//! https://github.com/philip82148/cpp-dump

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

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

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

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

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

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

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

// https://github.com/miscalculation53/library/tree/wip/template/template_float128.hpp

using namespace std;

#if __cplusplus > 202002L && defined(__STDCPP_FLOAT128_T__)
#include <stdfloat>
using f128 = std::float128_t;
#elif defined(__SIZEOF_FLOAT128__)
using f128 = __float128;
#else
#error "f128 requires a compiler with IEEE binary128 support, such as GCC on x86_64."
#endif
#define LIBRARY_HAS_FLOAT128 1

#if __cplusplus > 202002L && defined(__STDCPP_FLOAT128_T__) && \
    (defined(_GLIBCXX_HAVE_FLOAT128_MATH) || defined(_GLIBCXX_LDOUBLE_IS_IEEE_BINARY128))
#define LIBRARY_USE_QUADMATH 0
#else
#include <quadmath.h>
#define LIBRARY_USE_QUADMATH 1

//! https://gcc.gnu.org/onlinedocs/libquadmath/Math-Library-Routines.html
#define _p_F128_UNARY(name) \
  inline f128 name(f128 x) { return f128(name##q(__float128(x))); }
_p_F128_UNARY(acos)
_p_F128_UNARY(acosh)
_p_F128_UNARY(asin)
_p_F128_UNARY(asinh)
_p_F128_UNARY(atan)
_p_F128_UNARY(atanh)
_p_F128_UNARY(cbrt)
_p_F128_UNARY(ceil)
_p_F128_UNARY(cos)
_p_F128_UNARY(cosh)
_p_F128_UNARY(erf)
_p_F128_UNARY(erfc)
_p_F128_UNARY(exp)
_p_F128_UNARY(exp2)
_p_F128_UNARY(expm1)
_p_F128_UNARY(fabs)
_p_F128_UNARY(floor)
_p_F128_UNARY(lgamma)
_p_F128_UNARY(log)
_p_F128_UNARY(log10)
_p_F128_UNARY(log1p)
_p_F128_UNARY(log2)
_p_F128_UNARY(logb)
_p_F128_UNARY(nearbyint)
_p_F128_UNARY(rint)
_p_F128_UNARY(round)
_p_F128_UNARY(sin)
_p_F128_UNARY(sinh)
_p_F128_UNARY(sqrt)
_p_F128_UNARY(tan)
_p_F128_UNARY(tanh)
_p_F128_UNARY(tgamma)
_p_F128_UNARY(trunc)
#undef _p_F128_UNARY

template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline f128 abs(T x);

#define _p_F128_BINARY(name) \
  inline f128 name(f128 x, f128 y) { return f128(name##q(__float128(x), __float128(y))); } \
  template <class T, enable_if_t<is_arithmetic_v<T>, int> = 0> \
  inline f128 name(f128 x, T y) { return name(x, f128(y)); } \
  template <class T, enable_if_t<is_arithmetic_v<T>, int> = 0> \
  inline f128 name(T x, f128 y) { return name(f128(x), y); }
_p_F128_BINARY(atan2)
_p_F128_BINARY(copysign)
_p_F128_BINARY(fdim)
_p_F128_BINARY(fmax)
_p_F128_BINARY(fmin)
_p_F128_BINARY(fmod)
_p_F128_BINARY(hypot)
_p_F128_BINARY(nextafter)
_p_F128_BINARY(pow)
_p_F128_BINARY(remainder)
#undef _p_F128_BINARY

inline f128 fma(f128 x, f128 y, f128 z)
{ return f128(fmaq(__float128(x), __float128(y), __float128(z))); }
inline f128 hypot(f128 x, f128 y, f128 z)
{ return hypot(hypot(x, y), z); }
namespace template_float128_detail
{
template <class T>
constexpr bool arithmetic = is_arithmetic_v<T> || is_same_v<T, f128>;
}
#define _p_F128_TERNARY_MIXED(name) \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && template_float128_detail::arithmetic<U>, int> = 0> \
  inline f128 name(f128 x, T y, U z) { return name(x, f128(y), f128(z)); } \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && !is_same_v<T, f128> && template_float128_detail::arithmetic<U>, int> = 0> \
  inline f128 name(T x, f128 y, U z) { return name(f128(x), y, f128(z)); } \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && !is_same_v<T, f128> && template_float128_detail::arithmetic<U> && !is_same_v<U, f128>, int> = 0> \
  inline f128 name(T x, U y, f128 z) { return name(f128(x), f128(y), z); }
_p_F128_TERNARY_MIXED(fma)
_p_F128_TERNARY_MIXED(hypot)
#undef _p_F128_TERNARY_MIXED
inline f128 frexp(f128 x, int *exponent);
inline f128 ldexp(f128 x, int exponent)
{ return f128(ldexpq(__float128(x), exponent)); }
inline f128 scalbn(f128 x, int exponent)
{ return f128(scalbnq(__float128(x), exponent)); }
inline f128 scalbln(f128 x, long exponent)
{ return f128(scalblnq(__float128(x), exponent)); }
inline f128 modf(f128 x, f128 *integer)
{
  __float128 value;
  const auto fraction = modfq(__float128(x), &value);
  *integer = f128(value);
  return f128(fraction);
}
inline f128 remquo(f128 x, f128 y, int *quotient)
{ return f128(remquoq(__float128(x), __float128(y), quotient)); }
inline long lrint(f128 x) { return lrintq(__float128(x)); }
inline long lround(f128 x) { return lroundq(__float128(x)); }
inline long long llrint(f128 x) { return llrintq(__float128(x)); }
inline long long llround(f128 x) { return llroundq(__float128(x)); }
inline int ilogb(f128 x) { return ilogbq(__float128(x)); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isfinite(T x);
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isinf(T x);
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isnan(T x);
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool signbit(T x);
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isnormal(T x);
#endif

inline f128 nexttoward(f128 x, long double y)
{ return nextafter(x, f128(y)); }

namespace template_float128_detail
{
inline bool parse(string_view s, f128 &x)
{
  if (!s.empty() && s.front() == '+')
  {
    s.remove_prefix(1);
    if (s.empty() || s.front() == '-') return false;
  }
#if LIBRARY_USE_QUADMATH
  string token(s);
  char *end;
  errno = 0;
  const __float128 value = strtoflt128(token.c_str(), &end);
  if (end == token.data() || end != token.data() + token.size()) return false;
  if (errno == ERANGE && (value == 0 || isinfq(value))) return false;
  x = f128(value);
  return true;
#else
  auto [p, ec] = from_chars(s.data(), s.data() + s.size(), x);
  return ec == errc{} && p == s.data() + s.size();
#endif
}

inline string format(f128 x, chars_format style, int precision)
{
  string s(128, '\0');
  while (true)
  {
#if LIBRARY_USE_QUADMATH
    const char *spec = style == chars_format::fixed ? "%.*Qf" :
                       style == chars_format::scientific ? "%.*Qe" : "%.*Qg";
    const int n = quadmath_snprintf(s.data(), s.size(), spec, precision, __float128(x));
    if (n < 0) abort();
    if (size_t(n) < s.size())
    {
      s.resize(size_t(n));
      return s;
    }
    s.resize(size_t(n) + 1);
#else
    auto [p, ec] = to_chars(s.data(), s.data() + s.size(), x, style, precision);
    if (ec == errc{})
    {
      s.resize(size_t(p - s.data()));
      return s;
    }
    s.resize(s.size() * 2);
#endif
  }
}
} 

using ld = decltype(EPS);

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

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

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

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

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (size_t i = size_t(neg); i < s.size(); i++)
    res = 10 * res + unsigned(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 = u128(x);
  while (ux > 0)
  {
    res += char('0' + ux % 10);
    ux /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  if (!(is >> s)) return is;
  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 LIBRARY_REPEAT_LOOP(T,i,l,r,d) for (auto [i, library_repeat_end, library_repeat_step] = array<T, 3>{T(l), T(r), T(d)}; library_repeat_step > 0 ? i < library_repeat_end : i > library_repeat_end; i += library_repeat_step)

#define rep1(i,n) LIBRARY_REPEAT_LOOP(ll, i, 0, n, 1)
#define rep2(i,l,r) LIBRARY_REPEAT_LOOP(ll, i, l, r, 1)
#define rep3(i,l,r,d) LIBRARY_REPEAT_LOOP(ll, i, l, r, d)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i,n) LIBRARY_REPEAT_LOOP(int, i, 0, n, 1)
#define repi2(i,l,r) LIBRARY_REPEAT_LOOP(int, i, l, r, 1)
#define repi3(i,l,r,d) LIBRARY_REPEAT_LOOP(int, i, l, r, 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>
struct make_unsigned_ext
{
  using type = make_unsigned_t<T>;
};
template <>
struct make_unsigned_ext<i128>
{
  using type = u128;
};
template <>
struct make_unsigned_ext<u128>
{
  using type = u128;
};
template <class T>
using make_unsigned_ext_t = typename make_unsigned_ext<T>::type;

template <class T>
struct make_signed_ext
{
  using type = make_signed_t<T>;
};
template <>
struct make_signed_ext<u128>
{
  using type = i128;
};
template <class T>
using make_signed_ext_t = typename make_signed_ext<T>::type;

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;
}

template <class T>
constexpr make_unsigned_ext_t<T> unsigned_abs(T x);

template <class T, bool = is_integral_ext<T>>
struct unsigned_arithmetic { using type = T; };

template <class T>
struct unsigned_arithmetic<T, true>
{
  static_assert(!is_same_v<T, bool>);
  using type = common_type_t<unsigned int, make_unsigned_ext_t<T>>;
};

template <class T>
using unsigned_arithmetic_t = typename unsigned_arithmetic<T>::type;

// 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();

template <class T, class U>
inline bool chmin(T &a, U b);

template <class T, enable_if_t<is_same_v<T, i128> || is_same_v<T, u128>, int> = 0>
inline constexpr T abs(T x);

namespace template_math_detail
{
template <class T, class U>
constexpr common_type_t<T, U> gcd(T a, U b);

template <class T, class U>
constexpr common_type_t<T, U> lcm(T a, U b);
} 

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool>, int> = 0>
constexpr common_type_t<i128, T> gcd(i128 a, T b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool> && !is_same_v<T, i128> && !is_same_v<T, u128>, int> = 0>
constexpr common_type_t<T, i128> gcd(T a, i128 b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool>, int> = 0>
constexpr common_type_t<u128, T> gcd(u128 a, T b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool> && !is_same_v<T, i128> && !is_same_v<T, u128>, int> = 0>
constexpr common_type_t<T, u128> gcd(T a, u128 b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool>, int> = 0>
constexpr common_type_t<i128, T> lcm(i128 a, T b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool> && !is_same_v<T, i128> && !is_same_v<T, u128>, int> = 0>
constexpr common_type_t<T, i128> lcm(T a, i128 b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool>, int> = 0>
constexpr common_type_t<u128, T> lcm(u128 a, T b);

template <class T, enable_if_t<is_integral_ext<T> && !is_same_v<T, bool> && !is_same_v<T, i128> && !is_same_v<T, u128>, int> = 0>
constexpr common_type_t<T, u128> lcm(T a, u128 b);

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;
}

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b);
template <class T = ll, class A, class B, class M>
T mul_limited(A a, B b, M m);
template <class T = ll, class A, class B>
T mul_limited(A a, B b);
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m);
template <class T = ll, class A, class B>
T pow_limited(A a, B b);

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k);

//! https://misawa.github.io/others/avoid_errors/techniques_to_avoid_errors.html

template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base);

template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base, int n);
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base);
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base, int n);
// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

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

#define LMD(x,fx) ([&](const auto &x) { return fx; })
template <class F>
auto gen_vec(int n, const F &f);
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

namespace template_vector_detail
{
template <class T, size_t depth>
struct nested_vector
{
  using type = vc<typename nested_vector<T, depth - 1>::type>;
};
template <class T>
struct nested_vector<T, 0>
{
  using type = T;
};

template <class T, size_t depth, class V>
void populate_dimensions(typename nested_vector<T, depth>::type &result, const V *sizes, const T &init);

template <class T, size_t depth, class V>
typename nested_vector<T, depth>::type build_dimensions(const V *sizes, const T &init);
}

template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init);

template <class T, size_t d, size_t i = 0>
auto dvec(const ll (&sz)[d], const T &init);

template <class V, class T>
void fill(V &v, const T &value);

template <class T = ll>
optional<T> ctol(const char &c, const string &s);
template <class T = ll>
vc<T> stov(const string &s, char first);
template <class T = ll>
optional<vc<T>> stov(const string &s, const string &t);
template <class T>
string vtos(const vc<T> &v, char first);
template <class T>
string vtos(const vc<T> &v, const string &t);

template <class T>
vc<T> concat(const vvc<T> &vs);
template <class T>
vc<T> concat(const vc<T> &v);
template <class T, class... Ts>
vc<T> concat(vc<T> v, const vc<Ts> &...vs);

// 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();

template <class V>
auto SUM(const V &v);
template <class T, class V>
T SUM(const V &v);
template <class V>
auto MAX(const V &v);

template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p);

template <class T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs);

template <class V>
V reversed(const V &v);

template <class V, class Equal = equal_to<>>
void unique(V &v, Equal equal = {});

template <class V, class Compare = less<>, class Equal = equal_to<>>
void sortunique(V &v, Compare comp = {}, Equal equal = {});

template <class V, class U>
void rotate(V &v, U k);

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);
  static constexpr S e();
  template <class I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k);
};
template <class T, auto infty = nullptr>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b);
  static constexpr decltype(auto) e();
  template <class I>
  static constexpr S pow(const S &a, I k);
};
template <class T, auto infty = nullptr>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b);
  static constexpr S e();
  template <class I>
  static constexpr S pow(const S &a, I k);
};

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, class I>
typename M::S pow_monoid(typename M::S a, I k);

template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0);

template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0);

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;

template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {});

template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {});
#define DEFAULT_COMP ranges::less

template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, optional<T>>;
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, optional<T>>;
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, optional<T>>;
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, optional<T>>;

template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>;

template <class V, class Value>
inline auto lt_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, optional<typename V::const_iterator>>;
template <class V, class Value>
inline auto leq_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, optional<typename V::const_iterator>>;
template <class V, class Value>
inline auto gt_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, optional<typename V::const_iterator>>;
template <class V, class Value>
inline auto geq_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, optional<typename V::const_iterator>>;

namespace internal
{
template <class T>
bool binsearch_adjacent(T a, T b);
};

template <class T = ll, class Judge, class InitOk, class InitNg>
pair<T, T> binsearch(const Judge &judge, InitOk init_ok, InitNg init_ng, bool check_ok = true, bool check_ng = true);

// 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

#if defined(LOCAL) && LIBRARY_HAS_FLOAT128
namespace cpp_dump::_detail
{
  struct export_command;
  inline string export_var(const f128 &, const string &, size_t,
                           size_t, bool, const export_command &);
}
#endif

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

/*!
 * Portions adapted from https://github.com/sassan-dev/cpp-dump/tree/aabd98dd8a4b5ae3f62cf12387a68a9f6591197e
 *
 * MIT License
 *
 * Copyright (c) 2023 Ryota Sasaki
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in all
 * copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 * SOFTWARE.
 */

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

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

#if LIBRARY_HAS_FLOAT128
inline istream &operator>>(istream &is, f128 &x)
{
  string s;
  if (is >> s)
  {
    f128 value;
    if (template_float128_detail::parse(s, value)) x = value;
    else is.setstate(ios::failbit);
  }
  return is;
}

inline ostream &operator<<(ostream &os, f128 x)
{
  const auto field = os.flags() & ios::floatfield;
  const auto format = field == ios::fixed ? chars_format::fixed :
                      field == ios::scientific ? chars_format::scientific : chars_format::general;
  string s = template_float128_detail::format(x, format, int(os.precision()));
  if ((os.flags() & ios::showpos) && !signbit(x)) s.insert(s.begin(), '+');
  if (os.flags() & ios::uppercase)
    for (char &c : s) c = char(toupper(static_cast<unsigned char>(c)));
  return os << s;
}
#endif

namespace fastio
{
  constexpr uint32_t SIZ = 1 << 17;
  inline char ibuf[SIZ + 1], obuf[SIZ];
  inline uint32_t pil = 0, pir = 0, por = 0;

  __attribute__((noinline)) inline void load()
  {
    const uint32_t remaining = pir - pil;
    if (remaining && pil) memmove(ibuf, ibuf + pil, remaining);
    pil = 0;
    pir = remaining + uint32_t(fread(ibuf + remaining, 1, SIZ - remaining, stdin));
    ibuf[pir] = '\x7f';
  }

  inline bool ascii_space(unsigned char c)
  {
    return c == ' ' || unsigned(c - '\t') < 5;
  }

  inline void find_token()
  {
    for (;;)
    {
      const char *p = ibuf + pil;
      const char *end = ibuf + pir;
      while (ascii_space(*p)) ++p;
      pil = uint32_t(p - ibuf);
      if (p != end) return;
      load();
      if (!pir) abort();
    }
  }

  __attribute__((noinline)) inline void flush()
  {
    size_t written = 0;
    while (written < por)
    {
      const size_t n = fwrite(obuf + written, 1, por - written, stdout);
      if (!n) abort();
      written += n;
    }
    por = 0;
  }

  inline void append_bytes(const char *data, size_t size)
  {
    while (size)
    {
      const size_t n = min(size, size_t(SIZ - por));
      memcpy(obuf + por, data, n);
      por += uint32_t(n);
      data += n;
      size -= n;
      if (por == SIZ) flush();
    }
  }

  inline void append_char(char c)
  {
    if (por == SIZ) flush();
    obuf[por++] = c;
  }

  template <class T>
  using integer_magnitude = conditional_t<(sizeof(T) > sizeof(ull)), u128,
                            conditional_t<(sizeof(T) > sizeof(uint)), ull, uint>>;

  inline uint decode_eight(uint64_t columns)
  {
    columns &= 0x0f0f0f0f0f0f0f0fULL;
    const uint64_t pairs = (columns & 0x00ff00ff00ff00ffULL) * 10
                         + ((columns >> 8) & 0x00ff00ff00ff00ffULL);
    const uint64_t quads = (pairs & 0x0000ffff0000ffffULL) * 100
                         + ((pairs >> 16) & 0x0000ffff0000ffffULL);
    return uint(quads) * 10000 + uint(quads >> 32);
  }

  inline constexpr uint powers_of_ten[8] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000};

  template <class T>
  __attribute__((always_inline)) inline void rd1_integer(T &x);

  template <class T, enable_if_t<numeric_limits<T>::is_integer, int> = 0>
  __attribute__((always_inline)) inline void rd1(T &x);
  inline void rd1(char &x)
  {
    find_token();
    x = ibuf[pil++];
  }
  inline void rd1(string &x)
  {
    find_token();
    x.clear();
    for (;;)
    {
      const char *p = ibuf + pil;
      const char *end = ibuf + pir;
      const char *first = p;
      while (p != end && !ascii_space(*p)) ++p;
      x.append(first, size_t(p - first));
      pil = uint32_t(p - ibuf);
      if (p != end) return;
      load();
      if (!pir) return;
    }
  }

  template <class T>
  void rd1_real(T &x)
  {
    string token;
    rd1(token);
    const char *first = token.data();
    const char *last = first + token.size();
    if (first != last && *first == '+') ++first;
    auto result = from_chars(first, last, x, chars_format::general);
    if (result.ec != errc{} || result.ptr != last)
    {
      char *end;
      if constexpr (is_same_v<T, float>) x = strtof(token.c_str(), &end);
      else if constexpr (is_same_v<T, double>) x = strtod(token.c_str(), &end);
      else x = strtold(token.c_str(), &end);
      if (end != token.data() + token.size()) abort();
    }
  }
  inline void rd1(float &x) { rd1_real(x); }
  inline void rd1(double &x) { rd1_real(x); }
  inline void rd1(long double &x) { rd1_real(x); }
#if LIBRARY_HAS_FLOAT128
  inline void rd1(f128 &x)
  {
    string token;
    rd1(token);
    if (!template_float128_detail::parse(token, x)) abort();
  }
#endif

  template <class A, class B> void rd1(pair<A, B> &x);
  template <class... T> void rd1(tuple<T...> &x);
  template <class T, size_t N> void rd1(array<T, N> &x);
  template <class T> void rd1(vc<T> &x);

  template <class A, class B>
  void rd1(pair<A, B> &x);
  template <class... T>
  void rd1(tuple<T...> &x);
  template <class T, size_t N>
  void rd1(array<T, N> &x);
  template <class T>
  void rd1(vc<T> &x);

  inline constexpr auto decimal_triplets = []
  {
    array<array<char, 3>, 1000> result{};
    for (unsigned i = 0; i < 1000; ++i)
    {
      result[i][0] = char('0' + i / 100);
      result[i][1] = char('0' + i / 10 % 10);
      result[i][2] = char('0' + i % 10);
    }
    return result;
  }();

  inline void store_nine(char *out, uint x)
  {
    const uint q = x / 1000, r = q / 1000;
    memcpy(out, decimal_triplets[r].data(), 3);
    memcpy(out + 3, decimal_triplets[q - r * 1000].data(), 3);
    memcpy(out + 6, decimal_triplets[x - q * 1000].data(), 3);
  }

  inline char *prepend_small(char *end, uint x)
  {
    while (x >= 1000)
    {
      const uint q = x / 1000;
      end -= 3;
      memcpy(end, decimal_triplets[x - q * 1000].data(), 3);
      x = q;
    }
    const unsigned digits = 1 + unsigned(x >= 10) + unsigned(x >= 100);
    end -= digits;
    memcpy(end, decimal_triplets[x].data() + 3 - digits, digits);
    return end;
  }

  inline char *prepend_wide(char *end, ull x)
  {
    while (x >= 1000000000ULL)
    {
      const ull q = x / 1000000000ULL;
      end -= 9;
      store_nine(end, uint(x - q * 1000000000ULL));
      x = q;
    }
    return prepend_small(end, uint(x));
  }

  inline char *store_leading_triplet(char *out, uint x)
  {
    if (x >= 100)
    {
      memcpy(out, decimal_triplets[x].data(), 3);
      return out + 3;
    }
    if (x >= 10)
    {
      memcpy(out, decimal_triplets[x].data() + 1, 2);
      return out + 2;
    }
    *out = char('0' + x);
    return out + 1;
  }

  inline char *store_short(char *out, uint x)
  {
    if (x >= 1000000)
    {
      const uint q = x / 1000000;
      const uint r = x - q * 1000000;
      out = store_leading_triplet(out, q);
      memcpy(out, decimal_triplets[r / 1000].data(), 3);
      memcpy(out + 3, decimal_triplets[r % 1000].data(), 3);
      return out + 6;
    }
    if (x >= 1000)
    {
      const uint q = x / 1000;
      out = store_leading_triplet(out, q);
      memcpy(out, decimal_triplets[x - q * 1000].data(), 3);
      return out + 3;
    }
    return store_leading_triplet(out, x);
  }

  template <class U>
  inline char *store_ordinary(char *out, U x);

  template <class T, bool newline = false>
  __attribute__((always_inline)) inline void wt1_integer(T x);

  template <class T, enable_if_t<numeric_limits<T>::is_integer, int> = 0>
  __attribute__((always_inline)) inline void wt1(T x);
  inline void wt1(char x) { append_char(x); }
  inline void wt1(const char *x) { append_bytes(x, strlen(x)); }
  inline void wt1(string_view x) { append_bytes(x.data(), x.size()); }
  inline void wt1(const string &x) { append_bytes(x.data(), x.size()); }

  template <class T>
  void wt1_real(T x)
  {
    char text[numeric_limits<T>::max_exponent10 + 64];
    auto result = to_chars(text, text + sizeof(text), x, chars_format::fixed, 15);
    if (result.ec != errc{}) abort();
    append_bytes(text, size_t(result.ptr - text));
  }
  inline void wt1(float x) { wt1_real(x); }
  inline void wt1(double x) { wt1_real(x); }
  inline void wt1(long double x) { wt1_real(x); }
#if LIBRARY_HAS_FLOAT128
  inline void wt1(f128 x)
  {
    wt1(template_float128_detail::format(x, chars_format::fixed, 35));
  }
#endif

  template <class A, class B> void wt1(const pair<A, B> &x);
  template <class... T> void wt1(const tuple<T...> &x);
  template <class T, size_t N> void wt1(const array<T, N> &x);
  template <class T> void wt1(const vc<T> &x);

  template <class A, class B>
  void wt1(const pair<A, B> &x);
  template <class... T>
  void wt1(const tuple<T...> &x);
  template <class T, size_t N>
  void wt1(const array<T, N> &x);
  template <class T>
  void wt1(const vc<T> &x);

  template <class... T>
  void read(T &...x) { (rd1(x), ...); }
  template <class... T>
  void write(T &&...x) { (wt1(forward<T>(x)), ...); }
  template <class... T>
  void print(T &&...x)
  {
    if constexpr (sizeof...(T) == 1 &&
                  ((is_integral_v<decay_t<T>> && !is_same_v<decay_t<T>, char>) && ...))
      (wt1_integer<decay_t<T>, true>(forward<T>(x)), ...);
    else
    {
      size_t index = 0;
      ((index++ ? void(append_char(' ')) : void(), wt1(forward<T>(x))), ...);
      append_char('\n');
    }
  }
}

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

//! https://trap.jp/post/1224/

namespace internal
{

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

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

template <class... T>
void READVEC2nodump(int n, int m, vvc<T> &...v);

template <class... T>
void READJAGnodump(int n, vvc<T> &...vs);

}; 

#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)

template <class T>
void PRINTV(const vc<T> &v);
#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)

template <class T, class U, class P>
pair<T, U> &operator+=(pair<T, U> &a, const P &b);
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> a, const P &b);
template <class T, class U, class P>
pair<T, U> &operator-=(pair<T, U> &a, const P &b);
template <class T, class U, class P>
pair<T, U> operator-(pair<T, U> a, const P &b);
template <class T, class U>
pair<T, U> operator-(pair<T, U> a);

template <class T, size_t n, class A>
array<T, n> &operator+=(array<T, n> &a, const A &b);
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> a, const A &b);
template <class T, size_t n, class A>
array<T, n> &operator-=(array<T, n> &a, const A &b);
template <class T, size_t n, class A>
array<T, n> operator-(array<T, n> a, const A &b);
template <class T, size_t n>
array<T, n> operator-(array<T, n> a);

namespace internal
{

template <size_t... I, class A, class B>
auto &tuple_add_impl(A &a, const B &b, const index_sequence<I...>);
template <size_t... I, class A, class B>
auto &tuple_sub_impl(A &a, const B &b, const index_sequence<I...>);
template <size_t... I, class A>
auto &tuple_neg_impl(A &a, const index_sequence<I...>);

}; 

template <class... Ts, class Tp>
tuple<Ts...> &operator+=(tuple<Ts...> &a, const Tp &b);
template <class... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> a, const Tp &b);
template <class... Ts, class Tp>
tuple<Ts...> &operator-=(tuple<Ts...> &a, const Tp &b);
template <class... Ts, class Tp>
tuple<Ts...> operator-(tuple<Ts...> a, const Tp &b);
template <class... Ts>
tuple<Ts...> operator-(tuple<Ts...> a);

template <class T, class Add>
void offset(vc<T> &v, const Add &add);
template <class T, class Add>
void offset(vvc<T> &v, const Add &add);

template <class T, const size_t m>
array<vc<T>, m> unzip(const vc<array<T, m>> &vt);
template <class T, const size_t m>
vc<array<T, m>> zip(const array<vc<T>, m> &tv);

template <class T, class U>
pair<vc<T>, vc<U>> unzip(const vc<pair<T, U>> &vt);
template <class T, class U>
vc<pair<T, U>> zip(const pair<vc<T>, vc<U>> &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);

template <size_t... I, class Tp>
auto tv_to_vt_impl(const Tp &tv, index_sequence<I...>, size_t index);

};

template <class... Ts>
auto unzip(const vc<tuple<Ts...>> &vt);

template <class... Ts>
auto zip(const tuple<vc<Ts>...> &tv);

namespace internal
{

template <class... Ts>
auto zip_vectors(const vc<Ts> &...v);

};

#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;

template <class T = ll, class U1, class U2>
T randint(U1 l, U2 r);

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

namespace internal
{
template <bool does_sort, class V, class T>
void random_sample_range(V &res, T l, T r);
}; 

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

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

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): modular arithmetic operators.
 * Upstream: atcoder/modint.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

// 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>
  [[deprecated("larger_int: 128-bit integer is not widened; intermediate arithmetic may overflow")]]
  constexpr bool warn_no_wider_integer();
}

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

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): primality testing.
 * Upstream: atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

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 <class T>
constexpr bool isprime_constexpr(T n);

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 mod_type mod();
  static constexpr value_type umod();

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v);
  static constexpr value_type mul(value_type a, value_type b);
};

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

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): Barrett reduction.
 * Upstream: atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

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 void set_mod(mod_type m);
  static mod_type mod();
  static value_type umod();
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v);
  static value_type mul(value_type a, value_type b);
};

};
// 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); }
};

//! https://www.mathenachia.blog/even-mod-montgomery-impl/
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 void set_mod(mod_type m);
  static mod_type mod();
  static value_type umod();
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
  static value_type mul(value_type a, value_type b);
};

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 void set_mod(mod_type m);
  static mod_type mod();
  static value_type umod();
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
  static value_type mul(value_type a, value_type b);
};

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

template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b);

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:
    static constexpr M mod();

    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m));

    static mint raw(V v);

    modint_impl();

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

    M val() const;

    mint &operator+=(const mint &rhs);
    mint &operator-=(const mint &rhs);
    mint &operator*=(const mint &rhs);
    mint &operator/=(const mint &rhs);

    mint &operator++();
    mint &operator--();
    mint operator++(int);
    mint operator--(int);
    mint operator+() const;
    mint operator-() const;

    template <class T>
    mint pow(T n) const;
    mint inv() 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, 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;

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
{
  bool operator()(const mint &a, const mint &b) const;
};

template <class mint>
struct modint_hash
{
  auto operator()(const mint &x) const;
};
// 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:
  PowerTable();
  PowerTable(const mint &base);

  void reserve(int n);

  mint pow(int n);
};
// 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_;
  static void reserve(int n);
  static T inv(int n);

  static T P(int n, int k);
  static T C(int n, int k);
};
// 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

template <class T = ll, class U = larger_int_t<T>>
pair<T, T> svp2d(const pair<T, T> &a, const pair<T, T> &b);

template <class mint>
pair<decltype(mint(0).val()), decltype(mint(0).val())>
mint_to_rat(const mint &x);

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;
    }
  };

  struct mint_to_rat_fn
  {
    template <class T>
      requires requires(const T &x)
      {
        { T::mod() } -> std::convertible_to<ll>;
        { x.val() } -> std::convertible_to<ll>;
      }
    constexpr auto operator()(const T &x) const
        -> rat_value<typename decltype(mint_to_rat(x))::first_type>;
  };

  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>
  {
    template <class T>
    static constexpr bool can_convert();

    template <typename T>
      requires(!requires { std::views::all | std::declval<T>(); })
    constexpr decltype(auto) operator()(T &&t) const;
  };

  template <typename T>
    requires(!requires { std::views::all | std::declval<T>(); })
  constexpr decltype(auto) operator|(T &&t, rat_closure c);

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

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

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

void main2()
{
  CHAR(a, b);
  PRINT(b, a);
}

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() {}


// submit-code: hash=13ed6f8ab951 source=13ed6f8ab951033f65aac603031bc0e4ce7163d2eaa0a96c902dd563261629f0 input=db002f8e0d4d8bcb0422ea31f88c0f9ac0bfc257d0e8a9cf9199e75815778c04 body=f6099eab02f9118c772f27bc1173900867c2053dbb95b22acc1876451570251d