#define SINGLE_TESTCASE 

#define FAST_IO 

#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 ld = decltype(EPS);

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

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

using vb = vc<bool>;
using vl = vc<ll>;
using vpll = vc<pll>;
using vtlll = vc<tlll>;
using vtllll = vc<tllll>;
using vstr = vc<string>;
using vvb = vvc<bool>;
using vvl = vvc<ll>;

template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;
template <class T>
using pqg = priority_queue<T>;

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, 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();

  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().");
}

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

template <class T = ll, class U, class V, 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; })
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

// 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();

template <class T, auto x, enable_if_t<is_invocable_v<decltype(x)>, long> = 0>
const T &resolved_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 {};
  }
};
template <class T, auto infty = nullptr>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr decltype(auto) e() { return resolved_infty<T, infty>(); }
};
template <class T, auto infty = nullptr>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b) { return max(a, b); }
  static constexpr S e() { return -resolved_infty<T, infty>(); }
};

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

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

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = zip(tuple{__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 = 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;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};

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{998244353};
  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, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    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{(1LL << 61) - 1};
  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{(1LL << 61) - 1};
  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>
  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 <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
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>;
using modint1000000007 = static_modint32<1000000007>;
using modint = dynamic_modint32<-1>;
using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64 = dynamic_modint64<-1>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : 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>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_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;

template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};

template <class mint>
struct modint_less
{
};

template <class mint>
struct modint_hash
{
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

template <class mint>
struct PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;

public:

};
// 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)
    {
      return os << x.p << '/' << x.q;
    }
  };

  struct mint_to_rat_fn
  {
  };

  template <class T>
  struct rat_object
  {
    T value;
  };

  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 bi = Binomial<mint>;

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

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

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

// https://github.com/miscalculation53/library/tree/wip/graph/graph.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; }
    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() {}
  
  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)); }
  ConstRow operator[](int i) const
  { return ConstRow(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; }

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

template <class Cost = void>
struct Edge
{
  int from, to;
  Cost cost;
  int index;
  Edge() : from(-1), to(-1), index(-1) {}
  Edge(int s, int t, Cost c, int i = -1) : from(s), to(t), cost(c), index(i) {}
  
};

template <>
struct Edge<void>
{
  int from, to, index;
  static constexpr int cost = 1;
  Edge() : from(-1), to(-1), index(-1) {}
  Edge(int s, int t, int i = -1) : from(s), to(t), index(i) {}
  operator int() const { return to; }
  bool operator<(const Edge &) const { return false; }
  Edge rev() const { return Edge(to, from, index); }
};

template <bool is_directed, class Cost = void, bool is_erasable = false>
struct Graph
{
  using E = Edge<Cost>;

protected:
  template <class C, bool unweighted = is_void_v<C>>
  struct InternalEdge
  {
    int to, index;
    C cost;
  };
  template <class C>
  struct InternalEdge<C, true>
  {
    int to, index;
    static constexpr int cost = 1;
  };
  using GE = conditional_t<is_erasable, E, InternalEdge<Cost>>;
  using Weight = conditional_t<is_void_v<Cost>, int, Cost>;

  int n = 0, m = 0, era = 0;
  CSR<GE, is_erasable> g;
  vc<int> eid_to_elist_id;

  static E make_edge(int from, int to, const Weight &cost, int index)
  {
    if constexpr (is_void_v<Cost>)
      return E(from, to, index);
    else
      return E(from, to, cost, index);
  }

  struct OutEdgeRow;
  struct OutEdgeIter
  {
    using iterator_category = input_iterator_tag;
    using value_type = E;
    using difference_type = ptrdiff_t;
    using pointer = const E *;
    using reference = const E &;

  private:
    friend struct OutEdgeRow;
    int from = -1;
    typename vc<GE>::const_iterator it;
    mutable E e;
    OutEdgeIter(int from, typename vc<GE>::const_iterator it) : from(from), it(it) {}

  public:
    OutEdgeIter() = default;
    reference operator*() const
    {
      if constexpr (is_erasable)
        return *it;
      else
      {
        e = make_edge(from, it->to, it->cost, it->index);
        return e;
      }
    }
    OutEdgeIter &operator++()
    {
      ++it;
      return *this;
    }
    bool operator==(const OutEdgeIter &rhs) const { return it == rhs.it; }
    bool operator!=(const OutEdgeIter &rhs) const { return !(*this == rhs); }
  };

  struct OutEdgeRow
  {
  private:
    const Graph *g;
    int from, l, r;

    OutEdgeIter iter(int pos) const
    {
      return {from, g->g.get_elist().begin() + pos};
    }

  public:
    OutEdgeRow(const Graph *g, int from)
        : g(g), from(from), l(g->g.offset(from)), r(l + g->g[from].size()) {}
    OutEdgeIter begin() const { return iter(l); }
    OutEdgeIter end() const { return iter(r); }
    template <class I = ll>
    I size() const { return r - l; }
    bool empty() const { return l == r; }
    E back() const
    {
      assert(!empty());
      return (*this)[size() - 1];
    }
  };

  template <class F>
  void build(F input_edge)
  {
    vc<int> row_sizes(n);
    repi(i, m)
    {
      auto [u, v, w] = input_edge(i);
      assert(0 <= u && u < n && 0 <= v && v < n);
      row_sizes[u]++;
      if constexpr (!is_directed)
        if (u != v) row_sizes[v]++;
    }
    g = CSR<GE, is_erasable>(row_sizes);
    if constexpr (is_erasable)
      eid_to_elist_id.assign((is_directed ? 1 : 2) * m, -1);

    vc<int> cnt(n);
    repi(i, m)
    {
      auto [u, v, w] = input_edge(i);
      int j = cnt[u]++;
      int k = g.offset(u) + j;
      if constexpr (is_erasable)
        g[u][j] = make_edge(u, v, w, i);
      else if constexpr (is_void_v<Cost>)
        g[u][j] = GE{int(v), i};
      else
        g[u][j] = GE{int(v), i, w};
      if constexpr (is_erasable)
      {
        int id = is_directed ? i : 2 * i + (u <= v);
        eid_to_elist_id[id] = k;
      }
      if constexpr (!is_directed)
      {
        if (u != v)
        {
          j = cnt[v]++;
          k = g.offset(v) + j;
          if constexpr (is_erasable)
            g[v][j] = make_edge(v, u, w, i);
          else if constexpr (is_void_v<Cost>)
            g[v][j] = GE{int(u), i};
          else
            g[v][j] = GE{int(u), i, w};
          if constexpr (is_erasable)
            eid_to_elist_id[2 * i + (v <= u)] = k;
        }
      }
    }
  }

public:
  Graph() {}
  template <class I>
  Graph(int n, const vc<pair<I, I>> &es) : n(n), m(es.size()), era(0)
  {
    build(LMD(i, (tuple{es[i].first, es[i].second, Weight(1)})));
  }
  template <class I, class C = Cost, enable_if_t<!is_void_v<C>, int> = 0>
  Graph(int n, const vc<pair<I, I>> &es, const Weight &dflt_cost) : n(n), m(es.size()), era(0)
  {
    build(LMD(i, (tuple{es[i].first, es[i].second, dflt_cost})));
  }
  template <class I, class C, enable_if_t<is_same_v<C, Cost> && !is_void_v<C>, int> = 0>
  Graph(int n, const vc<tuple<I, I, C>> &es) : n(n), m(es.size()), era(0)
  {
    build(LMD(i, es[i]));
  }
  Graph(int n, const vc<E> &es) : n(n), m(es.size()), era(0)
  {
    build(LMD(i, (tuple{es[i].from, es[i].to, es[i].cost})));
  }

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

  auto out_edges(int v) const
  {
    return OutEdgeRow(this, v);
  }
  
  vc<E> edges() const
  {
    vc<E> res;
    res.reserve(num_of_edges<int>());
    if constexpr (is_directed)
    {
      repi(v, n) fec(e : out_edges(v))
      {
        res.eb(e);
      }
    }
    else
    {
      repi(v, n) fec(e : out_edges(v))
      {
        if (e.from <= e.to) res.eb(e);
      }
    }
    return res;
  }
  
private:
};

template <bool is_directed, bool is_erasable = false, class I>
Graph(int, const vc<pair<I, I>> &) -> Graph<is_directed, void, is_erasable>;
template <bool is_directed, bool is_erasable = false, class I, class Cost>
Graph(int, const vc<tuple<I, I, Cost>> &) -> Graph<is_directed, Cost, is_erasable>;
template <bool is_directed, bool is_erasable = false, class I, class Cost>
Graph(int, const vc<pair<I, I>> &, const Cost &) -> Graph<is_directed, Cost, is_erasable>;
template <bool is_directed, bool is_erasable = false, class Cost>
Graph(int, const vc<Edge<Cost>> &) -> Graph<is_directed, Cost, is_erasable>;

template <class Cost = void, bool is_erasable = false>
using GraphDirected = Graph<true, Cost, is_erasable>;
template <class Cost = void, bool is_erasable = false>
using GraphUndirected = Graph<false, Cost, is_erasable>;

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

template <class T>
struct MyQueue
{
private:
  vc<T> d;
  int pos = 0;

public:
  template <class I = ll>
  I size() const { return SZ<I>(d) - pos; }
  bool empty() const { return pos == SZ<int>(d); }
  void push(const T &t) { d.eb(t); }
  T &front() { return d[pos]; }
  void pop() { pos++; }

};

template <bool is_directed, class Cost, auto infty = nullptr>
struct ShortestPath
{
  using Dist = conditional_t<is_void_v<Cost>, ll, Cost>;

private:
  const Graph<is_directed, Cost> &g;
  static constexpr decltype(auto) inf() { return resolved_infty<Dist, infty>(); }

  vc<Dist> dists;
  vc<Edge<Cost>> prv;
  int source = -1;
  bool solved_all = false;

  void init_solve(int s, int t)
  {
    source = s;
    solved_all = (t == -1);
  }

public:
  explicit ShortestPath(const Graph<is_directed, Cost> &g) : g(g) {}
  ShortestPath(const Graph<is_directed, Cost> &&) = delete;

  vc<Dist> bfs(int s, int t = -1)
  {
    const int n = g.size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    MyQueue<int> que;
    dists[s] = 0;
    que.push(s);
    while (!que.empty())
    {
      auto v = que.front();
      que.pop();
      if (v == t)
        break;
      fe(e : g.out_edges(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
        {
          prv[e.to] = e;
          que.push(e.to);
        }
      }
    }
    return dists;
  }

  vc<Dist> bfs01(int s, int t = -1)
  {
    const int n = g.size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    vc<int> cur{int(s)}, nxt;
    vc<unsigned char> used(n, false);
    dists[s] = 0;
    while (!cur.empty() || !nxt.empty())
    {
      if (cur.empty()) cur.swap(nxt);
      int v = cur.back();
      cur.pop_back();
      if (used[v])
        continue;
      used[v] = true;
      if (v == t)
        break;
      fe(e : g.out_edges(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
        {
          prv[e.to] = e;
          if (e.cost == 0)
            cur.eb(e.to);
          else
            nxt.eb(e.to);
        }
      }
    }
    return dists;
  }

  vc<Dist> dial(int s, int max_cost, int t = -1)
  {
    static_assert(is_integral_ext<Dist>);
    const int n = g.size();
    assert(0 <= s && s < n);
    assert(0 <= max_cost);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});

    const int bcnt = max_cost + 1;
    vc<int> head(bcnt, -1), nxt(n, -2), pre(n, -2);
    int que_size = 0;
    auto erase = [&](int v, int b)
    {
      const int p = pre[v], q = nxt[v];
      if (p == -1)
        head[b] = q;
      else
        nxt[p] = q;
      if (q != -1)
        pre[q] = p;
      nxt[v] = pre[v] = -2;
      que_size--;
    };
    auto push = [&](int v, int b)
    {
      nxt[v] = head[b], pre[v] = -1;
      if (head[b] != -1)
        pre[head[b]] = v;
      head[b] = v, que_size++;
    };

    Dist cur = 0;
    dists[s] = 0;
    push(s, 0);
    while (que_size)
    {
      int b = int(cur % bcnt);
      while (head[b] == -1)
      {
        cur++;
        if (++b == bcnt)
          b = 0;
      }
      const int v = head[b];
      erase(v, b);
      assert(dists[v] == cur);
      if (v == t)
        break;
      fec(e : g.out_edges(v))
      {
        Dist nd = dists[v] + e.cost;
        if (nd < dists[e.to])
        {
          if (pre[e.to] != -2)
            erase(e.to, int(dists[e.to] % bcnt));
          dists[e.to] = nd, prv[e.to] = e;
          push(e.to, int(nd % bcnt));
        }
      }
    }
    return dists;
  }

  vc<Dist> dijkstra(int s, int t = -1)
  {
    const int n = g.size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    pql<pair<Dist, int>> pque;
    dists[s] = 0;
    pque.push({0, s});
    while (!pque.empty())
    {
      auto [d, v] = pque.top();
      pque.pop();
      if (v == t)
        break;
      if (dists[v] != d)
        continue;
      fec(e : g.out_edges(v))
      {
        Dist nd = dists[v] + e.cost;
        if (chmin(dists[e.to], nd))
        {
          prv[e.to] = e;
          pque.push({nd, e.to});
        }
      }
    }
    return dists;
  }

  vc<Dist> dijkstra_dense(int s, int t = -1)
  {
    const int n = g.size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    vc<bool> ok(n, false);
    dists[s] = 0;
    repi(_, n)
    {
      Dist mn = inf();
      int v = -1;
      repi(u, n) if (!ok[u] && chmin(mn, dists[u])) v = u;
      if (v == -1)
        break;
      if (v == t)
        break;
      ok[v] = true;
      fec(e : g.out_edges(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
          prv[e.to] = e;
      }
    }
    return dists;
  }

  vc<Dist> bellman_ford(int s)
  {
    const int n = g.size();
    assert(0 <= s && s < n);
    init_solve(s, -1);
    dists.assign(n, inf());
    prv.assign(n, {});
    dists[s] = 0;
    repi(t, 2 * n)
    {
      repi(v, n)
      {
        if (dists[v] == inf())
          continue;
        fec(e : g.out_edges(v))
        {
          Dist nd = dists[v] == -inf() ? -inf() : dists[v] + e.cost;
          if (dists[e.to] > nd)
          {
            prv[e.to] = e;
            if (t == n - 1)
              dists[e.to] = -inf();
            else
              dists[e.to] = nd;
          }
        }
      }
    }
    return dists;
  }

  vc<Dist> solve(int s, int t = -1)
  {
    if constexpr (is_void_v<Cost>)
      return bfs(s, t);
    else
    {
      bool neg = false;
      int zcnt = 0;
      Dist wplus1 = -inf();
      Dist max_cost = 0;
      bool wpluscnt_geq2 = false;
      repi(v, g.size()) fec(e : g.out_edges(v))
      {
        if (e.cost < 0)
        {
          neg = true;
          break;
        }
        chmax(max_cost, e.cost);
      }
      repi(v, g.size()) fec(e : g.out_edges(v))
      {
        if (e.cost == 0)
          zcnt++;
        if (e.cost > 0)
        {
          if (wplus1 < 0)
            wplus1 = e.cost;
          else if (wplus1 != e.cost)
          {
            wpluscnt_geq2 = true;
            break;
          }
        }
      }
      if (neg)
        return bellman_ford(s);
      else if (wpluscnt_geq2)
      {
        const ll n = g.size(), m = g.num_of_edges();
        if (n * n < (m << 4))
          return dijkstra_dense(s, t);
        else
        {
          if constexpr (is_integral_ext<Dist>)
          {
            const u128 c = u128(max_cost);
            const ull lg = max<ull>(1, bit_width(ull(n)) - 1);
            if (c <= u128(m) * lg / u128(n))
              return dial(s, int(c), t);
          }
          return dijkstra(s, t);
        }
      }
      else
      {
        if (zcnt == 0)
          return bfs(s, t);
        else
          return bfs01(s, t);
      }
    }
  }

  vc<Edge<Cost>> prev_edges() const
  {
    const int n = g.size();
    assert(SZ(dists) == n && "solve(s, t) is not called");
    return prv;
  }

};

template <bool is_directed, class Cost, auto infty = nullptr>
struct AllPairsShortestPath
{
  using Dist = typename ShortestPath<is_directed, Cost, infty>::Dist;
  using Matrix = vvc<Dist>;

private:
  using SP = ShortestPath<is_directed, Cost, infty>;
  const Graph<is_directed, Cost> &g;
  static constexpr decltype(auto) inf() { return resolved_infty<Dist, infty>(); }
  Matrix dist;
  vvc<int> prev;
  vc<Edge<Cost>> by_id;
  bool negative = false, solved = false;

  enum class Method { repeated_sssp, floyd_warshall, johnson };

  Method select_method() const
  {
    if constexpr (is_void_v<Cost>) return Method::repeated_sssp;
    bool neg = false, multiple_positive = false, have_positive = false;
    Dist positive = 0;
    repi(v, g.size()) fec(e : g.out_edges(v))
    {
      neg |= e.cost < Dist(0);
      if (e.cost > Dist(0))
      {
        if (have_positive && e.cost != positive) multiple_positive = true;
        positive = e.cost, have_positive = true;
      }
    }
    if (!neg && !multiple_positive) return Method::repeated_sssp;
    const ll n = g.size(), m = g.num_of_edges();
    if (n * n < 16 * m) return Method::floyd_warshall;
    if (neg) return Method::johnson;
    return Method::repeated_sssp;
  }

  void init()
  {
    const int n = g.size();
    negative = false, solved = true;
    dist.assign(n, vc<Dist>(n, inf()));
    prev.assign(n, vc<int>(n, -1));
    by_id.resize(g.num_of_edges());
    fec(e : g.edges()) by_id[e.index] = e;
  }
  const Matrix &fail()
  {
    negative = true;
    dist.clear(), prev.clear();
    return dist;
  }

public:
  explicit AllPairsShortestPath(const Graph<is_directed, Cost> &g) : g(g) {}
  AllPairsShortestPath(const Graph<is_directed, Cost> &&) = delete;

  const Matrix &solve()
  {
    const auto method = select_method();
    if (method == Method::floyd_warshall) return floyd_warshall();
    if (method == Method::johnson) return johnson();
    return repeated_sssp();
  }

  const Matrix &repeated_sssp()
  {
    init();
    fec(e : by_id) assert(e.cost >= Dist(0));
    SP sp(g);
    repi(s, g.size())
    {
      dist[s] = sp.solve(s);
      const auto row = sp.prev_edges();
      repi(t, g.size()) prev[s][t] = row[t].index;
    }
    return dist;
  }

  const Matrix &floyd_warshall()
  {
    init();
    const int n = g.size();
    repi(v, n) dist[v][v] = 0;
    repi(v, n) fec(e : g.out_edges(v))
    {
      if (chmin(dist[v][e.to], e.cost)) prev[v][e.to] = e.index;
      if (dist[v][v] < Dist(0)) return fail();
    }
    repi(k, n) repi(i, n)
    {
      if (dist[i][k] == inf()) continue;
      repi(j, n)
      {
        if (dist[k][j] == inf()) continue;
        
        const Dist &a = dist[i][k], &b = dist[k][j];
        Dist nd;
        if (b < Dist(0) && a < -inf() - b) nd = -inf();
        else if (b > Dist(0) && a > inf() - b) nd = inf();
        else nd = a + b;
        if (chmin(dist[i][j], nd)) prev[i][j] = prev[k][j];
        if (i == j && dist[i][j] < Dist(0)) return fail();
      }
    }
    return dist;
  }

  const Matrix &johnson()
  {
    if constexpr (is_void_v<Cost>)
      return repeated_sssp();
    else
    {
      init();
      const int n = g.size();
      
      vc<Dist> h(n, Dist(0));
      repi(iter, n)
      {
        auto next = h;
        bool changed = false;
        repi(v, n) fec(e : g.out_edges(v))
          if (chmin(next[e.to], h[v] + e.cost)) changed = true;
        if (!changed) break;
        if (iter == n - 1) return fail();
        h.swap(next);
      }
      auto reweighted = by_id;
      for (auto &e : reweighted) e.cost = e.cost + h[e.from] - h[e.to];
      Graph<is_directed, Cost> reweighted_graph(n, reweighted);
      SP sp(reweighted_graph);
      repi(s, n)
      {
        dist[s] = sp.dijkstra(s);
        const auto row = sp.prev_edges();
        repi(t, n)
        {
          if (dist[s][t] != inf()) dist[s][t] = dist[s][t] - h[s] + h[t];
          prev[s][t] = row[t].index;
        }
      }
      return dist;
    }
  }

};

void main2()
{
  LL(N, M);
  VEC(ll, N, P);
  VEC(tlll, M, UVW);
  offset(UVW, tlll{-1, -1, 0});

  rep(i, N)
  {
    UVW.eb(i, i + N, P[i]);
    UVW.eb(i + N, i, P[i]);
  }
  GraphDirected G(2 * N, UVW);
  AllPairsShortestPath apsp(G);
  auto dists = apsp.solve();
  dump(dists | cp::index());

  ll mn = INF, cnt = 0;
  rep(i, N, 2 * N) rep(j, N, 2 * N) if (i != j)
  {
    if (chmin(mn, dists[i][j]))
      cnt = 0;
    if (dists[i][j] == mn)
      cnt++;
  }
  PRINT(mn, cnt);
}

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() {}