#define SINGLE_TESTCASE 
// #define MULTI_TESTCASE
// #define AOJ_TESTCASE

#define FAST_IO 
// #define FAST_CIO
// #define INTERACTIVE

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

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

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

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

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

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

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

using vl = vc<ll>;
using vstr = vc<string>;

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

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

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

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

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

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

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

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

namespace default_infty_detail
{

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

template <class T, class U>
inline bool chmax(T &a, U b) { return a < b ? a = b, true : false; }

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

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

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

#define LMD(x,fx) ([&](const auto &x) { return fx; })
template <class F>
auto gen_vec(int n, const F &f)
{
  assert(n >= 0);
  vc<decltype(f(0))> res;
  res.reserve(n);
  repi(i, n) res.eb(f(i));
  return res;
}
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

template <class V, class T>
void fill(V &v, const T &value)
{
  if constexpr (is_assignable_v<decltype(*std::begin(v)), const T &>)
    std::fill(std::begin(v), std::end(v), value);
  else
    for (auto &row : v) fill(row, value);
}

template <class T, class... Ts>
vc<T> concat(vc<T> v, const vc<Ts> &...vs)
{
  (v.insert(v.end(), ALL(vs)), ...);
  return v;
}

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

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

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

template <class V>
auto MAX(const V &v) { return *max_element(ALL(v)); }

template <class T>
vc<T> perminv(const vc<T> &p)
{
  if (p.empty())
    return {};
  const int n = p.size();
  vc<T> q(MAX(p) + 1);
  repi(i, n) if (p[i] >= 0) q[p[i]] = i;
  return q;
}

template <class V>
V reversed(const V &v) { return V(v.rbegin(), v.rend()); }

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

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

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};

namespace internal
{
}

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); }

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

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

#define DEFAULT_COMP ranges::less

namespace internal
{
};

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

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

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

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

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

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

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

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

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

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) ((void)0)
#define local(...) 
#define oj(...) __VA_ARGS__
#define local_oj(a,b) (b)

template <class T, class Sequence>
vc<T> content(queue<T, Sequence> que);
template <class T, class Sequence, class Compare>
vc<T> content(priority_queue<T, Sequence, Compare> pque);
template <class T>
auto content(const T &obj);

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

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

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

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

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

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

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

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

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

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

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

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

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

template <class... T>
void rd1(tuple<T...> &tpl) {
  apply([](auto &...x) { (rd1(x), ...); }, tpl);
}

template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}

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

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

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

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

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

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

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

template <class T, class U>
void wt1(const pair<T, U> &val);
template <class... T>
void wt1(const tuple<T...> &tpl);
template <class T, size_t S>
void wt1(const array<T, S> &val);
template <class T>
void wt1(const vector<T> &val);

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

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

} 

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

namespace internal
{

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

template <class... T>
void READVECnodump(int n, vc<T> &...v)
{
  (v.resize(n), ...);
  READnodump(v...);
}

}; 

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

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

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

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

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

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

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

#define ENDL '\n'

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

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

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

namespace internal
{

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;
}

}; 

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 T, class Add>
void offset(vc<T> &v, const Add &add) { for (auto &vi : v) vi += add; }

template <class T, const size_t m>
array<vc<T>, m> unzip(const vc<array<T, m>> &vt)
{
  const size_t n = vt.size();
  array<vc<T>, m> tv;
  tv.fill(vc<T>(n));
  for (size_t i = 0; i < n; i++)
    for (size_t j = 0; j < m; j++)
      tv[j][i] = vt[i][j];
  return tv;
}

template <class T, class U>
pair<vc<T>, vc<U>> unzip(const vc<pair<T, U>> &vt)
{
  const size_t n = vt.size();
  pair<vc<T>, vc<U>> tv;
  tv.first.resize(n), tv.second.resize(n);
  for (size_t i = 0; i < n; i++)
    tie(tv.first[i], tv.second[i]) = vt[i];
  return tv;
}

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 <class... Ts>
auto unzip(const vc<tuple<Ts...>> &vt)
{
  const size_t n = vt.size();
  tuple<vc<Ts>...> tv;
  apply([&](auto &...v)
        { ((v.resize(n)), ...); }, tv);
  for (size_t i = 0; i < n; i++)
    internal::vt_to_tv_impl(tv, vt[i], make_index_sequence<tuple_size_v<decltype(tv)>>{}, i);
  return tv;
}

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = internal::zip_vectors(__VA_ARGS__)

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

mt19937_64 mt;

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

namespace internal
{
}; 

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

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

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

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

namespace larger_int_detail
{
}

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

public:
  using type = T;
};

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

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

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

#undef LARGER_INT

template <class T>
struct Rational;

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

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

namespace internal
{

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

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

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

};

namespace internal
{

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

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v);
};

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

namespace internal
{

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

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{};
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v);
};

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

namespace internal
{

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

struct montgomery64odd
{
  ull m = 0, im = 0, sq = 0;
  montgomery64odd() = default;
  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 = 0, mx = 0, imx = 0, d = 0, q = 0;
  uint b = 0;
  montgomery64() = default;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};

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

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

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

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

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

namespace internal
{

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

  private:
    V _v;

  public:

    modint_impl();

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v);

    M val() const;

    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }
    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }
    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }
    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }
    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }
    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }
    friend M safe_hash_key(const mint &x) { return x.val(); }
  
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

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

using modint998244353 = static_modint32<998244353>;

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

template <class T>
struct is_static_modint : false_type {};
template <int m, class PowInv>
struct is_static_modint<internal::modint_impl<internal::policy_static<m>, PowInv>> : true_type {};
template <ll m, class PowInv>
struct is_static_modint<internal::modint_impl<internal::policy_static<m>, PowInv>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

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

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

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

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

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

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

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

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

    friend ostream &operator<<(ostream &os, const rat_value &x)
    {
      os << x.p;
      if (x.q != 1)
        os << '/' << x.q;
      return os;
    }
  };

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

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {

  };

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

using mint = modint998244353;
// using mint = modint1000000007;
// using mint = static_modint<1000000000>;
// using mint = modint;

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

// https://github.com/miscalculation53/library/tree/wip/graph/tree/merge_tree.hpp

// https://github.com/miscalculation53/library/tree/wip/ds/uf/uf.hpp

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() {}

  void reset(int n)
  {
    assert(n >= 0);
    par.assign(n, -1);
    vdat.resize(n);
    gdat = typename UFData::GData(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)]; }
  bool same(int x, int y) { return leader(x) == leader(y); }
  
  template <class I = ll>
  I merge(int x, int y, const typename UFData::EWeight &w = 1)
  {
    x = leader(x), y = leader(y);
    if (x == y)
    {
      UFData::add_edge_same(*this, x, w);
      return x;
    }
    if (-par[x] < -par[y])
      swap(x, y);
    par[x] += par[y], par[y] = x;
    UFData::add_edge_diff(*this, x, y, w);
    return x;
  }

};

template <class EWeight_ = ll>
struct UFDataEmpty
{
  struct VData
  {
    VData() {}
    VData(int) {}
  };
  struct GData
  {
    GData() {}
    GData(int) {}
  };
  using EWeight = EWeight_;
  template <class UF>
  static void add_edge_diff(UF &, int, int, EWeight) {}
  template <class UF>
  static void add_edge_same(UF &, int, EWeight) {}
};

struct MergeTree
{
private:
  UnionFind<UFDataEmpty<>> uf;
  vc<int> top, par;

public:
  explicit MergeTree(int n = 0)
  {
    assert(n >= 0);
    uf.reset(n);
    top.resize(n);
    iota(ALL(top), 0);
    par.reserve(size_t(n) * 2);
    par.assign(n, -1);
  }

  vc<int> parents() const { return par; }

  bool same(int u, int v) { return uf.same(u, v); }
  
  int merge(int u, int v)
  {
    int x = uf.leader(u), y = uf.leader(v);
    if (x == y) return -1;
    int a = top[x], b = top[y], z = par.size();
    par.eb(-1);
    par[a] = par[b] = z;
    int r = uf.merge(x, y);
    top[r] = z;
    return z;
  }
};
// https://github.com/miscalculation53/library/tree/wip/graph/tree/rooted_tree.hpp

struct RootedTree
{
protected:
  static const int MSK = 1 << 30;
  int n;
  vc<int> p_head, dep, pre, post, preinv;
  
  inline int internal_head(int v) const { return (p_head[v] & MSK) ? v : p_head[v]; }
  
  inline int parent_of_head(int hv) const { return p_head[hv] & ~MSK; }

  void build(int rt, const vc<int> &siz, vc<int> &max_chi_siz)
  {
    
    vc<int> ord(n);
    {
      vc<int> cnt(n + 1);
      repi(i, n) cnt[siz[i]]++;
      cnt = cumlsum(cnt);
      repi(i, n) ord[n - 1 - (cnt[siz[i]]++)] = i;
    }
    
    {
      p_head[rt] = rt | MSK, post[rt] = n;
      vc<int> add(n, 1);
      repi(i, 1, n)
      {
        int v = ord[i];
        int p = p_head[v];
        dep[v] = dep[p] + 1;
        pre[v] = pre[p] + add[p];
        add[p] += siz[v];
        post[v] = pre[v] + siz[v];
        if (max_chi_siz[p] == siz[v])
        {
          max_chi_siz[p] = 0;
          p_head[v] = internal_head(p_head[v]);
        }
        else
          p_head[v] |= MSK;
      }
      preinv = perminv(pre);
    }
  }

public:
  RootedTree() {}
  template <class I>
  RootedTree(int n, const vc<I> &par)
  : n(n), p_head(n), dep(n), pre(n), post(n), preinv(n)
  {
    assert(n >= 1);
    assert(SZ(par) == n);
    int rt = -1;
    vc<int> deg(n), siz(n, 1), max_chi_siz(n);
    repi(i, n)
    {
      if (par[i] < 0 || par[i] == i)
      {
        assert(rt == -1 && "There are more than two roots");
        rt = i;
      }
      else
      {
        p_head[i] = par[i];
        deg[par[i]]++;
      }
    }
    assert(rt != -1 && "There is no root");
    repi(i, n)
    {
      int v = i;
      while (v != rt && deg[v] == 0)
      {
        int p = p_head[v];
        deg[p]--, deg[v]--;
        siz[p] += siz[v];
        chmax(max_chi_siz[p], siz[v]);
        v = p;
      }
    }
    build(rt, siz, max_chi_siz);
  }

  template <class I = ll>
  I size() const { return n; }
  
  int root() const { return preinv[0]; }

  int parent(int v) const
  {
    assert(0 <= v && v < n);
    assert(v != root());
    int ph = p_head[v];
    if (ph & MSK)
      return ph & ~MSK;
    else
      return preinv[pre[v] - 1];
  }

  vc<int> parents() const
  {
    vc<int> par(n, -1);
    int rt = root();
    repi(v, n) if (v != rt) par[v] = parent(v);
    return par;
  }

  int head(int v) const
  {
    assert(0 <= v && v < n);
    return internal_head(v);
  }

  int preorder(int v) const
  {
    assert(0 <= v && v < n);
    return pre[v];
  }
  
  int postorder(int v) const
  {
    assert(0 <= v && v < n);
    return post[v];
  }
  
  int preorder_select(int i) const
  {
    assert(0 <= i && i < n);
    return preinv[i];
  }

  vc<int> path_to_root(int v) const
  {
    assert(0 <= v && v < n);
    vc<int> res = {v};
    res.reserve(dep[v] + 1);
    int rt = root();
    while (v != rt)
    {
      int p = parent(v);
      res.eb(p);
      v = p;
    }
    return res;
  }
  
  vc<int> path(int u, int v) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    vc<int> pu = {u}, pv = {v};
    while (u != v)
    {
      if (dep[u] > dep[v])
      {
        int p = parent(u);
        pu.eb(p);
        u = p;
      }
      else
      {
        int p = parent(v);
        pv.eb(p);
        v = p;
      }
    }
    pv.pop_back();
    return concat(pu, reversed(pv));
  }

  const vc<int> &top_down_vertices() const { return preinv; }
  
  vc<int> bottom_up_vertices() const { return reversed(top_down_vertices()); }

  pair<string, vc<int>> dfs_ordered_vertices() const
  {
    string bp(2 * (n - 1), '?');
    vc<int> vs(2 * (n - 1) + 1);
    int rt = root();
    vs[0] = rt;
    repi(v, n)
    {
      if (v == rt)
        continue;
      int p = parent(v);
      int in_idx = 2 * pre[v] - dep[v] - 1;
      int out_idx = 2 * post[v] - dep[v] - 2;
      bp[in_idx] = '(', vs[in_idx + 1] = v;
      bp[out_idx] = ')', vs[out_idx + 1] = p;
    }
    return {bp, vs};
  }

  class ChildIterator;
  class ChildRange;

  class ChildIterator
  {
  private:
    const RootedTree* t;
    int cur;
    friend class RootedTree;
    friend class ChildRange;
    ChildIterator(const RootedTree* t, int cur) : t(t), cur(cur) {}

  public:
    using iterator_category = std::input_iterator_tag;
    using value_type        = int;
    using difference_type   = std::ptrdiff_t;
    using pointer           = int*;
    using reference         = int;
    int operator*() const { return t->preinv[cur]; }
    void operator++() { cur = t->post[t->preinv[cur]]; }
    bool operator!=(const ChildIterator& r) const { return cur != r.cur; }
    bool operator==(const ChildIterator& r) const { return cur == r.cur; }
  };

  class ChildRange
  {
  private:
    const RootedTree* t;
    int cur, end_pos;
    friend class RootedTree;
    ChildRange(const RootedTree* t, int cur, int end_pos) : t(t), cur(cur), end_pos(end_pos) {}

  public:
    ChildIterator begin() const { return ChildIterator(t, cur); }
    ChildIterator end() const { return ChildIterator(t, end_pos); }
    bool empty() const { return cur == end_pos; }
    vc<int> to_v() const { return vc<int>(begin(), end()); }
  };

  int heavy_child(int v) const
  {
    assert(0 <= v && v < n);
    if (post[v] - pre[v] == 1)
      return -1;
    return preinv[pre[v] + 1];
  }
  ChildRange children(int v) const
  {
    assert(0 <= v && v < n);
    return {this, pre[v] + 1, post[v]};
  }
  ChildRange light_children(int v) const
  {
    assert(0 <= v && v < n);
    int cur = pre[v] + 1;
    if (cur < post[v])
    {
      cur = post[preinv[cur]];
    }
    return {this, cur, post[v]};
  }

  pair<int, int> parent_child(int u, int v) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    return dep[u] < dep[v] ? pair{u, v} : pair{v, u};
  }
  
  int lca(int u, int v) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    while (true)
    {
      int hu = internal_head(u), hv = internal_head(v);
      if (hu == hv)
        break;
      if (dep[hu] > dep[hv])
        swap(u, v), swap(hu, hv);
      v = parent_of_head(hv);
    }
    return dep[u] < dep[v] ? u : v;
  }

  bool is_ancestor(int u, int v) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    return pre[u] <= pre[v] && pre[v] < post[u];
  }

  template <class Pred>
  pair<int, int> first_ancestor(int v, const Pred &pred) const
  {
    assert(0 <= v && v < n);
    int rt = root(), child = -1;
    while (true)
    {
      int h = internal_head(v);
      if (pred(h))
      {
        int ok = pre[h], ng = pre[v] + 1;
        while (ng - ok > 1)
        {
          int mid = ok + (ng - ok) / 2;
          if (pred(preinv[mid])) ok = mid;
          else ng = mid;
        }
        return {ok < pre[v] ? preinv[ok + 1] : child, preinv[ok]};
      }
      if (h == rt) return {rt, -1};
      child = h;
      v = parent_of_head(h);
    }
  }

  int la(int v, int k) const
  {
    assert(0 <= v && v < n);
    assert(0 <= k);
    if (k > dep[v])
      return -1;
    while (true)
    {
      int hv = internal_head(v);
      if (k <= dep[v] - dep[hv])
        break;
      k -= dep[v] - dep[hv] + 1;
      v = parent_of_head(hv);
    }
    return preinv[pre[v] - k];
  }

  int jump(int u, int v, int k) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    assert(0 <= k);
    int l = lca(u, v);
    int d_ul = dep[u] - dep[l];
    int d_vl = dep[v] - dep[l];
    if (k > d_ul + d_vl)
      return -1;
    if (k <= d_ul)
      return la(u, k);
    return la(v, d_ul + d_vl - k);
  }

  pair<int, int> subtree_interval(int v, bool edge = false) const
  {
    return {pre[v] + edge, post[v]};
  }
  class PathRange
  {
  private:
    
    static constexpr int capacity = 2 * numeric_limits<int>::digits;
    array<tuple<int, int, bool>, capacity> intervals;
    int count = 0;
    friend class RootedTree;
    PathRange() = default;

  public:
    using iterator = decltype(intervals)::const_iterator;
    iterator begin() const { return intervals.begin(); }
    iterator end() const { return intervals.begin() + count; }
    int size() const { return count; }
    bool empty() const { return count == 0; }
    vc<tuple<int, int, bool>> to_v() const { return {begin(), end()}; }
  };

  PathRange path_query(int u, int v, bool edge = false) const
  {
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);
    PathRange res;
    int down_begin = PathRange::capacity;
    while (true)
    {
      int hu = internal_head(u), hv = internal_head(v);
      if (hu == hv)
        break;
      if (dep[hu] > dep[hv])
      {
        res.intervals[res.count++] = {pre[hu], pre[u] + 1, true};
        u = parent_of_head(hu);
      }
      else
      {
        res.intervals[--down_begin] = {pre[hv], pre[v] + 1, false};
        v = parent_of_head(hv);
      }
    }

    if (dep[u] > dep[v])
    {
      int l = pre[v] + edge;
      int r = pre[u] + 1;
      if (l < r)
        res.intervals[res.count++] = {l, r, true};
    }
    else
    {
      int l = pre[u] + edge;
      int r = pre[v] + 1;
      if (l < r)
        res.intervals[res.count++] = {l, r, false};
    }

    while (down_begin < PathRange::capacity)
      res.intervals[res.count++] = res.intervals[down_begin++];
    return res;
  }
};

// https://github.com/miscalculation53/library/tree/wip/ds/meldable_set.hpp

struct MeldableSetPool
{
  class Set;

private:
  using Root = int;
  static constexpr Root null = -1;
  struct Node
  {
    ll count;
    union
    {
      ull bits;
      array<Root, 2> child;
    };
    explicit Node(ull bits) : count(__builtin_popcountll(bits)), bits(bits) {}
    Node(Root left, Root right, ll count) : count(count), child{left, right} {}
  };

  ll n = 0;
  int height = 0;
  vc<Node> nodes;

  static ull low_mask(int k) { return k == 64 ? ~0ULL : (1ULL << k) - 1; }
  ll count_of(Root root) const { return root == null ? 0 : nodes[root].count; }
  void check_root(Root root) const
  {
    assert(root == null || (0 <= root && root < SZ<int>(nodes)));
  }
  void pull(Root root)
  {
    nodes[root].count = count_of(nodes[root].child[0]) + count_of(nodes[root].child[1]);
  }

  Root make_path(ull key, int h)
  {
    Root root = nodes.size();
    nodes.eb(1ULL << (key & 63));
    key >>= 6;
    repi(i, h)
    {
      Root parent = nodes.size();
      if ((key >> i) & 1) nodes.eb(null, root, 1);
      else nodes.eb(root, null, 1);
      root = parent;
    }
    return root;
  }

  Root insert_impl(Root root, ull key, int h, bool &inserted)
  {
    if (root == null)
    {
      inserted = true;
      return make_path(key, h);
    }
    if (h == 0)
    {
      ull bit = 1ULL << (key & 63);
      inserted = !(nodes[root].bits & bit);
      nodes[root].bits |= bit;
      nodes[root].count += inserted;
      return root;
    }
    int b = (key >> (h + 5)) & 1;
    Root child = insert_impl(nodes[root].child[b], key, h - 1, inserted);
    nodes[root].child[b] = child;
    pull(root);
    return root;
  }

  Root erase_impl(Root root, ull key, int h, bool &erased)
  {
    if (root == null) return null;
    if (h == 0)
    {
      ull bit = 1ULL << (key & 63);
      erased = (nodes[root].bits & bit) != 0;
      nodes[root].bits &= ~bit;
      nodes[root].count -= erased;
    }
    else
    {
      int b = (key >> (h + 5)) & 1;
      nodes[root].child[b] = erase_impl(nodes[root].child[b], key, h - 1, erased);
      pull(root);
    }
    return nodes[root].count == 0 ? null : root;
  }

  Root merge_impl(Root a, Root b, int h)
  {
    if (a == null) return b;
    if (b == null) return a;
    if (h == 0)
    {
      nodes[a].bits |= nodes[b].bits;
      nodes[a].count = __builtin_popcountll(nodes[a].bits);
    }
    else
    {
      nodes[a].child[0] = merge_impl(nodes[a].child[0], nodes[b].child[0], h - 1);
      nodes[a].child[1] = merge_impl(nodes[a].child[1], nodes[b].child[1], h - 1);
      pull(a);
    }
    return a;
  }

  Root clone_impl(Root root, int h)
  {
    if (root == null) return null;
    Node node = nodes[root];
    Root res = nodes.size();
    nodes.eb(node);
    if (h > 0)
    {
      Root left = clone_impl(node.child[0], h - 1);
      Root right = clone_impl(node.child[1], h - 1);
      nodes[res].child = {left, right};
    }
    return res;
  }

public:
  explicit MeldableSetPool(ll n = 0, size_t expected_insertions = 0) : n(n)
  {
    assert(n >= 0);
    ull blocks = (ull(n) + 63) / 64;
    while ((1ULL << height) < blocks) height++;
    nodes.reserve(expected_insertions * (height + 1));
  }

  ll universe_size() const { return n; }
  size_t node_count() const { return nodes.size(); }
  void reserve(size_t count) { nodes.reserve(count); }

private:
  ll size(Root root) const { check_root(root); return count_of(root); }
  bool empty(Root root) const { check_root(root); return root == null; }

  Root singleton(ll key)
  {
    assert(0 <= key && key < n);
    return make_path(ull(key), height);
  }

  bool contains(Root root, ll key) const
  {
    check_root(root);
    assert(0 <= key && key < n);
    for (int h = height; h > 0 && root != null; h--)
      root = nodes[root].child[(ull(key) >> (h + 5)) & 1];
    return root != null && ((nodes[root].bits >> (key & 63)) & 1);
  }

  bool insert(Root &root, ll key)
  {
    check_root(root);
    assert(0 <= key && key < n);
    bool inserted = false;
    root = insert_impl(root, ull(key), height, inserted);
    return inserted;
  }

  bool erase(Root &root, ll key)
  {
    check_root(root);
    assert(0 <= key && key < n);
    bool erased = false;
    root = erase_impl(root, ull(key), height, erased);
    return erased;
  }

  void merge(Root &a, Root &b)
  {
    check_root(a), check_root(b);
    if (&a == &b) return;
    assert(a == null || b == null || a != b);
    a = merge_impl(a, b, height);
    b = null;
  }

  Root clone(Root root) { check_root(root); return clone_impl(root, height); }

  ll lt_cnt(Root root, ll key) const
  {
    check_root(root);
    if (key <= 0) return 0;
    if (key >= n) return count_of(root);
    ll res = 0;
    for (int h = height; h > 0 && root != null; h--)
    {
      int b = (ull(key) >> (h + 5)) & 1;
      if (b) res += count_of(nodes[root].child[0]);
      root = nodes[root].child[b];
    }
    if (root != null) res += __builtin_popcountll(nodes[root].bits & low_mask(key & 63));
    return res;
  }
  ll leq_cnt(Root root, ll key) const { return key >= n - 1 ? size(root) : lt_cnt(root, key + 1); }
  ll geq_cnt(Root root, ll key) const { return size(root) - lt_cnt(root, key); }
  ll gt_cnt(Root root, ll key) const { return size(root) - leq_cnt(root, key); }

  ll count(Root root, ll l, ll r) const
  {
    assert(0 <= l && l <= r && r <= n);
    return lt_cnt(root, r) - lt_cnt(root, l);
  }

  ll kth(Root root, ll k) const
  {
    check_root(root);
    assert(0 <= k && k < count_of(root));
    ull block = 0;
    for (int h = height; h > 0; h--)
    {
      ll left = count_of(nodes[root].child[0]);
      int b = k >= left;
      if (b) k -= left;
      root = nodes[root].child[b];
      block = (block << 1) | b;
    }
    ull bits = nodes[root].bits;
    int offset = 0;
    for (int step = 32; step > 0; step >>= 1)
    {
      int left = __builtin_popcountll(bits & low_mask(step));
      if (k >= left)
      {
        k -= left;
        bits >>= step;
        offset += step;
      }
    }
    return ll((block << 6) + offset);
  }

  ll geq_min(Root root, ll key) const
  {
    ll k = lt_cnt(root, key);
    return k == size(root) ? n : kth(root, k);
  }
  ll leq_max(Root root, ll key) const
  {
    ll k = leq_cnt(root, key);
    return k == 0 ? -1 : kth(root, k - 1);
  }
  ll gt_min(Root root, ll key) const { return key >= n - 1 ? n : geq_min(root, key + 1); }
  ll lt_max(Root root, ll key) const { return key <= 0 ? -1 : leq_max(root, key - 1); }
  ll min_element(Root root) const { return geq_min(root, 0); }
  ll max_element(Root root) const { return leq_max(root, n - 1); }

  template <class F>
  void enumerate(Root root, ll l, ll r, const F &f) const;

  vc<ll> content(Root root) const
  {
    vc<ll> res;
    res.reserve(size(root));
    enumerate(root, 0, n, [&](ll key) { res.eb(key); });
    return res;
  }

  class const_iterator
  {
    friend class Set;
    const MeldableSetPool *sets = nullptr;
    Root root = null;
    ll key = 0;
    const_iterator(const MeldableSetPool *sets, Root root, ll key) : sets(sets), root(root), key(key) {}

  public:
    using iterator_category = input_iterator_tag;
    using value_type = ll;
    using difference_type = ptrdiff_t;
    using reference = ll;
    using pointer = const ll *;
    const_iterator() = default;
    ll operator*() const { return key; }
    const ll *operator->() const { return &key; }
    const_iterator &operator++() { key = sets->gt_min(root, key); return *this; }
    const_iterator operator++(int) { auto old = *this; ++*this; return old; }
    bool operator==(const const_iterator &other) const { return sets == other.sets && root == other.root && key == other.key; }
    bool operator!=(const const_iterator &other) const { return !(*this == other); }
  };

  void clear(Root &root) { check_root(root); root = null; }

public:
  
  class Set
  {
    friend struct MeldableSetPool;
    MeldableSetPool *sets;
    Root root;
    Set(MeldableSetPool *sets, Root root) : sets(sets), root(root) {}

  public:
    Set(const Set &) = delete;
    Set &operator=(const Set &) = delete;
    Set(Set &&other) noexcept : sets(other.sets), root(exchange(other.root, null)) {}
    Set &operator=(Set &&other) noexcept
    {
      if (this != &other)
      {
        sets = other.sets;
        root = exchange(other.root, null);
      }
      return *this;
    }

    ll universe_size() const { return sets->universe_size(); }
    ll size() const { return sets->size(root); }
    bool empty() const { return sets->empty(root); }
    bool contains(ll key) const { return sets->contains(root, key); }
    bool insert(ll key) { return sets->insert(root, key); }
    bool erase(ll key) { return sets->erase(root, key); }
    void merge(Set &other)
    {
      assert(sets == other.sets);
      sets->merge(root, other.root);
    }
    Set clone() const { return Set(sets, sets->clone(root)); }

    ll lt_cnt(ll key) const { return sets->lt_cnt(root, key); }
    ll leq_cnt(ll key) const { return sets->leq_cnt(root, key); }
    ll geq_cnt(ll key) const { return sets->geq_cnt(root, key); }
    ll gt_cnt(ll key) const { return sets->gt_cnt(root, key); }
    ll count(ll l, ll r) const { return sets->count(root, l, r); }
    ll kth(ll k) const { return sets->kth(root, k); }
    ll geq_min(ll key) const { return sets->geq_min(root, key); }
    ll gt_min(ll key) const { return sets->gt_min(root, key); }
    ll leq_max(ll key) const { return sets->leq_max(root, key); }
    ll lt_max(ll key) const { return sets->lt_max(root, key); }
    ll min_element() const { return sets->min_element(root); }
    ll max_element() const { return sets->max_element(root); }

    template <class F>
    void enumerate(ll l, ll r, const F &f) const;
    vc<ll> content() const { return sets->content(root); }
    using iterator = MeldableSetPool::const_iterator;
    using const_iterator = MeldableSetPool::const_iterator;
    iterator begin() const { return iterator(sets, root, min_element()); }
    iterator end() const { return iterator(sets, root, universe_size()); }
    const_iterator cbegin() const { return begin(); }
    const_iterator cend() const { return end(); }
    void clear() { sets->clear(root); }

    void swap(Set &other) noexcept
    {
      std::swap(sets, other.sets), std::swap(root, other.root);
    }
    friend void swap(Set &a, Set &b) noexcept { a.swap(b); }
  };

  Set make_set() { return Set(this, null); }
  Set make_set(ll key) { return Set(this, singleton(key)); }

  void clear() { nodes.clear(); }
};

void main2()
{
  LL(N, M, Q);
  VEC(ll, N, A);
  VEC(tlll, M, UVW);
  offset(UVW, tlll{-1, -1, 0});
  ranges::sort(UVW, {}, LMD(t, get<2>(t)));
  UNZIP(UVW, U, V, W);

  vl ws(N, 0);
  MergeTree mg(N);
  fec([ u, v, w ] : UVW)
  {
    if (mg.same(u, v))
      continue;
    mg.merge(u, v);
    ws.eb(w);
  }
  const ll K = mg.parents().size();
  dump(mg.parents() | cp::index());
  RootedTree T(K, mg.parents());

  MeldableSetPool pool(N + 1, N);
  auto sts = GEN_VEC(K, v, pool.make_set());
  vl cnt(K);
  rep(v, N)
  {
    sts[v].insert(A[v]);
    cnt[v] = 1;
  }
  rep(v, N, K)
  {
    auto chi = T.children(v).to_v();
    swap(sts[v], sts[chi[0]]);
    sts[v].merge(sts[chi[1]]);
    cnt[v] = sts[v].size();
  }
  dump(cnt | cp::index());
  dump(ws | cp::index());

  rep(_, Q)
  {
    LL(s, c);
    s--;
    auto [u, v] = T.first_ancestor(s, LMD(v, cnt[v] >= c));
    dump(s, u, v);
    PRINT(v == -1 ? -1 : ws[v]);
  }
}

void test()
{

}

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

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