#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;
#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 fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
template <class T, class U>
inline bool chmin(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); }
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;
}
#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
template <class F>
auto gen_vec(const int &n, const F &f)
{
  vc<decltype(f(0))> res(n);
  repi(i, n) res[i] = f(i);
  return res;
}
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, class... Ts>
vc<T> concat(vc<T> v, const vc<Ts> &...vs)
{
  (v.insert(v.end(), ALL(vs)), ...);
  return v;
}
template <class V>
V reversed(const V &v) { return V(v.rbegin(), v.rend()); }
#if __cplusplus < 202002L
#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>
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();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
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;
}
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 T, const T infty = INF>
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 = INF>
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>
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;
}
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); });
}
#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
#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 popcount(ull x) { return __builtin_popcountll(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
#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;
}
void rd1(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}
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;
}
void wt1(const string s) {
  for (char c: s) wt1(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>
void PRINTV(const vc<T> &v) { for (auto &vi : v) PRINT(vi); }
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, const size_t m>
array<vc<T>, m> top(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>> top(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>> top(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>> top(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 top(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 top(const tuple<vc<Ts>...> &tv)
{
  size_t n = get<0>(tv).size();
  apply([&](auto &...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;
}
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 T>
struct CoordinateCompression
{
  vc<T> vals;
  CoordinateCompression() {}
  CoordinateCompression(const vc<T> &vec) : vals(sortuniqued(vec)) {}
  template <class I = ll>
  I get_id(const T &val) const
  {
    auto it = lower_bound(ALL(vals), val);
    if (it == vals.end() || *it != val)
      return -1;
    return it - vals.begin();
  }
  template <class I = ll>
  I size() const { return vals.size(); }
};
template <class Key, class Value>
struct FlatMap
{
  CoordinateCompression<Key> cc;
  vc<Value> vals;
  FlatMap() {}
  FlatMap(const vc<Key> &keys) : cc(keys), vals(cc.size()) {}
  Value &operator[](const Key &key)
  {
    const int i = cc.get_id(key);
    assert(i != -1);
    return vals[i];
  }
  Value &at(const Key &key) { return operator[](key); }
  template <class I = ll>
  inline I size() const { return cc.size(); }
  inline bool empty() const { return size() == 0; }
  struct Iterator
  {
  private:
    int i;
    const FlatMap &mp;
  public:
    Iterator(int i, const FlatMap &mp) : i(i), mp(mp) {}
    pair<Key, Value> operator*() const
    {
      assert(i != mp.cc.size());
      return pair{mp.cc.vals[i], mp.vals[i]};
    }
    Iterator &operator++()
    {
      i++;
      return *this;
    }
    bool operator!=(const Iterator &other) const { return i != other.i; }
  };
  Iterator begin() const { return Iterator(0, *this); }
  Iterator end() const { return Iterator(cc.size(), *this); }
};
template <class UFData, bool compress = true>
struct UnionFind
{
  friend UFData;
protected:
  vc<int> par;
  vc<typename UFData::VData> vdat;
public:
  typename UFData::GData gdat;
  UnionFind() {}
  UnionFind(int n) : par(n, -1), vdat(n), gdat(n)
  { repi(i, n) vdat[i] = typename UFData::VData(i); }
  virtual int leader(int x)
  {
    assert(0 <= x && x < SZ<int>(par));
    if (par[x] < 0)
      return x;
    if constexpr (compress)
      return par[x] = leader(par[x]);
    else
      return leader(par[x]);
  }
  template <class I = ll>
  I size(int x) { return -par[leader(x)]; }
};
template <class UFData>
struct UnionFindUndo : UnionFind<UFData, false>
{
  using UF = UnionFind<UFData, false>;
  friend UFData;
protected:
  using UF::par;
  using UF::vdat;
  vc<tuple<int, int, typename UFData::VData, typename UFData::GData, bool>> his;
public:
  UnionFindUndo() {}
  UnionFindUndo(int n) : UF(n) {}
  using UF::gdat;
  using UF::leader;
};
template <class Obj, class I, class Add, class Undo, class Run>
void offline_dynamic_connectivity(int tmax, const vc<tuple<I, I, Obj>> &span_obj, Add add, Undo undo, Run run)
{
  const int tmax_ceil = bit_ceil(tmax);
  vvc<Obj> nodes(tmax_ceil + tmax);
  fe([l, r, x] : span_obj)
  {
    assert(0 <= l && l <= r && r <= tmax);
    l += tmax_ceil, r += tmax_ceil;
    while (l < r)
    {
      if (l & 1)
        nodes[l++].eb(x);
      if (r & 1)
        nodes[--r].eb(x);
      l >>= 1, r >>= 1;
    }
  }
  auto dfs = [&](auto dfs, int i) -> void
  {
    fec(x : nodes[i]) add(x);
    repi(ni, 2 * i, 2 * i + 2)
    {
      if (ni < tmax_ceil + tmax)
        dfs(dfs, ni);
    }
    if (0 <= i - tmax_ceil && i - tmax_ceil < tmax)
      run(i - tmax_ceil);
    repi(_, nodes[i].size()) undo();
  };
  dfs(dfs, 1);
}
void init()
{
  oj(mt.seed(random_device()()));
}
void main2()
{
  LL(N, M, K);
  VEC(ll, N, A);
  vc<ll> L(N), R(N);
  rep(i, N) L.at(i) = max(0LL, i - M + 1), R.at(i) = min(N - M + 1, i + 1);
  vc<mint> cur(K);
  cur.at(0) = 1;
  vc<vc<mint>> sta;
  auto add = [&](ll a) -> void
  {
    sta.eb(cur);
    vc<mint> nxt(cur);
    rep(i, K) nxt.at((i + a) % K) += cur.at(i);
    swap(cur, nxt);
    dump("add", a, cur);
  };
  auto undo = [&]() -> void
  {
    swap(cur, sta.back());
    sta.pop_back();
    dump("undo", cur);
  };
  vc<mint> ans(N - M + 1);
  auto run = [&](ll i) -> void
  {
    ans.at(i) = cur.at(0) - 1;
    dump("run", i, ans.at(i));
  };
  offline_dynamic_connectivity(N - M + 1, top(tuple{L, R, A}), add, undo, run);
  PRINTV(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_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() {}