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

#ifndef LOCAL
#define FAST_IO
// #define FAST_CIO
// #define INTERACTIVE
#endif

#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

/**
 * @brief テンプレート（型）
 * @docs docs/template/template_types.md
 */

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

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

using namespace std;

/**
 * @brief 四倍精度浮動小数点数
 * @docs docs/template/template_types.md
 */

#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) { return f128(fabsq(__float128(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)
{ return f128(frexpq(__float128(x), 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) { return finiteq(__float128(x)); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isinf(T x) { return isinfq(__float128(x)); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isnan(T x) { return isnanq(__float128(x)); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool signbit(T x) { return signbitq(__float128(x)); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline bool isnormal(T x)
{ return isfinite(x) && abs(x) >= numeric_limits<f128>::min(); }
template <class T, enable_if_t<is_same_v<T, f128>, int> = 0>
inline int fpclassify(T x)
{
  if (isnan(x)) return FP_NAN;
  if (isinf(x)) return FP_INFINITE;
  if (x == 0) return FP_ZERO;
  return isnormal(x) ? FP_NORMAL : FP_SUBNORMAL;
}
#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
  }
}
} // namespace template_float128_detail

#ifndef EPS
#define EPS 1e-11
#endif
using ld = decltype(EPS);

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

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

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

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

#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  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;
}
#endif

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


/**
 * @brief テンプレート（rep）
 * @docs docs/template/template_rep.md
 */

#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

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


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


/**
 * @brief $128$ ビット整数を含めた整数判定
 * @docs docs/utils/is_integral_ext.md
 */

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


/**
 * @brief $128$ ビット整数に対応した make_singed と make_unsigned
 * @docs docs/utils/make_unsigned_ext.md
 */

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;

/**
 * @brief 整数の符号・絶対値・剰余を保つ型変換
 * @docs docs/utils/integer_arithmetic.md
 */

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)
{
  static_assert(is_integral_ext<T> && !is_same_v<T, bool>);
  using U = make_unsigned_ext_t<T>;
  return is_negative(x) ? U(U(0) - U(x)) : U(x);
}

// 整数は整数昇格後も unsigned となる型、それ以外は元の型。
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;

// mod 2^w (w は R の幅) で合同な値を返す。
template <class R, class T>
constexpr R wrapping_cast(T x)
{
  static_assert(is_integral_ext<R> && !is_same_v<R, bool>);
  static_assert(is_integral_ext<T> && !is_same_v<T, bool>);
  if constexpr (is_signed_ext<R>)
  {
    using U = make_unsigned_ext_t<R>;
    const U bits = U(x);
    if (bits <= U(numeric_limits<R>::max())) return R(bits);
    return R(-1) - R(U(~bits));
  }
  else return R(x);
}
// https://github.com/miscalculation53/library/tree/wip/utils/default_infty.hpp


#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif

/**
 * @brief 型ごとの既定の無限大
 * @docs docs/utils/default_infty.md
 */

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

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

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

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

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

/**
 * @brief テンプレート（演算）
 * @docs docs/template/template_math.md
 */

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, enable_if_t<is_same_v<T, i128> || is_same_v<T, u128>, int> = 0>
inline constexpr T abs(T x) { return is_negative(x) ? -x : x; }

namespace template_math_detail
{
template <class T, class U>
constexpr common_type_t<T, U> gcd(T a, U b)
{
  using R = common_type_t<T, U>;
  using W = make_unsigned_ext_t<R>;
  W x = W(unsigned_abs(a)), y = W(unsigned_abs(b));
  while (y != 0)
  {
    W r = x % y;
    x = y, y = r;
  }
  return R(x);
}

template <class T, class U>
constexpr common_type_t<T, U> lcm(T a, U b)
{
  using R = common_type_t<T, U>;
  using W = make_unsigned_ext_t<R>;
  W x = W(unsigned_abs(a)), y = W(unsigned_abs(b));
  return x == 0 || y == 0 ? R(0) : R(x / template_math_detail::gcd(x, y) * y);
}
} // namespace template_math_detail

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) { return template_math_detail::gcd(a, 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) { return template_math_detail::gcd(a, 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) { return template_math_detail::gcd(a, 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) { return template_math_detail::gcd(a, 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) { return template_math_detail::lcm(a, 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) { return template_math_detail::lcm(a, 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) { return template_math_detail::lcm(a, 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) { return template_math_detail::lcm(a, b); }

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 divceil(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)
{
  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)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;

  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B, class M>
T mul_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (b == 0)
    return 0;
  return T(a) > T(m) / T(b) ? T(m) : T(a) * T(b);
}
template <class T = ll, class A, class B>
T mul_limited(A a, B b) { return mul_limited<T>(a, b, default_infty<T>()); }
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (a <= 1 || b == 0)
    return min(ipow<T>(a, b), T(m));
  
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
    {
      if (res > T(m) / tmp)
        return m;
      res *= tmp;
    }
    b >>= 1;
    if (b == 0)
      break;
    if (tmp > T(m) / tmp)
      return m;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B>
T pow_limited(A a, B b) { return pow_limited<T>(a, b, default_infty<T>()); }

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;
  if (k == 2)
  {
    const T aa = T(a);
    T x = T(sqrtl((long double)a));
    while (x > aa / x)
      x--;
    while (x < numeric_limits<T>::max())
    {
      const T y = x + 1;
      if (y > aa / y)
        break;
      x = y;
    }
    return x;
  }

  auto isok = [&](T x) -> bool
  {
    if (x == 0)
      return true;
    T res = 1, k2 = k;
    while (true)
    {
      if (k2 & 1)
      {
        if (res > T(a) / x)
          return false;
        res *= x;
      }
      k2 >>= 1;
      if (k2 == 0)
        break;
      if (x > T(a) / x)
        return false;
      x *= x;
    }
    return res <= T(a);
  };

  T x = pow(a, 1.0 / k);
  bool up = true;
  while (!isok(x))
    up = false, x--;
  if (up)
  {
    while (x < numeric_limits<T>::max() && isok(x + 1))
      x++;
  }
  return x;
}
template <class T = ll, class A, class K>
constexpr T iroot_ceil(A a, K k)
{
  T x = iroot<T>(a, k);
  return ipow<T>(x, k) == a ? x : x + 1;
}

//! https://misawa.github.io/others/avoid_errors/techniques_to_avoid_errors.html
template <class D = decltype(EPS), class A>
int SGN(A a, D eps = EPS) { return int(a > eps) - int(a < -eps); }

// 位取り記数法と同じ順番（下位桁が後ろ）
// 0 に対しては {0} が返る
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base)
{
  assert(val >= 0);
  assert(base >= 2);
  if (val == 0)
    return {0};
  vc<T> a;
  while (val > 0)
  {
    a.emplace_back(val % base);
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
// 位取り記数法と同じ順番（下位桁が後ろ）
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base, int n)
{
  assert(val >= 0);
  assert(base >= 2);
  assert(n >= 0);
  vc<T> a(n);
  repi(i, n)
  {
    a[i] = val % base;
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  auto a = base_repr(val, base);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base, int n)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  assert(n >= 0);
  auto a = base_repr(val, base, n);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp


/**
 * @brief テンプレート（vector）
 * @docs docs/template/template_vector.md
 */

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

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

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)
{
  if constexpr (depth == 1)
  {
    typename nested_vector<T, depth>::type replacement(size_t(*sizes), init);
    result.swap(replacement);
  }
  else
  {
    result.resize(size_t(*sizes));
    for (auto &child : result) populate_dimensions<T, depth - 1>(child, sizes + 1, init);
  }
}

template <class T, size_t depth, class V>
typename nested_vector<T, depth>::type build_dimensions(const V *sizes, const T &init)
{
  if constexpr (depth == 0) return init;
  else
  {
    typename nested_vector<T, depth>::type result;
    populate_dimensions<T, depth>(result, sizes, init);
    return result;
  }
}
}

template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i >= d) return init;
  else return template_vector_detail::build_dimensions<T, d - i>(sz + i, init);
}

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

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

template <class T = ll>
optional<T> ctol(const char &c, const string &s)
{
  repi(i, SZ<int>(s)) if (s[i] == c) return T(i);
  return nullopt;
}
template <class T = ll>
vc<T> stov(const string &s, char first)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return s[i] - first; });
}
template <class T = ll>
optional<vc<T>> stov(const string &s, const string &t)
{
  vc<T> res;
  res.reserve(s.size());
  for (char c : s)
  {
    auto index = ctol<T>(c, t);
    if (!index) return nullopt;
    res.push_back(*index);
  }
  return res;
}
template <class T>
string vtos(const vc<T> &v, char first)
{
  string res = "";
  fe(vi : v) res += vi + first;
  return res;
}
template <class T>
string vtos(const vc<T> &v, const string &t)
{
  string res = "";
  fe(vi : v) res += t[vi];
  return res;
}

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

template <class T>
vc<T> merged(const vc<T> &a, const vc<T> &b)
{
  vc<T> res;
  merge(ALL(a), ALL(b), back_inserter(res));
  return res;
}

template <class T, class I>
T vecget(const vc<T> &v, I i, const T &dflt_negative = -default_infty<T>(), const T &dflt_positive = default_infty<T>())
{
  if (i < 0)
    return dflt_negative;
  if (i >= SZ<int>(v))
    return dflt_positive;
  return v[i];
}
// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif

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

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


/**
 * @brief 値または関数から値を取得
 * @docs docs/utils/resolved_value.md
 */

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

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

/**
 * @brief 既定値・値・関数から無限大を取得
 * @docs docs/utils/resolved_infty.md
 */

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>();
}

/**
 * @brief テンプレート（アルゴリズム）
 * @docs docs/template/template_algo.md
 */

template <class V>
auto SUM(const V &v)
{
  typename V::value_type s{};
  fec(vi : v) s += vi;
  return s;
}
template <class T, class V>
T SUM(const V &v)
{
  T s{};
  fec(vi : v) s += vi;
  return s;
}
template <class V>
auto MAX(const V &v) { return *max_element(ALL(v)); }
template <class V>
auto MIN(const V &v) { return *min_element(ALL(v)); }
template <class I = ll, class V>
optional<I> ARGMAX(const V &v)
{
  if (v.empty()) return nullopt;
  return I(max_element(ALL(v)) - v.begin());
}
template <class I = ll, class V>
optional<I> ARGMIN(const V &v)
{
  if (v.empty()) return nullopt;
  return I(min_element(ALL(v)) - v.begin());
}

template<class T = ll, class V>
T mex(const V &a)
{
  int n = a.size();
  vector<bool> exists(n, false);
  repi(i, n) if (0 <= a[i] && a[i] < n) exists[a[i]] = true;
  repi(x, n) if (!exists[x]) return x;
  return n;
}

// (0, 1. ..., n-1) の順列か判定
template <class I>
bool is_permutation(const vc<I> &p)
{
  const int n = p.size();
  vc<bool> b(n, false);
  repi(i, n)
  {
    if (!(0 <= p[i] && p[i] < n))
      return false;
    b[p[i]] = true;
  }
  return all_of(ALL(b), [](bool bi)
                { return bi; });
}

template <class T = ll>
vc<T> permid(const int &n, const int &base_index = 0)
{
  vc<T> p(n);
  repi(i, n) p[i] = i + base_index;
  return p;
}
template <class T>
vc<T> perminv(const vc<T> &p)
{
  if (p.empty())
    return {};
  const int n = p.size();
  vc<T> q(MAX(p) + 1);
  repi(i, n) if (p[i] >= 0) q[p[i]] = i;
  return q;
}
// a[p[i]] for all i
template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p)
{
  const int n = p.size();
  vc<T> res(n);
  repi(i, n)
  {
    assert(0 <= p[i] && p[i] < U(a.size()));
    res[i] = a[p[i]];
  }
  return res;
}
// p[q[r[i]]] for all i など
template <class T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs)
{
  return permuted(permuted(p, q), rs...);
}

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

#if __cplusplus < 202002L
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  sort(ALL(v), forward<Args>(args)...);
  return v;
}
#else
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  ranges::sort(v, forward<Args>(args)...);
  return v;
}
#endif

template <class V, class Equal = equal_to<>>
void unique(V &v, Equal equal = {}) { v.erase(std::unique(ALL(v), equal), v.end()); }
template <class V, class Equal = equal_to<>>
V uniqued(V v, Equal equal = {}) { unique(v, equal); return v; }

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

// 01234 -> 12340
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  if (n == 0)
    return;
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
// 01234 -> 12340
template <class V, class U>
V rotated(V v, U k) { rotate(v, k); return v; }

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 I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k) { return a * k; }
};
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 I>
  static constexpr S pow(const S &a, I k) { return k == 0 ? e() : a; }
};
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>(); }
  template <class I>
  static constexpr S pow(const S &a, I k) { return k == 0 ? e() : a; }
};

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)
{
  if constexpr (is_signed_ext<I>)
    assert(k >= 0);
  if constexpr (internal::HasMonoidPow<M, I>::value)
    return M::pow(a, k);
  else
  {
    typename M::S c = M::e();
    for (; k; k >>= 1)
    {
      if (k & 1)
        c = M::op(c, a);
      a = M::op(a, a);
    }
    return c;
  }
}

template <class G, class I>
typename G::S pow_group(typename G::S a, I k)
{
  if constexpr (is_signed_ext<I>)
  {
    if (k < 0)
    {
      a = G::inv(a);
      return G::op(pow_monoid<G>(a, -(k + 1)), a);
    }
  }
  return pow_monoid<G>(a, k);
}

// left_index が 0 なら、長さ n+1 で a.front() が e()
// left_index が 1 なら、長さ n で e() がない
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;
}
// right_index が 0 なら、長さ n+1 で a.back() が e()
// right_index が 1 なら、長さ n で e() がない
template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0)
{ return reversed(cuml<M>(reversed(v), right_index)); }
template <class T>
vc<T> cumlsum(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidAdd<T>>(v, left_index); }
template <class T>
vc<T> cumrsum(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidAdd<T>>(v, right_index); }
template <class T>
vc<T> cumlmin(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMin<T>>(v, left_index); }
template <class T>
vc<T> cumrmin(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMin<T>>(v, right_index); }
template <class T>
vc<T> cumlmax(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMax<T>>(v, left_index); }
template <class T>
vc<T> cumrmax(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMax<T>>(v, right_index); }

// デフォルトでは長さ n+1
// left_index, right_index をそれぞれ 1 にすると、左右が削除される
template <class T>
vc<T> adjd(const vc<T> &v, int left_index = 0, int right_index = 0)
{
  int n = v.size();
  assert(0 <= left_index && 0 <= right_index && left_index + right_index <= n + 1);
  vc<T> res(n + 1);
  if (n == 0)
  {
    res[0] = T{};
    res.erase(res.end() - right_index, res.end());
    res.erase(res.begin(), res.begin() + left_index);
    return res;
  }
  res[0] = v[0];
  repi(i, 1, n) res[i] = v[i] - v[i - 1];
  res[n] = -v[n - 1];
  res.erase(res.end() - right_index, res.end());
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}

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


/**
 * @brief テンプレート（二分探索）
 * @docs docs/template/template_binsearch.md
 */

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;

// --- LB, UB ---

#if __cplusplus < 202002L
struct identity
{
  template <class T>
  constexpr T &&operator()(T &&t) const noexcept
  { return forward<T>(t); }
};
namespace internal
{
  template <class T = ll, class V, class Judge>
  inline T bound_helper(const V &v, Judge judge)
  {
    int l = -1, r = v.size();
    while (r - l > 1)
    {
      int m = (l + r) / 2;
      if (judge(m))
        l = m;
      else
        r = m;
    }
    return r;
  }
};
// val <= v[i] となる最小の i (val 未満の値の個数)
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return comp(proj(*(v.begin() + i)), val); });
}
// val < v[i] となる最小の i (val 以下の値の個数)
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return !comp(val, proj(*(v.begin() + i))); });
}
#define DEFAULT_COMP less<>
#else
// val <= v[i] となる最小の i (val 未満の値の個数)
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 = {})
{ return ranges::lower_bound(v, val, comp, proj) - v.begin(); }
// val < v[i] となる最小の i (val 以下の値の個数)
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 = {})
{ return ranges::upper_bound(v, val, comp, proj) - v.begin(); }
#define DEFAULT_COMP ranges::less
#endif

// --- vector 等の lt, leq, gt, geq ---

// 条件を満たす添字を返す。左側は -1、右側は v.size() を番兵とする。
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>, T>
{
  static_assert(numeric_limits<T>::is_signed, "index type must represent -1");
  return LB<T>(v, val, comp, proj) - 1;
}
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>, T>
{
  static_assert(numeric_limits<T>::is_signed, "index type must represent -1");
  return UB<T>(v, val, comp, proj) - 1;
}
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>, T>
{ return UB<T>(v, val, comp, proj); }
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>, T>
{ return LB<T>(v, val, comp, proj); }
// v[i] < val となる i の個数
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>
{ return LB<T>(v, val, comp, proj); }
// v[i] <= val となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj); }
// val < v[i] となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - UB<T>(v, val, comp, proj); }
// val <= v[i] となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - LB<T>(v, val, comp, proj); }
// l <= v[i] < r となる i の個数
template <class T = ll, class V, class L, class R, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto in_cnt(const V &v, L l, R r, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{
  if (l > r)
    return 0;
  return lt_cnt<T>(v, r, comp, proj) - lt_cnt<T>(v, l, comp, proj);
}

// --- set 等の lt, leq, gt, geq ---

// 条件を満たす要素へのイテレータを返す。存在しなければ nullopt。
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>>
{
  auto it = v.lower_bound(val);
  if (it == v.begin()) return nullopt;
  return prev(it);
}
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>>
{
  auto it = v.upper_bound(val);
  if (it == v.begin()) return nullopt;
  return prev(it);
}
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>>
{
  auto it = v.upper_bound(val);
  if (it == v.end()) return nullopt;
  return it;
}
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>>
{
  auto it = v.lower_bound(val);
  if (it == v.end()) return nullopt;
  return it;
}

// --- 自作二分探索 ---

namespace internal
{
template <class T>
bool binsearch_adjacent(T a, T b)
{
  if (a < b)
    return a + 1 == b;
  if (b < a)
    return b + 1 == a;
  return false;
}
};

// (ok, ng)
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)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  while (!internal::binsearch_adjacent(ok, ng))
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}
template <class T = ld, class Judge, class InitOk, class InitNg>
T binsearch_real(const Judge &judge, InitOk init_ok, InitNg init_ng, int iteration_count = 100, bool check_ok = true, bool check_ng = true)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  repi(_, iteration_count)
  {
    T mid = (ok + ng) / 2;
    (judge(mid) ? ok : ng) = mid;
  }
  return ok;
}
// (ok, ng)
template <class T = ll, class Judge, class InitVal>
pair<T, T> expsearch(const Judge &judge, InitVal init_val, bool positive = true)
{
  T cur(init_val), step = 1;
  const bool cur_ok = judge(cur);
  auto advance = [&](T x, T d, bool pos) -> T
  {
    if (pos)
      return x > numeric_limits<T>::max() - d ? numeric_limits<T>::max() : x + d;
    else
      return x < numeric_limits<T>::lowest() + d ? numeric_limits<T>::lowest() : x - d;
  };
  T prv = advance(cur, 1, !positive);
  if (prv != cur && judge(prv) != cur_ok)
  {
    if (cur_ok)
      return {cur, prv};
    else
      return {prv, cur};
  }
  while (true)
  {
    T nxt = advance(cur, step, positive);
    assert(nxt != cur && "the boundary must exist in the searched direction");
    if (nxt == cur || judge(nxt) != cur_ok)
    {
      T ok = cur_ok ? cur : nxt;
      T ng = cur_ok ? nxt : cur;
      return binsearch<T>(judge, ok, ng, false, false);
    }
    cur = nxt;
    if (step > numeric_limits<T>::max() / 2)
      step = numeric_limits<T>::max();
    else
      step *= 2;
  }
}
// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp


/**
 * @brief テンプレート（ビット演算）
 * @docs docs/template/template_bit.md
 */

template <class T>
inline constexpr ull pow2(T k) { return 1ULL << k; }
template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

#if __cplusplus < 202002L
// x == 0 ならば 0、そうでなければ 1 + floor(log2(x))
// 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... 
inline constexpr ull bit_width(ull x) { return x == 0 ? 0 : 64 - __builtin_clzll(x); }
// 0, 1, 2, 2, 4, 4, 4, 4, 8, 8, ...
inline constexpr ull bit_floor(ull x) { return x == 0 ? 0ULL : 1ULL << (bit_width(x) - 1); }
// 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, ...
inline constexpr ull bit_ceil(ull x) { return x == 0 ? 1ULL : 1ULL << bit_width(x - 1); }
inline constexpr ull countr_zero(ull x) { assert(x != 0); return __builtin_ctzll(x); }
inline constexpr ull popcount(ull x) { return __builtin_popcountll(x); }
inline constexpr bool has_single_bit(ull x) { return popcount(x) == 1; }
#else
// 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... 
inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }
// 0, 1, 2, 2, 4, 4, 4, 4, 8, 8, ...
inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }
// 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, ...
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); }
#endif

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; }
template <class T>
inline void bset(T &x, uint k, bool b = 1) { b ? x |= (1ULL << k) : x &= ~(1ULL << k); }
template <class T>
inline void bflip(T &x, uint k) { x ^= (1ULL << k); }
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.
 */


#ifdef LOCAL
// cpp-dump 内の再帰的な出力からも、map 用のオーバーロードを参照できるようにする。
namespace cpp_dump::_detail
{
  struct export_command;
#define _p_LIBRARY_DECLARE_MAP_EXPORT(Map)                                  \
  template <class... Args>                                                 \
  string export_var(const std::Map<Args...> &, const string &, size_t,       \
                    size_t, bool, const export_command &);
  _p_LIBRARY_DECLARE_MAP_EXPORT(map)
  _p_LIBRARY_DECLARE_MAP_EXPORT(multimap)
  _p_LIBRARY_DECLARE_MAP_EXPORT(unordered_map)
  _p_LIBRARY_DECLARE_MAP_EXPORT(unordered_multimap)
#undef _p_LIBRARY_DECLARE_MAP_EXPORT
}

#include <cpp-dump.hpp> //! https://github.com/philip82148/cpp-dump

namespace cpp_dump::options
{
  inline string map_key_color = "\x1b[36m";
}

namespace cpp_dump::_detail
{
  template <class It, class Format>
  struct map_values_for_dump
  {
    It first, last;
    const Format &format;
    map_values_for_dump(It first, It last, const Format &format) : first(first), last(last), format(format) {}
    struct iterator
    {
      It it;
      const Format &format;
      decltype(auto) operator*() const { return format(it->second); }
      iterator &operator++() { ++it; return *this; }
      bool operator!=(const iterator &other) const { return it != other.it; }
    };
    iterator begin() const { return {first, format}; }
    iterator end() const { return {last, format}; }
  };

  inline string color_map_key(const string &text)
  {
    if (!use_es() || options::map_key_color.empty()) return text;
    string plain;
    for (size_t i = 0; i < text.size();)
    {
      if (text[i] == '\x1b' && i + 1 < text.size() && text[i + 1] == '[')
      {
        size_t end = i + 2;
        while (end < text.size() && (isdigit(static_cast<unsigned char>(text[end])) || text[end] == ';')) ++end;
        if (end < text.size() && text[end] == 'm')
        {
          i = end + 1;
          continue;
        }
      }
      plain += text[i++];
    }
    return es::apply(options::map_key_color, plain);
  }

  template <class Map, class Format>
  string export_library_map(
      const Map &map, const string &indent, size_t last_line_length,
      size_t current_depth, bool fail_on_newline, const export_command &command,
      const Format &format)
  {
    if (map.empty()) return es::bracket("{ }", current_depth);
    if (current_depth >= options::max_depth)
      return es::bracket("{ ", current_depth) + es::op("...") + es::bracket(" }", current_depth);

    const auto &key_command = command.next_for_map_key();
    const auto &value_command = command.next_for_map_value();
    const size_t next_depth = current_depth + 1;
    auto map_wrapper = [&]()
    {
      if constexpr (is_multimap<Map>) return _multimap_wrapper(map);
      else return _map_dummy_wrapper(map);
    }();

    bool multiline = options::cont_indent_style == types::cont_indent_style_t::always;
    if (options::cont_indent_style == types::cont_indent_style_t::when_nested)
      multiline = is_multimap<Map> || is_iterable_like<typename Map::key_type> || is_iterable_like<typename Map::mapped_type>;
    if (options::cont_indent_style == types::cont_indent_style_t::when_non_tuples_nested)
      multiline = is_multimap<Map>
          || (is_iterable_like<typename Map::key_type> && !is_tuple<typename Map::key_type>)
          || (is_iterable_like<typename Map::mapped_type> && !is_tuple<typename Map::mapped_type>);

    for (;; multiline = true)
    {
      if (multiline && fail_on_newline) return "\n";
      const string element_indent = multiline ? indent + "  " : indent;
      string output = es::bracket(multiline ? "{" : "{ ", current_depth);
      bool first = true, retry = false;
      auto skipped_map = command.create_skip_container(map_wrapper);
      for (const auto &[ellipsis, it, index] : skipped_map)
      {
        if (!first) output += es::op(multiline ? "," : ", ");
        first = false;
        if (multiline) output += "\n" + element_indent;
        if (ellipsis) output += es::op("...");
        else
        {
          const auto &[key, value] = *it;
          auto render = [&](const auto &x, const export_command &child_command)
          {
            return export_var(format(x), element_indent,
                              get_last_line_length(output, last_line_length),
                              next_depth, !multiline, child_command);
          };
          output += color_map_key(render(key, key_command));
          if constexpr (is_multimap<Map>)
            output += es::member(" (" + to_string(map.count(key)) + ")");
          output += es::op(": ");
          if constexpr (is_multimap<Map>)
          {
            auto [begin, end] = map.equal_range(key);
            // この view は所有しない。描画中だけ元の map を参照する。
            auto values = map_values_for_dump(begin, end, format);
            output += export_var(values, element_indent,
                                 get_last_line_length(output, last_line_length),
                                 next_depth, !multiline, value_command);
          }
          else output += render(value, value_command);
        }
        if (!multiline && (has_newline(output) || last_line_length + get_length(output) + 2 > options::max_line_width))
        {
          retry = true;
          break;
        }
      }
      if (retry) continue;
      return output + (multiline ? "\n" + indent : " ") + es::bracket("}", current_depth);
    }
  }

#define _p_LIBRARY_DEFINE_MAP_EXPORT(Map)                                                \
  template <class... Args>                                                             \
  string export_var(const std::Map<Args...> &value, const string &indent,                \
                    size_t last_line_length, size_t current_depth,                     \
                    bool fail_on_newline, const export_command &command)              \
  {                                                                                   \
    return export_library_map(value, indent, last_line_length, current_depth,           \
                              fail_on_newline, command,                               \
                              [](const auto &x) -> const auto & { return x; });        \
  }
  _p_LIBRARY_DEFINE_MAP_EXPORT(map)
  _p_LIBRARY_DEFINE_MAP_EXPORT(multimap)
  _p_LIBRARY_DEFINE_MAP_EXPORT(unordered_map)
  _p_LIBRARY_DEFINE_MAP_EXPORT(unordered_multimap)
#undef _p_LIBRARY_DEFINE_MAP_EXPORT
}
#endif

/**
 * @brief テンプレート（dump）
 * @docs docs/template/template_dump.md
 */

#ifdef LOCAL
namespace cpp_dump::_detail
{
#if LIBRARY_HAS_FLOAT128
  inline string export_var(
      const f128 &x, const string &, size_t,
      size_t, bool, const export_command &
  ) {
    return es::signed_number(template_float128_detail::format(
        x, chars_format::general, numeric_limits<f128>::max_digits10));
  }
#endif

  inline string export_var(
      const i128 &x, const string &, size_t,
      size_t, bool, const export_command &
  ) {
    return es::signed_number(i128tos(x));
  }

  template <class T>
  inline auto export_object_generic(
      const T &value, const string &indent, size_t last_line_length,
      size_t current_depth, bool fail_on_newline, const export_command &command
  ) -> decltype(value.dump_data(), string())
  {
    string class_name = es::class_name(get_typename<T>());
    _p_CPP_DUMP_DEFINE_EXPORT_OBJECT_COMMON1;
    apply(
        [&](const auto &...member)
        { (append_output(member.first, member.second.get()), ...); },
        value.dump_data()
    );
    _p_CPP_DUMP_DEFINE_EXPORT_OBJECT_COMMON2;
  }
} // namespace cpp_dump::_detail
#define _p_LIBRARY_CPP_DUMP_MEMBER(member) pair{string_view(#member), cref(member)}
#define CPP_DUMP_DEFINE_DATA(...)                                                      \
  auto dump_data() const                                                               \
  {                                                                                    \
    return make_tuple(_p_CPP_DUMP_EXPAND_VA(_p_LIBRARY_CPP_DUMP_MEMBER, __VA_ARGS__)); \
  }
#define dump(...) cpp_dump(__VA_ARGS__)
namespace cp = cpp_dump;
CPP_DUMP_SET_OPTION_GLOBAL(log_label_func, cp::log_label::line());
CPP_DUMP_SET_OPTION_GLOBAL(max_iteration_count, 100);
#define local(...) __VA_ARGS__
#define oj(...)
#define local_oj(a, b) (a)
CPP_DUMP_DEFINE_EXPORT_OBJECT_GENERIC(content());
#else
#define CPP_DUMP_DEFINE_DATA(...)
#define dump(...) ((void)0)
#define local(...)
#define oj(...) __VA_ARGS__
#define local_oj(a, b) (b)
#endif

template <class T, class Sequence>
vc<T> content(queue<T, Sequence> que)
{
  vc<T> res;
  while (!que.empty())
  {
    res.eb(que.front());
    que.pop();
  }
  return res;
}
template <class T, class Sequence, class Compare>
vc<T> content(priority_queue<T, Sequence, Compare> pque)
{
  vc<T> res;
  while (!pque.empty())
  {
    res.eb(pque.top());
    pque.pop();
  }
  return res;
}
template <class T>
auto content(const T &obj) { return obj.content(); }

/**
 * @brief テンプレート（入出力）
 * @docs docs/template/template_inout.md
 */

#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

// 十進変換とバッファ処理は公開 API の仕様から独立に設計。
#if defined FAST_IO and not defined LOCAL
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>>;

  // 各桁対、4 桁、8 桁の順に十進の重みをまとめる。
  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)
  {
    using U = integer_magnitude<T>;
    const char *p = ibuf + pil;
    for (;;)
    {
      while (ascii_space(*p)) ++p;
      if (p != ibuf + pir) break;
      pil = pir;
      load();
      if (!pir) abort();
      p = ibuf;
    }
    bool negative = false;
    if (*p == '+') ++p;
    else if constexpr (numeric_limits<T>::is_signed)
    {
      if (*p == '-') negative = true, ++p;
    }
    U value = 0;
    for (;;)
    {
      while (size_t(ibuf + pir - p) >= 8)
      {
        uint64_t columns;
        memcpy(&columns, p, 8);
#if __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__
        columns = __builtin_bswap64(columns);
#endif
        const uint64_t stop = ((columns + 0x4646464646464646ULL) |
                               (columns - 0x3030303030303030ULL)) & 0x8080808080808080ULL;
        if (stop)
        {
          if constexpr (sizeof(U) > sizeof(ull))
          {
            const unsigned digits = unsigned(__builtin_ctzll(stop)) / 8;
            if (digits)
            {
              uint tail;
              if (digits <= 3)
              {
                tail = unsigned(p[0] - '0');
                if (digits >= 2) tail = tail * 10 + unsigned(p[1] - '0');
                if (digits == 3) tail = tail * 10 + unsigned(p[2] - '0');
              }
              else tail = decode_eight(columns << (8 * (8 - digits)));
              value = value * powers_of_ten[digits] + tail;
              p += digits;
            }
            goto integer_complete;
          }
          else
          {
            if (stop & 0x80808080ULL)
            {
              if (!(stop & 0x808080ULL))
              {
                value = value * 1000 + uint(columns & 15) * 100
                      + uint(columns >> 8 & 15) * 10 + uint(columns >> 16 & 15);
                p += 3;
              }
              else if (!(stop & 0x8080ULL))
              {
                value = value * 100 + uint(columns & 15) * 10 + uint(columns >> 8 & 15);
                p += 2;
              }
              else if (!(stop & 0x80ULL))
              {
                value = value * 10 + uint(columns & 15);
                ++p;
              }
              goto integer_complete;
            }
            break;

          }
        }
        columns &= 0x0f0f0f0f0f0f0f0fULL;
        const uint64_t pairs = (columns & 0x00ff00ff00ff00ffULL) * 10
                             + ((columns >> 8) & 0x00ff00ff00ff00ffULL);
        const uint64_t quads = (pairs & 0x0000ffff0000ffffULL) * 100
                             + ((pairs >> 16) & 0x0000ffff0000ffffULL);
        value = value * 100000000 + uint(quads) * 10000 + uint(quads >> 32);
        p += 8;
      }
      while (unsigned(*p - '0') < 10)
      {
        value = value * 10 + unsigned(*p - '0');
        ++p;
      }
      pil = uint32_t(p - ibuf);
      if (pil != pir) break;
      load();
      p = ibuf;
      if (!pir) break;
    }
integer_complete:
    pil = uint32_t(p - ibuf);
    if constexpr (numeric_limits<T>::is_signed)
    {
      if (negative && value) x = T(-T(value - 1) - 1);
      else x = T(value);
    }
    else x = T(value);
  }

  template <class T, enable_if_t<numeric_limits<T>::is_integer, int> = 0>
  __attribute__((always_inline)) inline void rd1(T &x) { rd1_integer(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) { rd1(x.first), rd1(x.second); }
  template <class... T>
  void rd1(tuple<T...> &x)
  {
    apply([](auto &...element) { (rd1(element), ...); }, x);
  }
  template <class T, size_t N>
  void rd1(array<T, N> &x)
  {
    for (auto &element : x) rd1(element);
  }
  template <class T>
  void rd1(vc<T> &x)
  {
    if constexpr (is_same_v<T, bool>)
      for (size_t i = 0; i < x.size(); ++i) { bool value; rd1(value); x[i] = value; }
    else
      for (auto &element : x) rd1(element);
  }

  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)
  {
    if constexpr (sizeof(U) > sizeof(uint))
    {
      if (x >= 1000000000000000000ULL)
      {
        const uint hi = uint(x / 1000000000000000000ULL);
        const ull tail = x - ull(hi) * 1000000000000000000ULL;
        const uint middle = uint(tail / 1000000000ULL);
        out = store_leading_triplet(out, hi);
        store_nine(out, middle);
        store_nine(out + 9, uint(tail - ull(middle) * 1000000000ULL));
        return out + 18;
      }
    }
    if (x >= 1000000000)
    {
      const uint hi = uint(x / 1000000000);
      out = store_short(out, hi);
      store_nine(out, uint(x - U(hi) * 1000000000));
      return out + 9;
    }
    return store_short(out, uint(x));
  }

  template <class T, bool newline = false>
  __attribute__((always_inline)) inline void wt1_integer(T x)
  {
    using U = integer_magnitude<T>;
    U magnitude = U(x);
    bool negative = false;
    if constexpr (numeric_limits<T>::is_signed)
    {
      negative = x < 0;
      const U sign_mask = U(0) - U(negative);
      magnitude = (magnitude ^ sign_mask) + U(negative);
    }
    if constexpr (sizeof(U) <= sizeof(ull))
    {
      if (por > SIZ - 40) flush();
      char *out = obuf + por;
      // 負号の位置だけを進め、非負なら数字で上書きする。
      *out = '-';
      out += negative;
      out = store_ordinary(out, magnitude);
      if constexpr (newline) *out++ = '\n';
      por = uint32_t(out - obuf);
      return;
    }
    char text[48];
    char *const end = text + sizeof(text);
    char *first = end;
    if constexpr (sizeof(U) > sizeof(ull))
    {
      constexpr ull base = 1000000000000000000ULL;
      while (magnitude >= base)
      {
        const U q = magnitude / base;
        const ull tail = ull(magnitude - q * base);
        const uint hi = uint(tail / 1000000000ULL);
        first -= 18;
        store_nine(first, hi);
        store_nine(first + 9, uint(tail - ull(hi) * 1000000000ULL));
        magnitude = q;
      }
      first = prepend_wide(first, ull(magnitude));
    }
    else if constexpr (sizeof(U) > sizeof(uint)) first = prepend_wide(first, magnitude);
    else first = prepend_small(first, magnitude);
    if (negative) *--first = '-';
    append_bytes(first, size_t(end - first));
    if constexpr (newline) append_char('\n');
  }

  template <class T, enable_if_t<numeric_limits<T>::is_integer, int> = 0>
  __attribute__((always_inline)) inline void wt1(T x) { wt1_integer(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) { wt1(x.first), append_char(' '), wt1(x.second); }
  template <class... T>
  void wt1(const tuple<T...> &x)
  {
    apply([](const auto &...element)
    {
      size_t index = 0;
      ((index++ ? void(append_char(' ')) : void(), wt1(element)), ...);
    }, x);
  }
  template <class T, size_t N>
  void wt1(const array<T, N> &x)
  {
    for (size_t i = 0; i < N; ++i)
    {
      if (i) append_char(' ');
      wt1(x[i]);
    }
  }
  template <class T>
  void wt1(const vc<T> &x)
  {
    for (size_t i = 0; i < x.size(); ++i)
    {
      if (i) append_char(' ');
      wt1(x[i]);
    }
  }

  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');
    }
  }
}
#endif

#if defined FAST_IO and not defined LOCAL
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
#endif

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

// ---- 入力 ----
#if defined LOCAL or not defined FAST_IO
template <class T, class U>
istream &operator>>(istream &is, pair<T, U> &p)
{
  is >> p.first >> p.second;
  return is;
}
template <class... Ts>
istream &operator>>(istream &is, tuple<Ts...> &t)
{
  apply([&](auto &...a)
        { (is >> ... >> a); }, t);
  return is;
}
template <class T, size_t n>
istream &operator>>(istream &is, array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
template <class T>
istream &operator>>(istream &is, vc<T> &a)
{
  const size_t n = a.size();
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
#endif

namespace internal
{

#if defined LOCAL or not defined FAST_IO
template <class... Ts>
void CIN(Ts &...a) { (cin >> ... >> a); }
#endif

#if defined FAST_IO and not defined LOCAL
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }
#else
template <class... Ts>
void READnodump(Ts &...a) { CIN(a...); }
#endif

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

template <class... T>
void READVEC2nodump(int n, int m, vvc<T> &...v)
{
  (v.assign(size_t(n), vc<T>(size_t(m))), ...);
  READnodump(v...);
}

template <class... T>
void READJAGnodump(int n, vvc<T> &...vs)
{
  auto read_one = [&](auto &v)
  {
    v.resize(size_t(n));
    for (auto &row : v)
    {
      int k;
      READnodump(k);
      row.resize(size_t(k));
      READnodump(row);
    }
  };
  (read_one(vs), ...);
}

}; // namespace internal

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

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

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

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

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

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

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

// ----------

// ----- 出力 -----
#ifdef INTERACTIVE
#define ENDL endl
#else
#define ENDL '\n'
#endif

#if defined LOCAL or not defined FAST_IO
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p)
{
  os << p.first << ' ' << p.second;
  return os;
}

template <class... Ts>
ostream &operator<<(ostream &os, const tuple<Ts...> &t)
{
  if constexpr (sizeof...(Ts))
  {
    apply([&](const auto &...x)
          {
            int i = 0;
            ((os << (i++ ? " " : "") << x), ...);
          }, t);
  }
  return os;
}
template <class T, size_t n>
ostream &operator<<(ostream &os, const array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << a[i];
  }
  return os;
}
template <class T>
ostream &operator<<(ostream &os, const vc<T> &v)
{
  const size_t n = v.size();
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << v[i];
  }
  return os;
}

namespace internal
{

template <class... Ts>
void COUTW(const Ts &...a)
{
  if constexpr (sizeof...(Ts))
    (cout << ... << a);
}

template <class... Ts>
void COUTP(const Ts &...a)
{
  if constexpr (sizeof...(Ts))
  {
    int i = 0;
    ((cout << (i++ ? " " : "") << a), ...);
  }
  cout << ENDL;
}

}; // namespace internal
#endif

#if defined FAST_IO and not defined LOCAL
#define WRITE fastio::write
#define PRINT fastio::print
#else
#define WRITE internal::COUTW
#define PRINT internal::COUTP
#endif
#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

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

// ----- 基準ずらし -----
template <class T, class U, class P>
pair<T, U> &operator+=(pair<T, U> &a, const P &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> a, const P &b) { return a += b; }
template <class T, class U, class P>
pair<T, U> &operator-=(pair<T, U> &a, const P &b)
{
  a.first -= b.first;
  a.second -= b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator-(pair<T, U> a, const P &b) { return a -= b; }
template <class T, class U>
pair<T, U> operator-(pair<T, U> a)
{
  a.first = -a.first;
  a.second = -a.second;
  return a;
}

template <class T, size_t n, class A>
array<T, n> &operator+=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> a, const A &b) { return a += b; }
template <class T, size_t n, class A>
array<T, n> &operator-=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] -= b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator-(array<T, n> a, const A &b) { return a -= b; }
template <class T, size_t n>
array<T, n> operator-(array<T, n> a)
{
  for (auto &ai : a)
    ai = -ai;
  return a;
}

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 <size_t... I, class A, class B>
auto &tuple_sub_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) -= get<I>(b)), ...);
  return a;
}
template <size_t... I, class A>
auto &tuple_neg_impl(A &a, const index_sequence<I...>)
{
  ((get<I>(a) = -get<I>(a)), ...);
  return a;
}

}; // namespace internal

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... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> a, const Tp &b) { return a += b; }
template <class... Ts, class Tp>
tuple<Ts...> &operator-=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_sub_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts, class Tp>
tuple<Ts...> operator-(tuple<Ts...> a, const Tp &b) { return a -= b; }
template <class... Ts>
tuple<Ts...> operator-(tuple<Ts...> a)
{
  internal::tuple_neg_impl(a, make_index_sequence<sizeof...(Ts)>{});
  return a;
}

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; }
// ----------

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

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

namespace internal
{

template <size_t... I, class V, class Tp>
auto vt_to_tv_impl(V &tv, const Tp &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }

template <size_t... I, class Tp>
auto tv_to_vt_impl(const Tp &tv, index_sequence<I...>, size_t index)
{ return make_tuple(get<I>(tv)[index]...); }

};

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

template <class... Ts>
auto zip(const tuple<vc<Ts>...> &tv)
{
  size_t n = get<0>(tv).size();
  apply([&](auto &...v)
        { ((void(v), assert(v.size() == n)), ...); }, tv);
  vc<tuple<Ts...>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = internal::tv_to_vt_impl(tv, index_sequence_for<Ts...>{}, i);
  return vt;
}

namespace internal
{

template <class... Ts>
auto zip_vectors(const vc<Ts> &...v)
{
  if constexpr (sizeof...(Ts) == 2)
    return zip(pair{v...});
  else
    return zip(tuple{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


/**
 * @brief テンプレート（ランダム生成）
 * @docs docs/template/template_random.md
 */

mt19937_64 mt;

// [l, r] から等確率
template <class T = ll, class U1, class U2>
T randint(U1 l, U2 r)
{
  assert(T(l) <= T(r));
  return uniform_int_distribution<T>(T(l), T(r))(mt);
}
// [l, r) から等確率
template <class T = ll, class U1, class U2>
T randrange(U1 l, U2 r)
{
  assert(T(l) < T(r));
  return uniform_int_distribution<T>(T(l), T(r) - 1)(mt);
}

// [l, r) から一様ランダムな実数を返す
template <class T = double, class U1, class U2>
T randreal(U1 l, U2 r)
{
  assert(T(l) < T(r));
  return uniform_real_distribution<T>(T(l), T(r))(mt);
}

// 確率 p で true を返す
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)
{
  int k = res.size();
  T n = r - l;
  if (k <= 256)
  {
    repi(i, k)
    {
      T j = n - T(k) + T(i), x = randint<T>(0, j);
      if (find(res.begin(), res.begin() + i, x) != res.begin() + i)
        x = j;
      res[i] = x;
    }
  }
  else
  {
    unordered_set<T> used;
    used.reserve(2 * size_t(k));
    repi(i, k)
    {
      T j = n - T(k) + T(i), x = randint<T>(0, j);
      if (!used.insert(x).second)
        x = j, used.insert(x);
      res[i] = x;
    }
  }
  for (T &x : res) x += l;
  if constexpr (does_sort)
    sort(res.begin(), res.end());
  else
    shuffle(res.begin(), res.end(), mt);
}
}; // namespace internal

// [l, r) から相異なる k 個を選ぶ
// does_sort: ソートするかどうか
template <int k, bool does_sort, class T = ll, class U1, class U2>
array<T, k> random_sample_range_array(U1 l, U2 r)
{
  assert(T(r) - T(l) >= T(k));
  array<T, k> res;
  internal::random_sample_range<does_sort>(res, T(l), T(r));
  return res;
}
// [l, r) から相異なる k 個を選ぶ
// does_sort: ソートするかどうか
template <bool does_sort, class T = ll, class U1, class U2>
vc<T> random_sample_range_vector(U1 l, U2 r, int k)
{
  assert(k >= 0);
  assert(T(r) - T(l) >= T(k));
  vc<T> res(k);
  internal::random_sample_range<does_sort>(res, T(l), T(r));
  return res;
}
// 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


/**
 * @brief 積の計算に使う拡張型
 * @docs docs/utils/larger_int.md
 */

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()
  {
    return true;
  }
}

template <class T>
struct larger_int
{
private:
  static constexpr bool check()
  {
    // 型エイリアス経由でも、128 ビット整数を入力にしたときだけ警告する。
    if constexpr (is_same_v<T, i128> || is_same_v<T, u128>)
      return larger_int_detail::warn_no_wider_integer<T>();
    return true;
  }
  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)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_constexpr<int>(n);
  }

  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;

  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;

  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };

  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}

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() { return M; }
  static constexpr value_type umod() { return M; }

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

};
// 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)
  {
    assert(m >= 1 && "modint: set_mod(m) requires m >= 1");
    reducer = barrett32(m);
  }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v) { return v; }
  static value_type mul(value_type a, value_type b) { return reducer.mul(a, 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)
  {
    assert(m >= 1 && m % 2 == 1 && "modint: set_mod(m) requires a positive odd modulus");
    reducer = montgomery64odd(m);
  }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * 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)
  {
    assert(m >= 1 && "modint: set_mod(m) requires m >= 1");
    reducer = montgomery64(m);
  }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

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


/**
 * @brief 拡張ユークリッド互除法 (extgcd)
 * @docs docs/math/extgcd.md
 */

// g == gcd(x, y) >= 0, ax + by == g を満たす (g, x, y)
// max(|x|, |y|) <= max(|a|, |b|)
template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  
  // a*x1 + b*y1 == z1  ...(1)
  // a*x2 + b*y2 == z2  ...(2)
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    // (1)' = (2)
    // (2)' = (1) - q*(2)
    T q = z1 / z2;
    tie(x1, x2) = make_pair(x2, x1 - q * x2);
    tie(y1, y2) = make_pair(y2, y1 - q * y2);
    tie(z1, z2) = make_pair(z2, z1 - q * z2);
  }
  if (z1 < 0)
    x1 = -x1, y1 = -y1, z1 = -z1;
  return {z1, x1, y1};
}

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()
    {
      M m = Policy::mod();
      assert(m > 0 && "modint: modulus is not set; call mint::set_mod(m) before use");
      return m;
    }

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

    static void precompute()
    {
      if constexpr (!is_void_v<PowInv>)
        (void)PowInv::get();
    }

    static mint raw(V v)
    {
      mint x;
      x._v = v;
      return x;
    }

    modint_impl() : _v(0) { (void)mod(); }

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      const V m = mod();
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(m);
        if (x < 0)
          x += m;
        rem = x;
      }
      else
        rem = V(v % m);
      _v = Policy::init(rem);
    };

    M val() const { return Policy::val(_v); }

    mint &operator+=(const mint &rhs)
    {
      _v += rhs._v;
      if (_v >= Policy::umod())
        _v -= Policy::umod();
      return *this;
    }
    mint &operator-=(const mint &rhs)
    {
      _v -= rhs._v;
      if (_v >= Policy::umod())
        _v += Policy::umod();
      return *this;
    }
    mint &operator*=(const mint &rhs)
    {
      _v = Policy::mul(_v, rhs._v);
      return *this;
    }
    mint &operator/=(const mint &rhs)
    {
      return *this *= rhs.inv();
    }

    mint &operator++()
    {
      _v++;
      if (_v == Policy::umod())
        _v = 0;
      return *this;
    }
    mint &operator--()
    {
      if (_v == 0)
        _v = Policy::umod();
      _v--;
      return *this;
    }
    mint operator++(int)
    {
      mint res = *this;
      ++(*this);
      return res;
    }
    mint operator--(int)
    {
      mint res = *this;
      --(*this);
      return res;
    }
    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    template <class T>
    mint pow(T n) const
    {
      assert(n >= 0);
      if constexpr (!is_void_v<PowInv>)
        return raw(PowInv::get().pow(val(), n));
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }
    mint inv() const
    {
      if constexpr (!is_void_v<PowInv>)
      {
        return raw(PowInv::get().inv(val()));
      }
      else if constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(x);
      }
    }

    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }
    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }
    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }
    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }
    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }
    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }
    friend M safe_hash_key(const mint &x) { return x.val(); }
  
#if defined LOCAL or not defined FAST_IO
    friend std::istream &operator>>(std::istream &is, mint &x)
    {
      long long a;
      is >> a;
      x = a;
      return is;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint &x)
    {
      os << x.val();
      return os;
    }
#else
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }
#endif
  };

};

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
  {
    if constexpr (is_modint_v<mint>)
      return a.val() < b.val();
    else
      return a < b;
  }
};

template <class mint>
struct modint_hash
{
  auto operator()(const mint &x) const
  {
    if constexpr (is_modint_v<mint>)
      return std::hash<decltype(x.val())>{}(x.val());
    else
      return std::hash<mint>{}(x);
  }
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp


/**
 * @brief 累乗テーブル
 * @docs docs/math/modint/power_table.md
 */

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

public:
  PowerTable() {}
  PowerTable(const mint &base) : mod(mint::mod()), base(base), pw(1, 1) {}

  void reserve(int n)
  {
    if (mod != mint::mod())
    {
      mod = mint::mod();
      pw = {1};
    }
    int i = pw.size();
    if (n < i)
      return;
    pw.resize(n + 1);
    for (; i <= n; i++)
      pw[i] = pw[i - 1] * base;
  }

  mint pow(int n)
  {
    assert(n >= 0);
    reserve(n);
    return pw[n];
  }
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp



/**
 * @brief 二項係数
 * @docs docs/math/modint/binomial.md
 */

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)
  {
    if constexpr (is_dynamic_modint_v<T>)
    {
      if (mod != T::mod())
      {
        mod = T::mod();
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
      }
    }
    else
    {
      if (fac_.empty())
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    if (n < SZ(fac_))
      return;
    chmin(n, T::mod() - 1);
    int si = fac_.size();
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    repi(i, si, n + 1)
    {
      fac_[i] = fac_[i - 1] * T::raw(i);
      inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  static T fac(int n)
  {
    assert(n >= 0);
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n];
  }
  static T finv(int n)
  {
    assert(n < T::mod());
    if (n < 0)
      return 0;
    reserve(n);
    return finv_[n];
  }
  static T inv(int n)
  {
    n %= T::mod();
    if (n < 0)
      n += T::mod();
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }

  static T P(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n] * finv_[n - k];
  }
  static T C(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n] * finv_[k] * finv_[n - k];
  }
  static T H(int n, int k)
  {
    if (n == 0 && k == 0)
      return 1;
    return C(n + k - 1, k);
  }
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp


/**
 * @brief string を mint に
 * @docs docs/math/modint/stom.md
 */

template <class mint>
mint stom(string s)
{
  mint res = 0;
  fec(c : s)
  {
    res *= 10;
    res += c - '0';
  }
  return res;
}
// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp


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



/**
 * @brief 最短ベクトル問題（$2$ 次元）
 * @docs docs/math/svp2d.md
 */

// 入力: a = (a1, a2), b = (b1, b2)
// 出力: 整数 x, y を用いて v = (v1, v2) = xa + yb と表されるベクトルのうち、
//      ノルムの 2 乗 v1^2 + v2^2 が最小のもののひとつ (符号は不定)
// T は整数、U は T の 2 倍のビット数
// 初期値の √2 倍が T から溢れないなら正しく動く
template <class T = ll, class U = larger_int_t<T>>
pair<T, T> svp2d(const pair<T, T> &a, const pair<T, T> &b)
{
  assert((a != pair<T, T>{0, 0} && b != pair<T, T>{0, 0}));
  auto [a1, a2] = a;
  auto [b1, b2] = b;
  if constexpr (is_same_v<U, i128> && is_integral_v<T> && is_signed_v<T> && sizeof(T) <= sizeof(ll))
  {
    // 各入力が [-2^30, 2^30) なら、中間値も ll に収まる。
    constexpr ull offset = 1ULL << 30;
    if (((ull(a1) + offset) |
         (ull(a2) + offset) |
         (ull(b1) + offset) |
         (ull(b2) + offset)) < 2 * offset)
      return svp2d<T, ll>(a, b);
  }
  if ((U)a1 * a1 + (U)a2 * a2 < (U)b1 * b1 + (U)b2 * b2)
    swap(a1, b1), swap(a2, b2);
  while ((U)a1 * a1 + (U)a2 * a2 > (U)b1 * b1 + (U)b2 * b2)
  {
    swap(a1, b1), swap(a2, b2);
    const U dot = U(a1) * b1 + U(a2) * b2, norm = U(a1) * a1 + U(a2) * a2;
    U k = dot / norm, r = dot % norm;
    // 最も近い整数を選び、同距離なら大きい方を選ぶ。
    if (r > 0 && r >= norm - r) ++k;
    if (r < 0 && -r > norm + r) --k;
    b1 = T(U(b1) - k * a1), b2 = T(U(b2) - k * a2);
    if (b1 == 0 && b2 == 0)
      return {a1, a2};
  }
  return {a1, a2};
}

/**
 * @brief modint を有理数に
 * @docs docs/math/modint/to_rational.md
 */

template <class mint>
pair<decltype(mint(0).val()), decltype(mint(0).val())>
mint_to_rat(const mint &x)
{
  auto [p, q] = svp2d({x.val(), 1}, {mint::mod(), 0});
  if (q < 0)
    p = -p, q = -q;
  return {p, q};
}

#if __cplusplus >= 202302L
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>
    {
      auto [p, q] = mint_to_rat(x);
      return {p, q};
    }
  };

  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()
    {
#ifdef LOCAL
      if constexpr (_detail::is_value_with_command<T>)
        return can_convert<decltype(std::declval<T>().value)>();
      else
#endif
      if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;
        if constexpr (std::same_as<std::remove_cvref_t<T>, std::remove_cvref_t<Ref>>)
          return false;
        else
          return can_convert<Ref>();
      }
      else if constexpr (requires { std::tuple_size<std::remove_cvref_t<T>>::value; })
      {
        return []<size_t... I>(std::index_sequence<I...>)
        {
          if constexpr ((requires { std::get<I>(std::declval<T>()); } && ...))
            return (can_convert<decltype(std::get<I>(std::declval<T>()))>() || ...);
          else
            return false;
        }(std::make_index_sequence<std::tuple_size_v<std::remove_cvref_t<T>>>{});
      }
      else
        return std::invocable<mint_to_rat_fn, T>;
    }

    // 範囲アダプタ同士の合成は標準ライブラリの operator| に任せる。
    template <typename T>
      requires(!requires { std::views::all | std::declval<T>(); })
    constexpr decltype(auto) operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (can_convert<T>())
      {
#ifdef LOCAL
        // 表示設定の一時オブジェクトを保持し、元の値は参照したままにする。
        if constexpr (_detail::is_value_with_command<T>)
          return rat_object<std::remove_cvref_t<T>>{std::forward<T>(t)};
        else
#endif
        if constexpr (!std::ranges::range<T> || rat_is_map<std::remove_cvref_t<T>>)
          return rat_object<T>{std::forward<T>(t)};
        else
          return std::forward<T>(t) | std::views::transform(*this);
      }
      else
      {
        return std::forward<T>(t);
      }
    }
  };

#ifdef LOCAL
  template <class T>
  string export_object_generic(
      const rat_object<T> &object, const string &indent, size_t last_line_length,
      size_t current_depth, bool fail_on_newline, const _detail::export_command &command)
  {
    using U = std::remove_cvref_t<T>;
    const auto &value = object.value;
    auto render = [&](const auto &x)
    {
      return _detail::export_var(x, indent, last_line_length, current_depth, fail_on_newline, command);
    };
    if constexpr (_detail::is_value_with_command<U>)
      return _detail::export_var(rat_closure{}(value.value), indent, last_line_length,
                                current_depth, fail_on_newline, value.command);
    else if constexpr (!rat_closure::can_convert<const U &>())
      return render(value);
    else if constexpr (_detail::is_map<U>)
      return _detail::export_library_map(value, indent, last_line_length, current_depth,
                                        fail_on_newline, command, rat_closure{});
    else if constexpr (!std::ranges::range<U> && requires { std::tuple_size<U>::value; })
      return std::apply([&](const auto &...member)
      {
        return render(std::tuple{rat_object<const std::remove_reference_t<decltype(member)> &>{member}...});
      }, value);
    else
      return render(rat_closure{}(value));
  }

  // rat() の前後にある表示設定を、cpp-dump と同じ順序で合成する。
  template <class T>
    requires _detail::is_value_with_command<T>
  auto operator|(rat_object<T> object, const _detail::export_command &command)
  {
    return rat_object<T>{object.value | command};
  }

  template <class T>
    requires _detail::is_value_with_command<T>
  auto operator|(rat_object<T> object, _detail::export_command &&command)
  {
    return rat_object<T>{object.value | std::move(command)};
  }
#endif

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

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

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


// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_basic_ops.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_base.hpp


/**
 * @brief 代数的構造の struct（基本）
 * @docs docs/algebra/algebra_base.md
 */

template <class S_, auto op_, auto e_>
struct Monoid
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
};

template <class S_, auto op_, auto e_, auto inv_>
struct Group
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
  static constexpr auto inv = inv_;
};

template <class S_, auto add_, auto e0_, auto mul_, auto e1_>
struct SemiRing
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_>
struct Ring
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_, auto inv_>
struct Field
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
  static constexpr auto inv = inv_;
};

template <class M, class = void>
struct HasInverse : false_type {};
template <class M>
struct HasInverse<M, void_t<decltype(M::inv(declval<const typename M::S &>()))>> : true_type {};

template <class M>
struct OppositeMonoid
{
  using S = typename M::S;
  static constexpr S op(const S &a, const S &b) { return M::op(b, a); }
  static constexpr auto e = M::e;
};
template <class G>
struct OppositeGroup
{
  using S = typename G::S;
  static constexpr S op(const S &a, const S &b) { return G::op(b, a); }
  static constexpr auto e = G::e;
  static constexpr auto inv = G::inv;
};
template <class M>
struct NormalAndOppositeMonoid
{
  struct S
  {
    typename M::S normal;
    typename M::S opposite;
    S() {}
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename M::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args)
        : normal(std::forward<Args>(args)...), opposite(normal) {}
    S rev() { return {opposite, normal}; }
    S(const typename M::S &normal, const typename M::S &opposite) : normal(normal), opposite(opposite) {}
  };
  static constexpr S op(const S &a, const S &b) { return {M::op(a.normal, b.normal), M::op(b.opposite, a.opposite)}; }
  static constexpr S e() { return {M::e(), M::e()}; }
};
template <class G>
struct NormalAndOppositeGroup
{
  struct S
  {
    typename G::S normal;
    typename G::S opposite;
    S() {}
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename G::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args)
        : normal(std::forward<Args>(args)...), opposite(normal) {}
    S rev() { return {opposite, normal}; }
    S(const typename G::S &normal, const typename G::S &opposite) : normal(normal), opposite(opposite) {}
  };
  static constexpr S op(const S &a, const S &b) { return {G::op(a.normal, b.normal), G::op(b.opposite, a.opposite)}; }
  static constexpr S e() { return {G::e(), G::e()}; }
  static constexpr S inv(const S &a) { return {G::inv(a.normal), G::inv(a.opposite)}; }
};

template <class SR>
using MonoidOfSemiRingAdd = Monoid<typename SR::S, SR::add, SR::e0>;
template <class SR>
using MonoidOfSemiRingMul = Monoid<typename SR::S, SR::mul, SR::e1>;
template <class R>
using GroupOfRingAdd = Group<typename R::S, R::add, R::e0, R::minus>;
template <class K>
using GroupOfFieldMul = Group<typename K::S, K::mul, K::e1, K::inv>;

// Madd は可換
template <class Madd, class Mmul>
struct SemiRingFromMonoidMonoid
{
  static_assert(is_same_v<typename Madd::S, typename Mmul::S>, "Madd::S and Mmul::S must be identical");
  using S = typename Madd::S;
  static constexpr auto add = Madd::op;
  static constexpr auto e0 = Madd::e;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

// Gadd は可換
template <class Gadd, class Mmul>
struct RingFromGroupMonoid
{
  static_assert(is_same_v<typename Gadd::S, typename Mmul::S>, "Gadd::S and Mmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

// Gadd, Gmul は可換
template <class Gadd, class Gmul>
struct FieldFromGroupGroup
{
  static_assert(is_same_v<typename Gadd::S, typename Gmul::S>, "Gadd::S and Gmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Gmul::op;
  static constexpr auto e1 = Gmul::e;
  static constexpr auto inv = Gmul::inv;
};

#ifdef LOCAL
CPP_DUMP_DEFINE_EXPORT_OBJECT_GENERIC(normal, opposite);
#endif

/**
 * @brief 代数的構造（四則演算と min, max）
 * @docs docs/algebra/algebra_basic_ops.md
 */

template <class T, class = void>
struct has_e1 : false_type {};
template <class T>
struct has_e1<T, void_t<decltype(T::e1())>> : true_type {};
template <class T>
inline constexpr bool has_e1_v = has_e1<T>::value;

template <class T>
struct MonoidMul
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e()
  {
    if constexpr (has_e1_v<S>)
      return S::e1();
    else
      return 1;
  }
};

template <class T>
struct GroupAddSub
{
  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 S{};
  }
  static constexpr S inv(S a) { return -a; }
  template <class I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k) { return a * k; }
};
template <class T>
struct GroupMulDiv
{
  using S = T;
  static constexpr S op(S a, S b) { return a * b; }
  static constexpr S e()
  {
    if constexpr (has_e1_v<S>)
      return S::e1();
    else
      return S(1);
  }
  static constexpr S inv(S a) { return e() / a; }
};

template <class T, auto infty = nullptr>
using SemiRingMinPlus = SemiRingFromMonoidMonoid<MonoidMin<T, infty>, MonoidAdd<T>>;
template <class T, auto infty = nullptr>
using SemiRingMaxPlus = SemiRingFromMonoidMonoid<MonoidMax<T, infty>, MonoidAdd<T>>;
template <class T>
using RingAddSubMul = RingFromGroupMonoid<GroupAddSub<T>, MonoidMul<T>>;
template <class T>
using FieldAddSubMulDiv = FieldFromGroupGroup<GroupAddSub<T>, GroupMulDiv<T>>;

/**
 * @brief Fenwick Tree
 * @docs docs/ds/fenwick_tree/fenwick_tree.md
 */

// G は可換群 (prefix だけなどであれば可換モノイドでも OK)
template <class G>
struct FenwickTree
{
  using S = typename G::S;

private:
  int n;
  vc<S> dat;

public:
  FenwickTree() {}
  FenwickTree(int n) : n(n), dat(n + 1, G::e()) {}
  FenwickTree(const vc<S> &v) : FenwickTree(v.size())
  {
    repi(i, n) dat[i + 1] = v[i];
    repi(i, 1, n + 1)
    {
      int p = i + (i & -i);
      if (p <= n)
        dat[p] = G::op(dat[p], dat[i]);
    }
  }

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

  // [0, r)
  // 逆元は必要ない
  S sum(int r) const
  {
    assert(0 <= r && r <= n);
    S s = G::e();
    while (r > 0)
    {
      s = G::op(s, dat[r]);
      r -= r & -r;
    }
    return s;
  }
  // [l, r)
  // 逆元が必要
  S sum(int l, int r) const
  {
    assert(0 <= l && l <= r && r <= n);
    // 両端の prefix に共通する節点までで打ち切る。
    S right = G::e(), left = G::e();
    while (l < r)
    {
      right = G::op(right, dat[r]);
      r -= r & -r;
    }
    while (r < l)
    {
      left = G::op(left, dat[l]);
      l -= l & -l;
    }
    return G::op(G::inv(left), right);
  }
  // 逆元が必要
  S get(int i) const
  {
    assert(0 <= i && i < n);
    return sum(i, i + 1);
  }

  // 逆元は必要ない
  void add(int i, S x)
  {
    assert(0 <= i && i < n);
    i++;
    while (i <= n)
    {
      dat[i] = G::op(dat[i], x);
      i += i & -i;
    }
  }
  // 逆元が必要
  void set(int i, S x) { add(i, G::op(G::inv(get(i)), x)); }

private:
  pair<int, S> prefix_lt_raw(S w) const
  {
    if (w <= G::e())
      return {-1, G::e()};
    int k = bit_floor(n);
    int x = 0;
    S v = G::e();
    while (k > 0)
    {
      if (x + k <= n)
      {
        S nv = G::op(v, dat[x + k]);
        if (nv < w)
          v = nv, x += k;
      }
      k >>= 1;
    }
    return {x, v};
  }

public:
  // 非負の通常の整数和を想定する。該当する接頭辞がなければ nullopt。
  template <class I = ll>
  optional<pair<I, S>> lt_max_id_sum(S w) const
  {
    auto result = prefix_lt_raw(w);
    return result.first < 0 ? nullopt : optional<pair<I, S>>(pair<I, S>{I(result.first), std::move(result.second)});
  }
  template <class I = ll>
  I lt_max(S w) const
  {
    static_assert(numeric_limits<I>::is_signed, "index type must represent -1");
    return prefix_lt_raw(w).first;
  }
  template <class I = ll>
  I geq_min(S w) const { return prefix_lt_raw(w).first + 1; }
  template <class I = ll>
  I leq_max(S w) const { return lt_max<I>(w + 1); }
  template <class I = ll>
  I gt_min(S w) const { return geq_min<I>(w + 1); }

  // 要素が [0, size()) の多重集合を管理するのに使ったとき、多重集合のサイズを返す
  inline S size_of_multiset() const { return sum(n); }

  // 要素が [0, size()) の多重集合を管理するのに使ったとき、値 x は何番目から何番目か [l, r)
  inline pair<S, S> order_in_multiset(int x) const { return {sum(x), sum(x + 1)}; }
  // 要素が [0, size()) の多重集合を管理するのに使ったとき、k 番目の値
  // 範囲外の k は nullopt
  template <class I = ll>
  inline optional<I> kth_in_multiset(S k) const
  {
    if (k < 0) return nullopt;
    int key = leq_max<int>(k);
    return key < n ? optional<I>(I(key)) : nullopt;
  }
  // 要素が [0, size()) の多重集合を管理するのに使ったとき、x 未満で最大の要素の順位 (左番兵 -1)
  template <class T, class I = ll>
  inline I lt_max_in_multiset(T x) const
  {
    static_assert(numeric_limits<I>::is_signed, "index type must represent -1");
    S count = sum(clamp(x, T(0), T(n)));
    return I(count) - 1;
  }
  // x 以下で最大の要素の順位 (左番兵 -1)
  template <class T, class I = ll>
  inline I leq_max_in_multiset(T x) const { return lt_max_in_multiset<T, I>(x + 1); }
  // x 以上で最小の要素の順位 (右番兵 size_of_multiset())
  template <class T, class I = ll>
  inline I geq_min_in_multiset(T x) const { return sum(clamp(x, T(0), T(n))); }
  // x 超過で最小の要素の順位 (右番兵 size_of_multiset())
  template <class T, class I = ll>
  inline I gt_min_in_multiset(T x) const { return geq_min_in_multiset<T, I>(x + 1); }

  vc<S> content() const
  {
    vc<S> res(dat.begin() + 1, dat.end());
    // 線形構築で親へ加えた値を、逆順に取り除く。
    for (int i = n; i > 0; i--)
    {
      const int p = i + (i & -i);
      if (p <= n) res[p - 1] = G::op(res[p - 1], G::inv(res[i - 1]));
    }
    return res;
  }
};

void main2()
{
  LL(N);
  VEC(ll, N, A);
  FenwickTree<GroupAddSub<ll>> fw(A);
  set<ll> st{-1, N};
  rep(i, N) if (A[i] == 0) st.insert(i);

  LL(Q);
  rep(_, Q)
  {
    LL(x, b);
    x--;
    fw.set(x, b);
    st.erase(x);
    ll l = *lt_max(st, x).value(), r = *gt_min(st, x).value();
    dump(fw.content(), st, l, r);
    PRINT(fw.sum(l + 1, r));
    fw.set(x, A[x]);
    if (A[x] == 0)
      st.insert(x);
  }
}

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";
      #ifdef LOCAL
      cerr << s;
      #endif
      /* コードテストで確認する際にコメントアウトを外す
      cerr << val;
      //*/
    };
  
    #if defined FAST_IO and not defined LOCAL
    CERR("\n[FAST_IO]\n\n", "32");
    #endif
    #if defined FAST_CIO and not defined LOCAL
    CERR("\n[FAST_CIO]\n\n", "32");
    #endif
    #if (defined FAST_CIO or defined INTERACTIVE) and not defined LOCAL
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    #endif
    cout << fixed << setprecision(20);
  
    init();
    #ifdef LOCAL
    test();
    #endif
  
    #if defined AOJ_TESTCASE or (defined LOCAL and defined SINGLE_TESTCASE)
    CERR("\n[AOJ_TESTCASE]\n\n", "35");
    while (true)
    {
      dump("new testcase");
      main2();
    }
    #elif defined SINGLE_TESTCASE
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
    #elif defined MULTI_TESTCASE
    CERR("\n[MULTI_TESTCASE]\n\n", "33");
    local(while (true))
    {
      dump("T");
      IN(uint, T);
      while (T--)
      {
        dump("new testcase");
        main2();
      }
    }
    #endif
  }
};
Main<init, main2, test> main_dummy;
int main() {}


// submit-code: hash=9ddc5a74935f source=9ddc5a74935fabdd5eea58ffda60d9925c26c5e118b679b98481a2f3b0c803ab input=fd06333d752761ecc7960434c8e4661027f827110aad04745e2e54772b1aa162 body=fd06333d752761ecc7960434c8e4661027f827110aad04745e2e54772b1aa162