#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>
namespace {
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 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/integer_arithmetic.hpp

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

template <class T>
constexpr bool is_negative(T x)
{
  static_assert(is_integral_ext<T>);
  if constexpr (is_signed_ext<T>) return x < 0;
  return false;
}

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

namespace default_infty_detail
{
  template <class T, class = void>
  struct has_infty : false_type {};

  template <class T>
  struct has_infty<T, void_t<decltype(T::infty())>> : true_type {};

  template <class T>
  const T &custom_value()
  {
    static const T value = T::infty();
    return value;
  }

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

template <class T>
constexpr decltype(auto) default_infty()
{
  if constexpr (default_infty_detail::has_infty<T>::value)
    return default_infty_detail::custom_value<T>();
  else if constexpr (is_same_v<T, i128> || is_same_v<T, u128>)
    return T(INF) * T(INF);
  else if constexpr (is_integral_ext<T> && !is_same_v<T, bool>)
  {
    if constexpr (sizeof(T) >= sizeof(ll))
      return T(INF);
    else if constexpr (numeric_limits<T>::digits >= 31)
      return (T(1) << 30) - 1;
    else
      return T(numeric_limits<T>::max() / 2);
  }
  else if constexpr (numeric_limits<T>::has_infinity)
    return numeric_limits<T>::infinity();
  else
    static_assert(default_infty_detail::unsupported<T>, "No default infinity for this type; specify infty explicitly or define T::infty().");
}

namespace template_math_detail
{

} 

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)
{
  const T x = T(a), y = T(b);
  if constexpr (is_signed_ext<T>)
    if (y == T(-1)) return T(0);
  const T r = x % y;
  return r != T(0) && is_negative(x) != is_negative(y) ? T(r + y) : r;
}

// 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; })
#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);
}

// 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 T, auto x, enable_if_t<!is_invocable_v<decltype(x)>, int> = 0>
constexpr T resolved_value()
{
  return T(x);
}

template <class T, auto x, enable_if_t<is_invocable_v<decltype(x)>, long> = 0>
const T &resolved_value()
{
  static const T value = T(x());
  return value;
}

template <class T, auto x = nullptr>
constexpr decltype(auto) resolved_infty()
{
  if constexpr (is_same_v<decltype(x), nullptr_t>)
    return default_infty<T>();
  else
    return resolved_value<T, x>();
}

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)

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/maxflow.hpp

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

template <class T, bool is_erasable = false>
struct CSR
{
protected:
  int n, m;
  
  vc<int> start;
  vc<T> elist;
  vc<int> len;
  inline int get_last(int i) const
  {
    if constexpr (is_erasable)
      return start[i] + len[i];
    else
      return start[i + 1];
  }

  template <class Iter>
  struct RowBase
  {
    using iterator = Iter;
    using reference = typename iterator_traits<iterator>::reference;

  private:
    iterator begi, endi;

  public:
    RowBase(const iterator &begi, const iterator &endi) : begi(begi), endi(endi) {}
    inline iterator begin() const { return begi; }
    inline iterator end() const { return endi; }
    template <class I = ll>
    inline I size() const { return endi - begi; }
    inline bool empty() const { return size() == 0; }

    inline reference operator[](int i) const { return *(begi + i); }

    inline reference back() const
    {
      assert(!empty());
      return *prev(endi);
    }

  };
  using Row = RowBase<typename vc<T>::iterator>;
  using ConstRow = RowBase<typename vc<T>::const_iterator>;

public:
  CSR() : n(0), m(0), start{0} {}
  
  CSR(const vc<int> &row_sizes)
      : n(row_sizes.size())
  {
    fec(s : row_sizes) assert(s >= 0);
    start = cumlsum(row_sizes);
    m = start.back();
    elist.resize(m);
    if constexpr (is_erasable)
      len = row_sizes;
  }
  
  template <class I>
  CSR(int n, const vc<pair<I, T>> &ies) : n(n), m(ies.size()), start(n, 0), elist(m)
  {
    if constexpr (is_erasable)
      len.resize(n);
    fec([ i, e ] : ies)
    {
      assert(0 <= i && i < n);
      start[i]++;
    }
    start = cumlsum(start);
    if constexpr (is_erasable)
      repi(i, n) len[i] = start[i + 1] - start[i];
    auto cnt = start;
    repi(j, m)
    {
      cauto & [ i, e ] = ies[j];
      int &k = cnt[i];
      elist[k] = e;
      k++;
    }
  }
  
  CSR(const vvc<T> &vv) : n(vv.size()), start(n + 1, 0)
  {
    m = 0;
    fec(row : vv) m += row.size();
    elist.resize(m);
    if constexpr (is_erasable)
      len.resize(n);
    for (int i = 0, j = 0; i < n; i++)
    {
      start[i] = j;
      if constexpr (is_erasable)
        len[i] = vv[i].size();
      fec(e : vv[i])
      {
        elist[j] = e;
        j++;
      }
    }
    start.back() = m;
  }

  Row operator[](int i) { return Row(elist.begin() + start[i], elist.begin() + get_last(i)); }

  int offset(int i) const
  {
    assert(0 <= i && i <= n);
    return start[i];
  }

  template <class I = ll>
  I size() const { return n; }

  vc<T> &get_elist() { return elist; }
};

template <class Cap>
struct MaxFlow
{
  static_assert(is_integral_ext<Cap> && !is_same_v<Cap, bool>);

  struct edge
  {
    int from, to;
    Cap cap, flow;
    CPP_DUMP_DEFINE_DATA(from, to, cap, flow);
  };

private:
  struct ResidualEdge
  {
    int to, rev;
    Cap cap;
  };

  int n = 0;
  mutable CSR<ResidualEdge> g;
  mutable vc<int> pos;
  mutable vc<edge> pending;
  
  mutable CSR<edge> originals;
  mutable vc<int> merged_pos;
  mutable vc<Cap> merged_flow;
  mutable int built = 0;
  mutable bool dirty = false;

  bool merge_parallel_edges() const
  {
    
    array<uint64_t, 1024> seen{};
    const int m = num_of_edges<int>(), samples = min(m, 1024);
    int repeated = 0, omitted = 0;
    for (int j = 0; j < samples; j++)
    {
      
      const int i = m <= 1024 ? j : int(ll(j / 16) * m / 64) + j % 16;
      const auto e = get_edge(i);
      if (e.from == e.to || e.cap == 0) { omitted++; continue; }
      const uint64_t key = (uint64_t(e.from) << 32) | uint32_t(e.to);
      const int slot = (key * 0x9e3779b97f4a7c15ULL) >> 54;
      if (seen[slot] == key) repeated++;
      seen[slot] = key;
    }
    
    return 2 * omitted <= samples && repeated >= max(1, samples / 64);
  }

  Cap finish_flow(Cap result)
  {
    const auto &data = g.get_elist();
    for (int k = 0; k < int(merged_pos.size()); k++)
    {
      const Cap current = data[data[merged_pos[k]].rev].cap;
      const Cap previous = merged_flow[k];
      if (current == previous) continue;
      Cap remaining = current > previous ? current - previous : previous - current;
      for (auto &e : originals[k])
      {
        const Cap f = min<Cap>(remaining, current > previous ? e.cap - e.flow : e.flow);
        if (current > previous) e.flow += f;
        else e.flow -= f;
        remaining -= f;
        if (remaining == 0) break;
      }
      assert(remaining == 0);
      merged_flow[k] = current;
    }
    return result;
  }

  void check_flow(int s, int t, Cap limit) const
  {
    assert(0 <= s && s < n && 0 <= t && t < n && s != t);
    assert(Cap(0) <= limit);
  }

  Cap flow_bound(int s, int t, Cap limit) const
  {
    auto cut_bound = [&](int v, bool reverse)
    {
      Cap sum = 0;
      for (const auto &e : g[v]) if (e.to != v)
      {
        const Cap c = reverse ? g.get_elist()[e.rev].cap : e.cap;
        if (c >= limit - sum) return limit;
        sum += c;
      }
      return sum;
    };
    limit = min(limit, cut_bound(s, false));
    return min(limit, cut_bound(t, true));
  }

  int dinic_budget_factor(int s, int t, Cap progress) const
  {
    Cap unit = 0;
    auto check = [&](const vc<bool> &use)
    {
      auto selected = [&](int v) { return use.empty() || use[v]; };
      unit = [&]
      {
        Cap result = 0;
        for (int v = 0; v < n; v++) if (selected(v))
          for (const auto &e : g[v]) if (e.to != v && selected(e.to))
          {
            Cap a = e.cap;
            while (a != 0)
            {
              const Cap b = result % a;
              result = a, a = b;
            }
            if (result == 1) return result;
          }
        return result;
      }();
      if (unit == 0) return true;
      for (int v = 0; v < n; v++) if (v != s && v != t && selected(v))
      {
        int in = 0, out = 0;
        auto add = [&](int &sum, Cap cap)
        {
          if (sum > 8) return;
          const Cap value = unit == 1 ? cap : cap / unit;
          sum = value > Cap(8 - sum) ? 9 : sum + int(value);
        };
        for (const auto &e : g[v]) if (e.to != v && selected(e.to))
        {
          add(out, e.cap), add(in, g.get_elist()[e.rev].cap);
          if (in > 8 && out > 8) return false;
        }
      }
      return true;
    };
    if (check({})) return 8;
    
    auto use = min_cut(s);
    vc<bool> reaches_t(n, false);
    vc<int> que(n);
    int front = 0, back = 0;
    reaches_t[t] = true, que[back++] = t;
    while (front < back)
    {
      const int v = que[front++];
      for (const auto &e : g[v])
        if (!reaches_t[e.to] && g.get_elist()[e.rev].cap > 0)
          reaches_t[e.to] = true, que[back++] = e.to;
    }
    for (int v = 0; v < n; v++) use[v] = use[v] && reaches_t[v];
    
    auto small_remaining_cut = [&](int v, bool reverse)
    {
      Cap remaining = progress;
      for (const auto &e : g[v]) if (e.to != v && use[e.to])
      {
        const Cap cap = reverse ? g.get_elist()[e.rev].cap : e.cap;
        if (cap > remaining) return false;
        remaining -= cap;
      }
      return true;
    };
    if (small_remaining_cut(s, false) || small_remaining_cut(t, true)) return 8;
    if (check(use)) return 8;
    
    vc<bool> large_reachable(n, false);
    front = back = 0;
    large_reachable[s] = true, que[back++] = s;
    while (front < back)
    {
      const int v = que[front++];
      for (const auto &e : g[v]) if (use[e.to] && !large_reachable[e.to])
      {
        const Cap value = unit == 1 ? e.cap : e.cap / unit;
        if (value <= 8) continue;
        if (e.to == t) return 1;
        large_reachable[e.to] = true, que[back++] = e.to;
      }
    }
    int cut = 0;
    for (int v = 0; v < n; v++) if (large_reachable[v])
      for (const auto &e : g[v]) if (use[e.to] && !large_reachable[e.to])
      {
        const Cap value = unit == 1 ? e.cap : e.cap / unit;
        if (value > Cap(32 - cut)) return 1;
        cut += int(value);
      }
    return 64;
  }

  template <bool adaptive>
  Cap run_dinic(int s, int t, Cap limit, ll &budget, bool &finished, int *structure = nullptr)
  {
    vc<int> level(n), iter(n), que(n), path;
    vc<Cap> bound, sent;
    Cap total = 0;
    ll work = 0;
    finished = true;
    while (total < limit)
    {
      fill(ALL(level), -1);
      if constexpr (adaptive) work += n;
      int front = 0, back = 0;
      level[s] = 0;
      que[back++] = s;
      while (front < back && level[t] == -1)
      {
        const int v = que[front++];
        for (const auto &e : g[v])
        {
          if constexpr (adaptive) work++;
          if (e.cap == 0 || level[e.to] != -1) continue;
          level[e.to] = level[v] + 1;
          if (e.to == t) break;
          que[back++] = e.to;
        }
      }
      if (level[t] == -1) return total;
      
      ll phase_budget = budget * (level[t] <= 16 ? 8 : 2);
      if constexpr (adaptive)
        if (work >= phase_budget)
        {
          if (structure && *structure == 0)
          {
            *structure = total >= limit - total ? 8 : dinic_budget_factor(s, t, total);
            budget *= *structure, phase_budget *= *structure;
          }
          if (work >= phase_budget) { finished = false; return total; }
        }

      fill(ALL(iter), 0);
      if constexpr (adaptive) work += n;
      bool stop = false;
      if (level[t] <= 64)
      {
        auto dfs = [&](auto &self, int v, Cap up) -> Cap
        {
          if (v == s) return up;
          Cap res = 0;
          const int level_v = level[v];
          const auto row = g[v];
          for (int &i = iter[v]; i < int(row.size()); i++)
          {
            if constexpr (adaptive)
              if (++work >= phase_budget) { stop = true; return res; }
            auto &e = row[i];
            if (level_v <= level[e.to] || g.get_elist()[e.rev].cap == 0) continue;
            const Cap f = self(self, e.to, min<Cap>(up - res, g.get_elist()[e.rev].cap));
            e.cap += f, g.get_elist()[e.rev].cap -= f;
            res += f;
            if (res == up || stop) return res;
          }
          level[v] = n;
          return res;
        };
        total += dfs(dfs, t, limit - total);
        if (stop) { finished = false; return total; }
        continue;
      }
      if (path.empty()) path.resize(n), bound.resize(n), sent.resize(n);
      int depth = 0;
      path[0] = t, bound[0] = limit - total, sent[0] = 0;
      while (true)
      {
        const int v = path[depth];
        if (v != s && sent[depth] < bound[depth] && !stop)
        {
          int &i = iter[v];
          const auto row = g[v];
          while (i < int(row.size()))
          {
            if constexpr (adaptive)
              if (++work >= phase_budget) { stop = true; break; }
            const auto &e = row[i];
            if (level[e.to] < level[v] && g.get_elist()[e.rev].cap > 0) break;
            i++;
          }
          if (!stop && i < int(row.size()))
          {
            const auto &e = row[i];
            const Cap next_bound = min<Cap>(bound[depth] - sent[depth], g.get_elist()[e.rev].cap);
            path[++depth] = e.to;
            bound[depth] = next_bound, sent[depth] = 0;
            continue;
          }
          if (!stop) level[v] = n;
        }

        const Cap f = v == s ? bound[depth] : sent[depth];
        if (depth == 0)
        {
          total += f;
          break;
        }
        const int p = path[--depth];
        const auto row = g[p];
        auto &e = row[iter[p]];
        e.cap += f;
        g.get_elist()[e.rev].cap -= f;
        sent[depth] += f;
        if (sent[depth] < bound[depth]) iter[p]++;
      }
      if (stop) { finished = false; return total; }
    }
    return total;
  }

public:
  MaxFlow() = default;
  explicit MaxFlow(int n) : n(n)
  {
    assert(0 <= n);
  }

  template <class I = ll>
  I size() const { return n; }
  template <class I = ll>
  I num_of_edges() const { return built + pending.size(); }

  int add_edge(int from, int to, Cap cap)
  {
    assert(0 <= from && from < n && 0 <= to && to < n);
    assert(Cap(0) <= cap);
    const int id = num_of_edges<int>();
    pending.push_back({from, to, cap, 0});
    return id;
  }

  void build() const
  {
    if (pending.empty() && !dirty && g.template size<int>() == n) return;
    const int m = num_of_edges<int>();
    if (!merge_parallel_edges())
    {
      vc<int> degree(n, 0);
      int omitted = 0;
      for (int i = 0; i < m; i++)
      {
        const auto e = get_edge(i);
        if (e.from == e.to || e.cap == 0) omitted++;
        else degree[e.from]++, degree[e.to]++;
      }
      CSR<ResidualEdge> next_graph(degree);
      CSR<edge> next_originals(vc<int>{omitted});
      vc<int> next_pos(m), count(n, 0);
      auto &data = next_graph.get_elist();
      int special = 0;
      for (int i = 0; i < m; i++)
      {
        const auto e = get_edge(i);
        if (e.from == e.to || e.cap == 0)
        {
          next_originals.get_elist()[special] = e;
          next_pos[i] = ~special++;
          continue;
        }
        const int a = next_graph.offset(e.from) + count[e.from]++;
        const int b = next_graph.offset(e.to) + count[e.to]++;
        data[a] = {e.to, b, Cap(e.cap - e.flow)};
        data[b] = {e.from, a, e.flow};
        next_pos[i] = a;
      }
      swap(g, next_graph), swap(originals, next_originals);
      pos.swap(next_pos);
      vc<int>().swap(merged_pos), vc<Cap>().swap(merged_flow);
      built = pos.size(), dirty = false;
      vc<edge>().swap(pending);
      return;
    }
    vc<int> degree(n, 0);
    for (int i = 0; i < m; i++)
    {
      const auto e = get_edge(i);
      if (e.from != e.to && e.cap > 0) degree[e.from]++;
    }
    CSR<int> outgoing(degree);
    vc<int> count(n, 0);
    for (int i = 0; i < m; i++)
    {
      const auto e = get_edge(i);
      if (e.from != e.to && e.cap > 0)
        outgoing.get_elist()[outgoing.offset(e.from) + count[e.from]++] = i;
    }
    vc<edge> combined;
    vc<int> group(m, -1), last(n, -1), multiplicity;
    combined.reserve(outgoing.get_elist().size());
    multiplicity.reserve(outgoing.get_elist().size());
    for (int v = 0; v < n; v++) for (int i : outgoing[v])
    {
      const auto e = get_edge(i);
      int k = last[e.to];
      
      if (k == -1 || combined[k].from != v || e.cap > numeric_limits<Cap>::max() - combined[k].cap)
      {
        k = combined.size();
        last[e.to] = k;
        combined.eb(e), multiplicity.eb(1);
      }
      else
      {
        combined[k].cap += e.cap, combined[k].flow += e.flow;
        multiplicity[k]++;
      }
      group[i] = k;
    }
    fill(ALL(degree), 0);
    vc<int> special_row(combined.size(), -1), row_sizes;
    vc<int> next_merged_pos;
    vc<Cap> next_merged_flow;
    for (int k = 0; k < int(combined.size()); k++)
    {
      const auto &e = combined[k];
      degree[e.from]++, degree[e.to]++;
      if (multiplicity[k] > 1)
      {
        special_row[k] = row_sizes.size();
        row_sizes.eb(multiplicity[k]);
        next_merged_flow.eb(e.flow);
      }
    }
    row_sizes.eb(m - outgoing.get_elist().size());
    CSR<edge> next_originals(row_sizes);
    CSR<ResidualEdge> next_graph(degree);
    vc<int> next_pos(m), combined_pos(combined.size(), -1);
    next_merged_pos.resize(next_merged_flow.size());
    fill(ALL(count), 0);
    auto &data = next_graph.get_elist();
    for (int i = 0; i < m; i++)
    {
      const int k = group[i];
      if (k == -1 || combined_pos[k] != -1) continue;
      const auto &e = combined[k];
      const int a = next_graph.offset(e.from) + count[e.from]++;
      const int b = next_graph.offset(e.to) + count[e.to]++;
      data[a] = {e.to, b, Cap(e.cap - e.flow)};
      data[b] = {e.from, a, e.flow};
      combined_pos[k] = a;
      if (special_row[k] != -1) next_merged_pos[special_row[k]] = a;
    }
    count.assign(row_sizes.size(), 0);
    for (int i = 0; i < m; i++)
    {
      const int k = group[i];
      if (k != -1 && special_row[k] == -1) next_pos[i] = combined_pos[k];
      else
      {
        const int row = k == -1 ? int(row_sizes.size()) - 1 : special_row[k];
        const int index = next_originals.offset(row) + count[row]++;
        next_originals.get_elist()[index] = get_edge(i);
        next_pos[i] = ~index;
      }
    }
    swap(g, next_graph);
    swap(originals, next_originals);
    pos.swap(next_pos);
    merged_pos.swap(next_merged_pos), merged_flow.swap(next_merged_flow);
    built = pos.size();
    dirty = false;
    vc<edge>().swap(pending);
  }

  edge get_edge(int i) const
  {
    assert(0 <= i && i < num_of_edges<int>());
    if (i >= built) return pending[i - built];
    if (pos[i] < 0) return originals.get_elist()[~pos[i]];
    const auto &data = g.get_elist();
    const auto &e = data[pos[i]], &r = data[e.rev];
    return {r.to, e.to, Cap(e.cap + r.cap), r.cap};
  }

private:
  
  template <bool bounded, class F>
  Cap run_push_relabel(int s, int t, Cap flow_limit, ll budget, bool &finished, F resume)
  {
    const int source = n, N = n + 1, max_h = 2 * N;
    
    Cap return_capacity = flow_limit;

    vc<Cap> excess(N, 0);
    vc<int> height(N), iter(N), count(max_h + 1), head(max_h + 1, -1);
    vc<int> next(N), que(N);
    vc<unsigned char> active(N, false);
    int highest = -1;
    ll work = 0;
    ll spent = 0;
    bool interrupted = false;
    if constexpr (bounded) spent = 8LL * N;
    auto exhausted = [&]
    {
      if constexpr (bounded)
        if (spent >= budget)
        {
          const ll extension = resume();
          if (extension == 0) { interrupted = true; return true; }
          budget = spent + extension;
        }
      return false;
    };
    const ll global_budget = 2LL * g.get_elist().size() + 6LL * N;

    auto activate = [&](int v)
    {
      if (v == source || v == t || excess[v] == 0 || active[v]) return;
      assert(height[v] < max_h);
      next[v] = head[height[v]], head[height[v]] = v;
      active[v] = true;
      highest = max(highest, height[v]);
    };
    auto rebuild_active = [&](int current)
    {
      if constexpr (bounded) spent += 4LL * N;
      fill(ALL(head), -1);
      fill(ALL(active), false);
      highest = -1;
      for (int v = 0; v < n; v++) if (v != current) activate(v);
    };
    auto global_relabel = [&]
    {
      if constexpr (bounded) spent += 4LL * N;
      fill(ALL(height), max_h);
      fill(ALL(iter), 0);
      
      int front = 0, back = 0;
      height[t] = 0, que[back++] = t;
      while (front < back)
      {
        const int v = que[front++];
        for (const auto &e : g[v])
        {
          if constexpr (bounded) spent++;
          if (height[e.to] == max_h && g.get_elist()[e.rev].cap > 0)
            height[e.to] = height[v] + 1, que[back++] = e.to;
        }
      }
      
      front = back = 0;
      height[source] = N;
      if (height[s] == max_h && return_capacity > 0)
        height[s] = N + 1, que[back++] = s;
      while (front < back)
      {
        const int v = que[front++];
        for (const auto &e : g[v])
        {
          if constexpr (bounded) spent++;
          if (height[e.to] == max_h && g.get_elist()[e.rev].cap > 0)
            height[e.to] = height[v] + 1, que[back++] = e.to;
        }
      }
      fill(ALL(count), 0);
      for (int h : height) count[h]++;
      rebuild_active(-1);
      work = 0;
    };

    excess[s] = flow_limit;
    global_relabel();
    while (highest >= 0 && !exhausted())
    {
      if constexpr (bounded) spent++;
      if (head[highest] == -1) { highest--; continue; }
      const int v = head[highest];
      head[highest] = next[v], active[v] = false;
      const auto row = g[v];
      while (excess[v] > 0 && !exhausted())
      {
        int &i = iter[v];
        const int degree = row.size();
        if (i == degree + (v == s))
        {
          const int old = height[v];
          int h = max_h;
          if (v == s)
          {
            work++;
            if constexpr (bounded) spent++;
            if (return_capacity > 0) h = N + 1;
          }
          for (const auto &e : g[v])
          {
            work++;
            if constexpr (bounded) spent++;
            if (e.cap > 0) h = min(h, height[e.to] + 1);
          }
          assert(h < max_h);
          height[v] = h, i = 0;
          count[old]--, count[h]++;
          if (old < N && count[old] == 0)
          {
            if constexpr (bounded) spent += n;
            
            for (int u = 0; u < n; u++) if (old < height[u] && height[u] < N)
            {
              count[height[u]]--;
              height[u] = N + 1, count[N + 1]++, iter[u] = 0;
            }
            rebuild_active(v);
          }
          continue;
        }
        work++;
        if constexpr (bounded) spent++;
        if (i == degree)
        {
          if (return_capacity > 0 && height[v] == N + 1)
          {
            const Cap f = min(excess[v], return_capacity);
            return_capacity -= f;
            excess[v] -= f, excess[source] += f;
          }
          else i++;
          continue;
        }
        auto &e = row[i];
        if (e.cap > 0 && height[v] == height[e.to] + 1)
        {
          const Cap f = min(excess[v], e.cap);
          e.cap -= f, g.get_elist()[e.rev].cap += f;
          excess[v] -= f, excess[e.to] += f;
          activate(e.to);
        }
        else i++;
      }
      if (excess[v] > 0) break;
      if (work >= global_budget) global_relabel();
    }
    finished = !interrupted;
    const Cap result = excess[t];
    return result;
  }

public:

  Cap flow(int s, int t, Cap flow_limit = numeric_limits<Cap>::max())
  {
    check_flow(s, t, flow_limit);
    if (flow_limit == 0) return 0;
    build();
    flow_limit = flow_bound(s, t, flow_limit);
    if (flow_limit == 0) return 0;
    
    bool finished;
    ll budget = n + ll(g.get_elist().size());
    if (n < 128 || g.get_elist().size() < 1024)
      return finish_flow(run_dinic<false>(s, t, flow_limit, budget, finished));
    int structure = 0;
    Cap total = run_dinic<true>(s, t, flow_limit, budget, finished, &structure);
    if (finished || total == flow_limit) return finish_flow(total);
    if (structure == 0)
    {
      structure = total >= flow_limit - total ? 8 : dinic_budget_factor(s, t, total);
      if (structure > 1)
      {
        budget *= structure;
        total += run_dinic<true>(s, t, flow_limit - total, budget, finished);
        if (finished || total == flow_limit) return finish_flow(total);
      }
    }

    vc<Cap> saved;
    saved.reserve(g.get_elist().size());
    for (const auto &e : g.get_elist()) saved.eb(e.cap);
    CSR<ResidualEdge> dinic_graph;
    Cap extra = 0;
    auto resume = [&]() -> ll
    {
      if (dinic_graph.template size<int>() == 0)
      {
        dinic_graph = g;
        
        int index = 0;
        for (auto &e : dinic_graph.get_elist()) e.cap = saved[index++];
        vc<Cap>().swap(saved);
      }
      swap(g, dinic_graph);
      budget *= 2;
      bool done;
      extra += run_dinic<true>(s, t, flow_limit - total - extra, budget, done);
      swap(g, dinic_graph);
      return done ? 0 : 8 * budget;
    };
    const Cap f = run_push_relabel<true>(s, t, flow_limit - total, 8 * budget, finished, resume);
    if (finished) return finish_flow(total + f);
    swap(g, dinic_graph);
    return finish_flow(total + extra);
  }

  vc<bool> min_cut(int s) const
  {
    assert(0 <= s && s < n);
    build();
    vc<bool> seen(n, false);
    vc<int> que(n);
    int front = 0, back = 0;
    seen[s] = true, que[back++] = s;
    while (front < back)
    {
      const int v = que[front++];
      for (const auto &e : g[v]) if (e.cap > 0 && !seen[e.to])
        seen[e.to] = true, que[back++] = e.to;
    }
    return seen;
  }
};

void main2()
{
  LL(N, M, S, T);
  S--, T--;
  VEC(tlll, M, UVW);
  offset(UVW, tlll{-1, -1, 0});
  UNZIP(UVW, U, V, W);

  MaxFlow<ll> mf(N);
  fec([ u, v, w ] : UVW) mf.add_edge(u, v, w);

  PRINT(mf.flow(S, T));
}

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() {}