, c.size()) c_[i] = c[i].val();
  return c_;
}
void main2()
{
  LL(N, M, K);
  bi::reserve(1000);
  vc<mint> f(N + 1);
  rep(i, 1, N + 1) f[i] = bi::finv_[i];
  vc<mint> g(N + 1);
  auto dp = dvec({M + 2, M + 2, N + 1}, mint(0));
  dp[M + 1][M + 1][0] = bi::fac(N);
  rep(k, M + 1, K + 1) dp[M + 1][M + 1][0] *= k;
  rep(k, M, 0, -1)
  {
    rep(x, M + 2)
    {
      dp[k][x] = convolution(f, dp[k + 1][x]);
      dp[k][x].resize(N + 1);
    }
    rep(n, N + 1) rep(x, M + 2)
    {
      if (dp[k + 1][x][n] == 0)
        continue;
      {
        ll nn = n;
        ll ck = nn - n;
        ll nx = ck == 0 ? k : x;
        mint coef = bi::finv(ck);
        dp[k][nx][nn] += coef * dp[k + 1][x][n];
      }
    }
    rep(nx, M + 2) rep(nn, N + 1) dp[k][nx][nn] *= (k <= K ? nx : 1);
  }
  mint ans = 0;
  rep(x, M + 2) ans += dp[1][x][N];
  PRINT(ans);
}
void test()
{
}
template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      /* コードテストで確認する際にコメントアウトを外す
      cerr << val;
      //*/
    };
    CERR("\n[FAST_CIO]\n\n", "32");
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(20);
    init();
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}