#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;
#ifndef EPS
#define EPS 1e-11
#endif
using ld = decltype(EPS);
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;
#define vc vector
template <class T>
using vvc = vc<vc<T>>;
template <class T>
using vvvc = vc<vc<vc<T>>>;
using vb = vc<bool>;
using vl = vc<ll>;
using vpll = vc<pll>;
using vtlll = vc<tlll>;
using vtllll = vc<tllll>;
using vstr = vc<string>;
using vvb = vvc<bool>;
using vvl = vvc<ll>;
template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;
template <class T>
using pqg = priority_queue<T>;
#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  i128 res = 0;
  if (s.front() == '-')
  {
    for (int i = 1; i < (int)s.size(); i++)
      res = 10 * res + s[i] - '0';
    res = -res;
  }
  else
  {
    for (auto &&c : s)
      res = 10 * res + c - '0';
  }
  return res;
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  if (x < 0)
    x = -x, sign = "-";
  while (x > 0)
  {
    res += '0' + x % 10;
    x /= 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__)
#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif
#ifndef EPS
#define EPS 1e-11
#endif
template <class T = ll>
inline constexpr T divfloor(cauto &a, cauto &b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll>
inline constexpr T safemod(cauto &a, cauto &b) { return T(a) - T(b) * divfloor<T>(a, b); }
template <class T = ll>
constexpr T ipow(cauto &a, auto 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;
}
#define ALL(a) (a).begin(), (a).end()
template <class T = ll>
inline T SZ(cauto &x) { return x.size(); }
template <class T, size_t d, size_t i = 0>
auto dvec(cauto (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}
#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif
template <class T = ll>
struct max_op
{
  T operator()(const T &a, const T &b) const { return max(a, b); }
};
template <class T = ll>
struct min_op
{
  T operator()(const T &a, const T &b) const { return min(a, b); }
};
template <class T, const T val>
struct const_fn
{
  T operator()() const { return val; }
};
using max_e = const_fn<ll, -INF>;
using min_e = const_fn<ll, INF>;
using zero_fn = const_fn<ll, 0LL>;
#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif
#if __cplusplus < 202002L
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  sort(ALL(v), forward<Args>(args)...);
  return v;
}
#else
#endif
template <class V>
void unique(V &v) { v.erase(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;
  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;
}
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 = ll>
struct direct_product
{
private:
  vc<T> a;
public:
  direct_product(const vc<T> &a) : a(a)
  {
    assert(!a.empty());
    fec(ai : a) assert(ai >= 1);
  }
  struct Iterator
  {
  private:
    vc<T> b;
    const direct_product &prod;
  public:
    Iterator(const vc<T> &b, const direct_product &prod) : b(b), prod(prod) {}
    vc<T> operator*() const { return b; }
    Iterator& operator++()
    {
      b.back()++;
      repi(i, SZ<int>(prod.a) - 1, 0, -1)
      {
        if (b[i] == prod.a[i])
        {
          b[i] = 0;
          b[i - 1]++;
        }
        else
          break;
      }
      return *this;
    }
    bool operator!=(const Iterator &other) const { return b != other.b; }
  };
  Iterator begin() const { return Iterator(vc<T>(a.size(), 0), *this); }
  Iterator end() const
  {
    vc<T> c(a.size(), 0);
    c[0] = a[0];
    return Iterator(c, *this);
  }
};
const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};
#if __cplusplus < 202002L
template <class T = ll, class V, class... Args>
inline T LB(const V &v, Args&&... args)
{ return lower_bound(ALL(v), forward<Args>(args)...) - v.begin(); }
template <class T = ll, class V, class... Args>
inline T UB(const V &v, Args&&... args)
{ return upper_bound(ALL(v), forward<Args>(args)...) - v.begin(); }
#else
#endif
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;
inline constexpr ull MASK(auto 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 countr_zero(ll x) { assert(x != 0); return countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return has_single_bit((ull)x); }
#endif
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>
struct bsubsets
{
private:
  T x;
public:
  bsubsets(T x) : x(x) {}
  struct Iterator
  {
  private:
    T y;
    bool is_end;
    const bsubsets &bs;
  public:
    Iterator(T y, bool is_end, const bsubsets &bs) : y(y), is_end(is_end), bs(bs) {}
    T operator*() const { return y; }
    Iterator& operator++()
    {
      if (y == 0)
        is_end = true;
      y = (y - 1) & bs.x;
      return *this;
    }
    bool operator!=(const Iterator &other) const { return y != other.y || is_end != other.is_end; }
  };
  Iterator begin() const { return Iterator(x, false, *this); }
  Iterator end() const { return Iterator(x, true, *this); }
};
template <class T>
struct bsupsets
{
private:
  int n;
  T x;
public:
  bsupsets(int n, T x) : n(n), x(x) {}
  struct Iterator
  {
  private:
    T y;
    const bsupsets &bs;
  public:
    Iterator(T y, const bsupsets &bs) : y(y), bs(bs) {}
    T operator*() const { return y; }
    Iterator& operator++()
    {
      y = (y + 1) | bs.x;
      return *this;
    }
    bool operator!=(const Iterator &other) const { return y != other.y; }
  };
  Iterator begin() const { return Iterator(x, *this); }
  Iterator end() const { return Iterator((T(1) << n) | x, *this); }
};
#ifdef LOCAL
#include <cpp-dump.hpp> // https://github.com/philip82148/cpp-dump
namespace cpp_dump::_detail
{
} // namespace cpp_dump::_detail
#define dump(...) cpp_dump(__VA_ARGS__)
namespace cp = cpp_dump;
CPP_DUMP_SET_OPTION_GLOBAL(log_label_func, cp::log_label::line());
CPP_DUMP_SET_OPTION_GLOBAL(max_iteration_count, 10000);
#define local(...) __VA_ARGS__
#else
#define dump(...)
#define local(...)
#endif
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(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}
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_real(T &x) {
  string s;
  rd1(s);
  x = stod(s);
}
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(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }
template <class T, 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 wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(const string s) {
  for (char c: s) wt1(c);
}
void wt1(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt1(s[i]);
}
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 <typename T>
void wt1_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt1(s);
}
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }
template <class T, 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]);
  }
}
} // namespace fastio
#if defined FAST_IO and not defined LOCAL
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
#endif
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); }
#if defined FAST_IO and not defined LOCAL
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }
#else
template <class... Ts>
void READnodump(Ts &...a) { CIN(a...); }
#endif
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__)
#ifdef INTERACTIVE
#define ENDL endl
#else
#define ENDL '\n'
#endif
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 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
#if defined FAST_IO and not defined LOCAL
#define WRITE fastio::write
#define PRINT fastio::print
#else
#define WRITE internal::COUTW
#define PRINT internal::COUTP
#endif
#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)
#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)
template <class T, class U>
pair<T, U> operator+=(pair<T, U> &a, cauto &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U>
pair<T, U> operator+(pair<T, U> &a, cauto &b) { return a += b; }
template <class T, size_t n>
array<T, n> operator+=(array<T, n> &a, cauto &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n>
array<T, n> operator+(array<T, n> &a, cauto &b) { return a += b; }
namespace internal
{
template <size_t... I>
auto tuple_add_impl(auto &a, cauto &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}
}; // namespace internal
template <class... Ts>
tuple<Ts...> operator+=(tuple<Ts...> &a, cauto &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts>
tuple<Ts...> operator+(tuple<Ts...> &a, cauto &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>
auto vt_to_tv_impl(auto &tv, cauto &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }
template <size_t... I>
auto tv_to_vt_impl(cauto &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;
}
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 ll powmod64_constexpr(ll x, ll n, ll m)
{
  if (m == 1)
    return 0;
  ull _m = (ull)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    u128 y128(y);
    if (n & 1)
      r = (y128 * r) % _m;
    y = (y128 * 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;
}
constexpr bool isprime64_constexpr(ll n)
{
  if (n <= INT_MAX)
    return isprime32_constexpr(n);
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod64_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = (u128(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);
template <ll n>
constexpr bool isprime64 = isprime64_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));
  }
};
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;
  // sq = (2^64)^2 % m = (2^128 - m) % m = (-m % 2^128) % m
  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;  // m == 2^b * mx, mx is odd
    imx = inv64(mx);
    d = powmod64_constexpr((mx + 1) / 2, b, mx);  // 2^{-b} mod mx
    u128 sq = -u128(mx) % mx;  // 2^128 mod mx
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    ull p = x & MASK(b);  // x mod 2^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 <class T = ll>
constexpr tuple<T, T, T> extgcd(const T &a, const T &b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  // a*x1 + b*y1 == z1  ...(1)
  // a*x2 + b*y2 == z2  ...(2)
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    // (1)' = (2)
    // (2)' = (1) - q*(2)
    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
{
  using mint = static_modint;
private:
  uint _v;
  static constexpr uint umod() { return m; }
  static constexpr bool prime = internal::isprime32<m>;
public:
  static constexpr int mod() { return m; }
  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++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = 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 mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    ull z = _v;
    z *= rhs._v;
    _v = (uint)(z % umod());
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(ll 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 (prime)
    {
      assert(_v != 0);
      return pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<int>(_v, m);
      assert(g == 1);
      return 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; }
};
template <int id>
struct dynamic_modint
{
  using mint = dynamic_modint;
private:
  uint _v;
  static internal::barrett32 bt;
  static uint umod() { return bt.umod(); }
public:
  static int mod() { return (int)(bt.umod()); }
  static void set_mod(int m)
  {
    assert(m >= 1);
    bt = internal::barrett32(m);
  }
  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++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = 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 mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(ll 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
  {
    auto [g, x, y] = extgcd<int>(_v, mod());
    assert(g == 1);
    return 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; }
};
template <int id>
internal::barrett32 dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
template <int m>
istream &operator>>(istream &is, static_modint<m> &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <int m>
ostream &operator<<(ostream &os, const static_modint<m> &x)
{
  os << x.val();
  return os;
}
template <int id>
istream &operator>>(istream &is, dynamic_modint<id> &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <int id>
ostream &operator<<(ostream &os, const dynamic_modint<id> &x)
{
  os << x.val();
  return os;
}
template <int m>
void rd1(static_modint<m> &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <int m>
void wt1(const static_modint<m> &x)
{
  fastio::wt1(x.val());
}
template <int id>
void rd1(dynamic_modint<id> &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <int id>
void wt1(const dynamic_modint<id> &x)
{
  fastio::wt1(x.val());
}
using mint = modint998244353;
template <ll m>
struct static_modint64
{
  using mint = static_modint64;
private:
  ull _v;
  static constexpr ull umod() { return m; }
  static constexpr bool prime = internal::isprime64<m>;
public:
  static constexpr ll mod() { return m; }
  static_modint64() : _v(0) {}
  template <class T>
  static_modint64(T v)
  {
    if constexpr (is_unsigned_v<T>)
    {
      _v = (ull)(v % umod());
    }
    else
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (ull)x;
    }
  }
  ll val() const { return (ll)_v; }
  mint& operator++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = 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 mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    u128 z = _v;
    z *= rhs._v;
    _v = (ull)(z % umod());
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(ll 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 (prime)
    {
      assert(_v != 0);
      return pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<ll>(_v, m);
      assert(g == 1);
      return 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; }
};
template <int id>
struct dynamic_modint64_odd
{
  using mint = dynamic_modint64_odd;
private:
  ull _v;  // montgomery expression
  static internal::montgomery64odd mg;
  static ull umod() { return mg.umod(); }
public:
  static ll mod() { return (ll)(mg.umod()); }
  static void set_mod(ll m)
  {
    assert(m >= 1 && m % 2 == 1);
    mg = internal::montgomery64odd(m);
  }
  dynamic_modint64_odd() : _v(0) {}
  dynamic_modint64_odd(i128 v)
  { _v = mg.inv_reduce(v); }
  ll val() const { return (ll)mg.reduce(_v); }
  mint& operator++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = 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 mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(ll 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
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return 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; }
};
template <int id>
internal::montgomery64odd dynamic_modint64_odd<id>::mg((1LL << 61) - 1);
template <int id>
struct dynamic_modint64
{
  using mint = dynamic_modint64;
private:
  ull _v;  // montgomery expression
  static internal::montgomery64 mg;
  static ull umod() { return mg.umod(); }
public:
  static ll mod() { return (ll)(mg.umod()); }
  static void set_mod(ll m)
  {
    assert(m >= 1);
    mg = internal::montgomery64(m);
  }
  dynamic_modint64() : _v(0) {}
  dynamic_modint64(i128 v)
  { _v = mg.inv_reduce(v); }
  ll val() const { return (ll)mg.reduce(_v); }
  mint& operator++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = 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 mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(ll 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
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return 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; }
};
template <int id>
internal::montgomery64 dynamic_modint64<id>::mg((1LL << 61) - 1);
using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64_odd = dynamic_modint64_odd<-1>;
using modint64 = dynamic_modint64<-1>;
template <int m>
istream &operator>>(istream &is, static_modint64<m> &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <int m>
ostream &operator<<(ostream &os, const static_modint64<m> &x)
{
  os << x.val();
  return os;
}
template <int id>
istream &operator>>(istream &is, dynamic_modint64_odd<id> &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <int id>
ostream &operator<<(ostream &os, const dynamic_modint64_odd<id> &x)
{
  os << x.val();
  return os;
}
template <int id>
istream &operator>>(istream &is, dynamic_modint64<id> &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <int id>
ostream &operator<<(ostream &os, const dynamic_modint64<id> &x)
{
  os << x.val();
  return os;
}
template <int m>
void rd1(static_modint64<m> &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <int m>
void wt1(const static_modint64<m> &x)
{
  fastio::wt1(x.val());
}
template <int id>
void rd1(dynamic_modint64_odd<id> &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <int id>
void wt1(const dynamic_modint64_odd<id> &x)
{
  fastio::wt1(x.val());
}
template <int id>
void rd1(dynamic_modint64<id> &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <int id>
void wt1(const dynamic_modint64<id> &x)
{
  fastio::wt1(x.val());
}
template <class P>
struct PrimePower
{
  P p;
  int e;
  P pe;
  PrimePower() {}
  PrimePower(P p, int e = 1) : p(p), e(e), pe(ipow(p, e)) {}
  PrimePower(P p, int e, P pe) : p(p), e(e), pe(pe) {}
  template <class P2>
  PrimePower(const PrimePower<P2> &pp) : p(pp.p), e(pp.e), pe(pp.pe) {}
};
#ifdef LOCAL
CPP_DUMP_DEFINE_EXPORT_OBJECT(PrimePower<int>, p, e, pe);
CPP_DUMP_DEFINE_EXPORT_OBJECT(PrimePower<ll>, p, e, pe);
#endif
tuple<int, ll, ll> ord_pow_div(ll n, ll m)
{
  assert(m >= 2);
  if (m == 2)
  {
    int e = countr_zero(n);
    return {e, 1LL << e, n >> e};
  }
  if (n % m != 0)
    return {0, 1, n};
  n /= m;
  if (n % m != 0)
    return {1, m, n};
  n /= m;
  ll m2 = m * m;
  auto [f, m2f, nn] = ord_pow_div(n, m2);
  int e = 2 + 2 * f;
  ll me = m2f * m2;
  if (nn % m == 0)
    e++, me *= m, nn /= m;
  return {e, me, nn};
}
namespace internal
{
template <class mint>
bool is_prime_impl(ll n, cauto &bases)
{
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = (n - 1) >> countr_zero(n - 1);
  mint::set_mod(n);
  for (ll a : bases)
  {
    ll t = d;
    mint y = mint(a).pow(t);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y *= y;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
}; // namespace internal
bool is_prime(ll n)
{
  static constexpr array<ll, 3> bases32 = {2, 7, 61};
  static constexpr array<ll, 7> bases64 = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases32);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases64);
  }
}
namespace internal
{
template <class mint>
ll get_prime_factor_impl(ll n)
{
  mint::set_mod(n);
  int m = pow(n, .125);
  mt19937 _mt;
  while (true)
  {
    int c = 1 + _mt() % 100;
    mint x = 2, y = 2, prod = 1;
    ll g = 1;
    while (g == 1)
    {
      repi(i, m)
      {
        x = x * x + c;
        y = y * y + c, y = y * y + c;
        prod *= x - y;
      }
      g = gcd(prod.val(), n);
    }
    if (g == n)
      continue;
    if (is_prime(g))
      return g;
    else if (is_prime(n / g))
      return n / g;
    else
      return get_prime_factor_impl<mint>(g);
  }
}
ll get_prime_factor(ll n)
{
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
}
}; // namespace internal
vc<PrimePower<ll>> factorize(ll n)
{
  vc<PrimePower<ll>> res;
  repi(p, 2, 100)
  {
    if (n % p == 0)
    {
      auto [e, pe, nn] = ord_pow_div(n, p);
      res.emplace_back(PrimePower<ll>(p, e, pe));
      n = nn;
    }
  }
  while (n > 1)
  {
    if (is_prime(n))
    {
      res.emplace_back(n);
      break;
    }
    ll p = internal::get_prime_factor(n);
    auto [e, pe, nn] = ord_pow_div(n, p);
    res.emplace_back(PrimePower<ll>(p, e, pe));
    n = nn;
  }
  sort(ALL(res), [&](cauto &pp1, cauto &pp2)
       { return pp1.p < pp2.p; });
  return res;
}
template <class S_, auto op_, auto e_>
struct Monoid
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
};
template <class S_, auto op_, auto e_, auto inv_>
struct Group
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
  static constexpr auto inv = inv_;
};
template <class S_, auto add_, auto e0_, auto mul_, auto e1_>
struct SemiRing
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};
template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_>
struct Ring
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};
template <class SR>
using MonoidOfSemiRingAdd = Monoid<typename SR::S, SR::add, SR::e0>;
template <class SR>
using MonoidOfSemiRingMul = Monoid<typename SR::S, SR::mul, SR::e1>;
template <class R>
using GroupOfRingAdd = Group<typename R::S, R::add, R::e0, R::minus>;
template <class 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>
struct MonoidMul
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e() { return 1; }
};
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 T>
struct GroupAddSub
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e() { return 0; }
  static constexpr S inv(S a) { return -a; }
};
template <class T>
struct GroupMulDiv
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e() { return 1; }
  static constexpr S inv(S a) { return 1 / a; }
};
template <class T, const T infty = INF>
struct SemiRingMinPlus
{
  using S = T;
  static constexpr S add(S a, S b) { return min(a, b); }
  static constexpr S e0() { return infty; }
  static constexpr S mul(S a, S b) { return a + b; }
  static constexpr S e1() { return 0; }
};
template <class T, const T infty = INF>
struct SemiRingMaxPlus
{
  using S = T;
  static constexpr S add(S a, S b) { return max(a, b); }
  static constexpr S e0() { return -infty; }
  static constexpr S mul(S a, S b) { return a + b; }
  static constexpr S e1() { return 0; }
};
template <class T>
struct RingAddSubMul
{
  using S = T;
  static constexpr S add(S a, S b) { return a + b; }
  static constexpr S minus(S a) { return -a; }
  static constexpr S e0() { return 0; }
  static constexpr S mul(S a, S b) { return a * b; }
  static constexpr S e1() { return 1; }
};
template <class M>
vc<typename M::S> zeta_supset(const vc<typename M::S> &a)
{
  if (a.empty())
    return {};
  assert(has_single_bit(a.size()));
  const int n = countr_zero(a.size());
  auto b = a;
  repi(i, n) repi(s, 1 << n)
  {
    if (!btest(s, i))
    {
      int t = s;
      bset(t, i);
      b[s] = M::op(b[s], b[t]);
    }
  }
  return b;
}
void init() {}
void main2()
{
  LL(T, M);
  auto facM = factorize(M);
  ll K = facM.size();
  rep(_, T)
  {
    LL(N, B, C, D);
    VEC(ll, N, A);
    vc<mint> W(N);
    W.at(0) = B;
    rep(i, 1, N) W.at(i) = C * W.at(i - 1) + D;
    dump(W);
    vc<mint> f(1 << K, 1);
    rep(i, N)
    {
      if (M % A.at(i) != 0)
        continue;
      ll s = 0;
      rep(j, K)
      {
        ll e2 = get<0>(ord_pow_div(A.at(i), facM.at(j).p));
        if (e2 == facM.at(j).e)
          bset(s, j, 0);
        else
          bset(s, j, 1);
      }
      f.at(s) *= 1 + W.at(i);
      dump(i, A.at(i), 1 + W.at(i), s | cp::bin(K));
    }
    dump(f);
    auto g = zeta_supset<MonoidMul<mint>>(f);
    dump(g);
    mint ans = 0;
    rep(s, 1 << K)
    {
      ans += ipow(-1, popcount(s)) * g.at(s);
    }
    if (M == 1)
      ans--;
    PRINT(ans);
  }
}
void test()
{
  /*
  local(
    rep(testcase, 100000)
    {
      cout << endl;
      dump(testcase);
      // ----- generate cases -----
      ll N = 1 + rand() % 5;
      vl A(N);
      rep(i, N) A.at(i) = 1 + rand() % 10;
      // --------------------------
      // ------ check output ------
      #define INPUT A
      auto god = naive(INPUT);
      auto ans = solve(INPUT);
      if (god != ans)
      {
        dump(INPUT);
        dump(god, ans);
        exit(0);
      }
      // --------------------------
    }
    dump("ok");
  );
  //*/
}
int main()
{
  cauto CERR = [](string val, string color)
  {
    string s = "\033[" + color + "m" + val + "\033[m";
    #ifdef LOCAL
    cerr << s;
    #endif
    /* コードテストで確認する際にコメントアウトする
    cerr << val;
    //*/
  };
  #if defined FAST_IO and not defined LOCAL
  CERR("\n[FAST_IO]\n\n", "32");
  #endif
  #if defined FAST_CIO and not defined LOCAL
  CERR("\n[FAST_CIO]\n\n", "32");
  cin.tie(0);
  ios::sync_with_stdio(false);
  #endif
  cout << fixed << setprecision(20);
  test();
  init();
  #if defined AOJ_TESTCASE or (defined LOCAL and defined SINGLE_TESTCASE)
  CERR("\n[AOJ_TESTCASE]\n\n", "35");
  while (true)
  {
    dump("new testcase");
    main2();
  }
  #elif defined SINGLE_TESTCASE
  CERR("\n[SINGLE_TESTCASE]\n\n", "36");
  main2();
  #elif defined MULTI_TESTCASE
  CERR("\n[MULTI_TESTCASE]\n\n", "33");
  dump("T");
  IN(uint, T);
  while (T--)
  {
    dump("new testcase");
    main2();
  }
  #endif
}