#define INF 4'000'000'000'000'000'037LL
#include <bits/stdc++.h>
using namespace std;
namespace {
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
#define vc vector
template <class T>
using vvc = vc<vc<T>>;
using vpll = vc<pll>;
#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
#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 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 fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
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>
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>
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); }
#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
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 V>
auto MAX(const V &v) { return *max_element(ALL(v)); }
#if __cplusplus < 202002L
#else
#endif
template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e() { return 0; }
};
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 T>
vc<T> cumlsum(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidAdd<T>>(v, left_index); }
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); }
#if __cplusplus < 202002L
inline constexpr ull bit_width(ull x) { return x == 0 ? 0 : 64 - __builtin_clzll(x); }
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); }
#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 msb_pos(ull x) { assert(x != 0); return bit_width(x) - 1; }
#define dump(...)
#define oj(...) __VA_ARGS__
namespace fastio {
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;
}
template <typename T>
void rd1_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}
void rd1(ll &x) { rd1_integer(x); }
template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <size_t N = 0, typename T>
void rd1_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd1(x);
    rd1_tuple<N + 1>(t);
  }
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  rd1_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd1(array<T, N> &x) {
  for (auto &d: x) rd1(d);
}
template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd1(h), read(t...);
}
void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <size_t N = 0, typename T>
void wt1_tuple(const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt1(' '); }
    const auto x = std::get<N>(t);
    wt1(x);
    wt1_tuple<N + 1>(t);
  }
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  wt1_tuple(tpl);
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
void print() { wt1('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt1(head);
  if (sizeof...(Tail)) wt1(' ');
  print(forward<Tail>(tail)...);
}
} // namespace fastio
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
namespace internal
{
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(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 LL(...) IN(ll, __VA_ARGS__)
#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define VEC(T, n, ...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define PRINT fastio::print
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; }
namespace internal
{
};
mt19937_64 mt;
namespace internal
{
constexpr ll powmod32_constexpr(ll x, ll n, int m)
{
  if (m == 1)
    return 0;
  uint _m = (uint)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
constexpr bool isprime32_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;
  constexpr ll bases[3] = {2, 7, 61};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod32_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
template <int n>
constexpr bool isprime32 = isprime32_constexpr(n);
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));
  }
};
}
namespace internal
{
#define REF static_cast<mint &>(*this)
#define CREF static_cast<const mint &>(*this)
#define VAL *static_cast<const mint *>(this)
template <class mint>
struct modint_base
{
  mint &operator+=(const mint &rhs)
  {
    mint &self = REF;
    self._v += rhs._v;
    if (self._v >= self.umod())
      self._v -= self.umod();
    return self;
  }
  mint &operator-=(const mint &rhs)
  {
    mint &self = REF;
    self._v -= rhs._v;
    if (self._v >= self.umod())
      self._v += self.umod();
    return self;
  }
  mint &operator/=(const mint &rhs)
  {
    mint &self = REF;
    return self = self * rhs.inv();
  }
  mint &operator++()
  {
    mint &self = REF;
    self._v++;
    if (self._v == self.umod())
      self._v = 0;
    return self;
  }
  mint &operator--()
  {
    mint &self = REF;
    if (self._v == 0)
      self._v = self.umod();
    self._v--;
    return self;
  }
  mint operator++(int)
  {
    mint res = VAL;
    ++REF;
    return res;
  }
  mint operator--(int)
  {
    mint res = VAL;
    --REF;
    return res;
  }
  mint operator+() const { return VAL; }
  mint operator-() const { return mint() - VAL; }
  mint pow(ll n) const
  {
    assert(n >= 0);
    mint x = VAL, 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 mint(lhs).eq(rhs); }
  friend bool operator!=(const mint &lhs, const mint &rhs)
  { return mint(lhs).neq(rhs); }
private:
  bool eq(const mint &rhs) { return REF._v == rhs._v; }
  bool neq(const mint &rhs) { return REF._v != rhs._v; }
};
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void rd1(T &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void wt1(const T &x) { fastio::wt1(x.val()); }
template <class T = ll>
constexpr tuple<T, T, T> extgcd(const T &a, const 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};
}
template <int m>
struct static_modint : internal::modint_base<static_modint<m>>
{
  using mint = static_modint;
private:
  friend struct internal::modint_base<static_modint<m>>;
  uint _v;
  static constexpr uint umod() { return m; }
  static constexpr bool prime = internal::isprime32<m>;
public:
  static constexpr int mod() { return m; }
  static mint raw(int v)
  {
    mint x;
    x._v = v;
    return x;
  }
  static_modint() : _v(0) {}
  template <class T>
  static_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }
  int val() const { return (int)_v; }
  mint& operator*=(const mint &rhs)
  {
    ull z = _v;
    z *= rhs._v;
    _v = (uint)(z % umod());
    return *this;
  }
  mint inv() const
  {
    if (prime)
    {
      assert(_v != 0);
      return CREF.pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<int>(_v, m);
      assert(g == 1);
      return x;
    }
  }
};
template <int id>
struct dynamic_modint : internal::modint_base<dynamic_modint<id>>
{
  using mint = dynamic_modint;
private:
  friend struct internal::modint_base<dynamic_modint<id>>;
  uint _v;
  static internal::barrett32 bt;
  static uint umod() { return bt.umod(); }
public:
  static int mod() { return (int)(bt.umod()); }
  static mint raw(int v)
  {
    mint x;
    x._v = v;
    return x;
  }
  dynamic_modint() : _v(0) {}
  template <class T>
  dynamic_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }
  int val() const { return (int)_v; }
  mint& operator*=(const mint &rhs)
  {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint inv() const
  {
    auto [g, x, y] = extgcd<int>(_v, mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::barrett32 dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint<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_modint<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <class T>
struct PowerTable
{
private:
  decltype(T::mod()) mod;
  T base;
  vc<T> pw;
public:
  PowerTable() {}
  PowerTable(T base) : mod(T::mod()), base(base), pw(1, 1) {}
  void reserve(int n)
  {
    if (mod != T::mod())
    {
      mod = T::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;
  }
  T pow(int n)
  {
    reserve(n);
    return pw[n];
  }
};
template <class T>
struct Binomial
{
private:
  static decltype(T::mod()) mod;
  static vc<T> fac_, finv_, inv_;
public:
  static void reserve(int n)
  {
    if (mod != T::mod())
    {
      mod = T::mod();
      fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    int i = fac_.size();
    chmin(n, T::mod() - 1);
    if (n < i)
      return;
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    for (; i <= n; i++)
    {
      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 inv(T n)
  {
    assert(n != 0);
    reserve(n.val());
    return inv_[n.val()];
  }
};
template <class T> decltype(T::mod()) Binomial<T>::mod{};
template <class T> vc<T> Binomial<T>::fac_{};
template <class T> vc<T> Binomial<T>::finv_{};
template <class T> vc<T> Binomial<T>::inv_{};
using mint = modint998244353;
template <class mint, class T = ll, class U1, class U2, size_t n>
constexpr pair<mint, mint> crt_mod_constexpr(const array<U1, n> &rs, const array<U2, n> &ms)
{
  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};
}
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 (powmod32_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, rank2 + 1> root;  // root[i]^(2^i) == 1
  std::array<mint, rank2 + 1> iroot; // root[i] * iroot[i] == 1
  std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
  std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;
  std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
  std::array<mint, std::max(0, 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 constexpr (is_static_modint_v<mint>)
  {
    if constexpr (!internal::isprime32<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_modint<id>> &) {}
template <int id>
void intt(vc<dynamic_modint<id>> &) {}
template <int mod>
void ntt(vc<static_modint<mod>> &a)
{
  using mint = static_modint<mod>;
  int n = int(a.size());
  int h = countr_zero((unsigned int)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 <int mod>
void intt(vc<static_modint<mod>> &a)
{
  using mint = static_modint<mod>;
  int n = int(a.size());
  int h = countr_zero((unsigned int)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;
    }
  }
  mint in = mint(n).inv();
  fem(ai : a) ai *= in;
}
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);
  a.resize(z), b.resize(z);
  ntt(a), ntt(b);
  repi(i, z) a[i] *= b[i];
  intt(a);
  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_modint<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 <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>
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_modint<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_modint<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_;
}
ll K;
struct Mn
{
  using S = vc<mint>;
  static constexpr S op(const S &f, const S &g)
  {
    auto h = convolution(f, g);
    vc<mint> res(K);
    rep(i, SZ(h)) res.at(i % K) += h.at(i);
    return res;
  }
  static constexpr S e()
  {
    vc<mint> res(K);
    res.at(0) = 1;
    return res;
  }
};
template <class T>
struct CSR
{
private:
  int n;
  vc<int> start;
  vc<T> elist;
  struct Row
  {
    using iterator = typename vc<T>::const_iterator;
  private:
    iterator begi, endi;
  public:
    Row(const iterator &begi, const iterator &endi) : begi(begi), endi(endi) {}
    inline iterator begin() const { return begi; }
    inline iterator end() const { return endi; }
    template <class I = ll>
    inline I size() const { return endi - begi; }
    inline bool empty() const { return size() == 0; }
    inline T get(int i) const
    {
      assert(0 <= i && i < size());
      return *(begi + i);
    }
    inline T back() const
    {
      assert(!empty());
      return *prev(endi);
    }
  };
public:
  CSR() {}
  template <class I>
  CSR(int n, const vc<pair<I, T>> &ies) : n(n), elist(ies.size())
  {
    assert(n >= 0);
    start.assign(n, 0);
    fec([ i, e ] : ies)
    {
      assert(0 <= i && i < n);
      start[i]++;
    }
    start = cumlsum(start);
    auto cnt = start;
    fec([ i, e ] : ies) elist[cnt[i]++] = e;
  }
  CSR(const vvc<T> &vv) : n(vv.size()), start(n + 1)
  {
    int m = 0;
    fec(row : vv) m += row.size();
    elist.resize(m);
    int k = 0;
    repi(i, n)
    {
      start[i] = k;
      fec(e : vv[i]) elist[k++] = e;
    }
    start.back() = m;
  }
  Row row(int i) const
  {
    if (!(0 <= i && i < n))
      return Row(elist.begin(), elist.begin());
    return Row(elist.begin() + start[i], elist.begin() + start[i + 1]);
  }
  template <class I = ll>
  I size() const { return n; }
  vvc<T> to_vv() const
  {
    vvc<T> res(n);
    repi(i, n) res[i] = {elist.begin() + start[i], elist.begin() + start[i + 1]};
    return res;
  }
};
struct GroupIndex
{
private:
  int n, m;
  CSR<int> csr;
public:
  GroupIndex() {}
  template <class T>
  GroupIndex(const vc<T> &a) : n(a.size()), m(a.empty() ? 0 : MAX(a) + 1)
  {
    vc<pair<int, int>> ies(n);
    repi(i, n)
    {
      assert(0 <= a[i]);
      ies[i] = {a[i], i};
    }
    csr = CSR(m, ies);
  }
  auto idxs(int val) const { return csr.row(val); }
  template <class I = ll>
  I lt_max(int val, int i) const
  {
    auto is = idxs(val);
    ll j = ::lt_max(is, i);
    return j == -1 ? -1 : is.get(j);
  }
  template <class I = ll>
  I leq_max(int val, int i) const
  {
    auto is = idxs(val);
    ll j = ::leq_max(is, i);
    return j == -1 ? -1 : is.get(j);
  }
  template <class I = ll>
  I gt_min(int val, int i) const
  {
    auto is = idxs(val);
    ll j = ::gt_min(is, i);
    return j == is.size() ? n : is.get(j);
  }
  template <class I = ll>
  I geq_min(int val, int i) const
  {
    auto is = idxs(val);
    ll j = ::geq_min(is, i);
    return j == is.size() ? n : is.get(j);
  }
  template <class I = ll>
  I lt_cnt(int val, int i) const { return ::lt_cnt(idxs(val), i); }
  template <class I = ll>
  I leq_cnt(int val, int i) const { return ::leq_cnt(idxs(val), i); }
  template <class I = ll>
  I gt_cnt(int val, int i) const { return ::gt_cnt(idxs(val), i); }
  template <class I = ll>
  I geq_cnt(int val, int i) const { return ::geq_cnt(idxs(val), i); }
  template <class I = ll>
  I in_cnt(int val, int l, int r) const { return ::in_cnt(idxs(val), l, r); }
  template <class I = ll>
  vvc<I> to_vv() const
  {
    auto res = csr.to_vv();
    vvc<I> res2(res.size());
    rep(i, res.size()) res2[i] = vc<I>(ALL(res[i]));
    return res2;
  }
};
template <class M, class I>
vc<pair<typename M::S, typename M::S>> dc_range_prod_left_right
(const vc<typename M::S> &v, const vc<pair<I, I>> &lrs)
{
  using S = typename M::S;
  const int n = v.size(), q = lrs.size();
  vc<pair<S, S>> res(q);
  vc<int> mids(q, n + 1);
  rep(qi, q)
  {
    auto [l, r] = lrs[qi];
    if (l == r)
    {
      res[qi] = pair{M::e(), M::e()};
      continue;
    }
    if (l + 1 == r)
    {
      res[qi] = pair{v[l], M::e()};
      continue;
    }
    const int j = msb_pos(l ^ (r - 1));
    mids[qi] = (l | (1 << j)) & ~((1 << j) - 1);
  }
  vc<S> dat(n);
  GroupIndex grp(mids);
  repi(mid, 1, n)
  {
    auto qis = grp.idxs(mid);
    int l = n + 1, r = -1;
    fec(qi : qis)
    {
      auto [l_, r_] = lrs[qi];
      chmin(l, l_), chmax(r, r_);
    }
    dat[mid - 1] = v[mid - 1];
    repi(i, mid - 2, l - 1, -1) dat[i] = M::op(v[i], dat[i + 1]);
    dat[mid] = v[mid];
    repi(i, mid + 1, r) dat[i] = M::op(dat[i - 1], v[i]);
    fec(qi : qis)
    {
      auto [l_, r_] = lrs[qi];
      res[qi] = {dat[l_], dat[r_ - 1]};
    }
  }
  return res;
}
void init()
{
  oj(mt.seed(random_device()()));
}
void main2()
{
  LL(N, M, K_);
  K = K_;
  VEC(ll, N, A);
  vvc<mint> fs(N);
  rep(i, N)
  {
    vc<mint> f(K);
    f.at(0) += 1;
    f.at(A.at(i)) += 1;
    fs.at(i) = f;
  }
  vc<pll> LR(N - M + 1);
  rep(i, N - M + 1) LR.at(i) = {i, i + M};
  auto res = dc_range_prod_left_right<Mn>(fs, LR);
  fec([f, g] : res)
  {
    mint ans = 0;
    rep(i, K) ans += f.at(i) * g.at((K - i) % K);
    PRINT(ans - 1);
  }
}
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_IO]\n\n", "32");
    cout << fixed << setprecision(20);
    test();
    init();
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}