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

// #define SINGLE_TESTCASE
// #define AOJ_TESTCASE

// #define FAST_CIO
// #define INTERACTIVE

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

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

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

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

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

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

using namespace std;

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

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

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

#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 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>;
}
#endif

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

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

using ld = decltype(EPS);

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

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

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

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

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

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

#define overload4(_1,_2,_3,_4,name,...) name
#define LIBRARY_REPEAT_LOOP(T,i,l,r,d) for (auto [i, library_repeat_end, library_repeat_step] = array<T, 3>{T(l), T(r), T(d)}; library_repeat_step > 0 ? i < library_repeat_end : i > library_repeat_end; i += library_repeat_step)

#define repi1(i,n) LIBRARY_REPEAT_LOOP(int, i, 0, n, 1)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

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

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

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

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

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

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

template <class T>
struct make_unsigned_ext
{
  using type = make_unsigned_t<T>;
};
template <>
struct make_unsigned_ext<i128>
{
  using type = u128;
};
template <>
struct make_unsigned_ext<u128>
{
  using type = u128;
};
template <class T>
using make_unsigned_ext_t = typename make_unsigned_ext<T>::type;

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

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

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

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

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

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

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

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

  template <class T>
  const T &custom_value();

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

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

namespace template_math_detail
{

} 

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b)
{
  const T x = T(a), y = T(b);
  if constexpr (is_signed_ext<T>)
    if (y == T(-1)) return T(0);
  const T r = x % y;
  return r != T(0) && is_negative(x) != is_negative(y) ? T(r + y) : r;
}

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

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

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

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;
};

}

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

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

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

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

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

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

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

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};
template <class T, auto infty = nullptr>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr decltype(auto) e() { return resolved_infty<T, infty>(); }
};
template <class T, auto infty = nullptr>
struct MonoidMax
{
  using S = T;
};

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
  {
  };
}

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

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

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;
}
};

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 = 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

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

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

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

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

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

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

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

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

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

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) ((void)0)
#define local(...) 
#define oj(...) __VA_ARGS__

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

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

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

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

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

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

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

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

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

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

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

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

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

        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 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;
      
      *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);
  inline void wt1(char x) { append_char(x); }
  inline void wt1(const char *x) { append_bytes(x, strlen(x)); }
  inline void wt1(string_view x) { append_bytes(x.data(), x.size()); }
  inline void wt1(const string &x) { append_bytes(x.data(), x.size()); }

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

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

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

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

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

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

namespace internal
{

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

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

}; 

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

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

#define LL(...) IN(ll, __VA_ARGS__)

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

#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)

#define PRINT fastio::print

namespace internal
{

}; 

namespace internal
{

};

namespace internal
{

};

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

mt19937_64 mt;

namespace internal
{
}; 

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

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

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

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

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

namespace larger_int_detail
{
}

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

public:
  using type = T;
};

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

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

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

#undef LARGER_INT

template <class T>
struct Rational;

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

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

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

namespace internal
{

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

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

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

};

namespace internal
{

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

  static constexpr bool is_prime = isprime_constexpr(M);

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

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

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

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

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

namespace internal
{

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

struct montgomery64odd
{
  ull m = 0, im = 0, sq = 0;
  montgomery64odd() = default;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

//! 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 value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

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

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

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

namespace internal
{

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

  private:
    V _v;

  public:

    modint_impl();

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

    M val() const;

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

  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;
using modint1000000007 = static_modint32<1000000007>;
using modint = dynamic_modint32<-1>;
using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64 = dynamic_modint64<-1>;

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

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

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

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

template <class mint>
struct modint_less
{
};

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

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

public:

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

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

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

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

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

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

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

  struct mint_to_rat_fn
  {
  };

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

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

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

  };

}

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/cumulative_sum/imos.hpp

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

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

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 auto e = M::e;
};
template <class G>
struct OppositeGroup
{
  using S = typename G::S;
  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;
  };
};
template <class G>
struct NormalAndOppositeGroup
{
  struct S
  {
    typename G::S normal;
    typename G::S 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>;

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;
};

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;
};

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;
};

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;
};

template <class T>
struct GroupAddSub
{
  using S = T;
};
template <class T>
struct GroupMulDiv
{
  using S = T;
};

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>>;

template <class G>
struct Imos
{
  using S = typename G::S;

private:
  vc<S> d;

public:

};
// https://github.com/miscalculation53/library/tree/wip/ds/segtree/sparse_lazy_segtree.hpp

template <class AM>
struct SparseLazySegmentTree
{
  using S = typename AM::S;
  using F = typename AM::F;

private:
  template <class A, class = void>
  struct HasPow : false_type {};
  template <class A>
  struct HasPow<A, void_t<decltype(A::pow(declval<const S &>(), ll{}))>> : true_type {};

  static S repeat(S x, ll len)
  {
    if constexpr (HasPow<AM>::value)
      return AM::pow(x, len);
    else
    {
      S res = AM::e();
      while (len)
      {
        if (len & 1) res = AM::op(res, x);
        len >>= 1;
        if (len) x = AM::op(x, x);
      }
      return res;
    }
  }

  struct Entry
  {
    ll l, r;
    S unit, val;
    Entry(ll l, ll r, const S &unit) : l(l), r(r), unit(unit), val(repeat(unit, r - l)) {}
  };
  struct Node
  {
    Entry entry;
    S prod;
    F lazy;
    ll lo, hi;
    array<int, 2> chi;
    bool has_lazy;
    Node(const Entry &entry)
        : entry(entry), prod(entry.val), lazy(AM::id()), lo(entry.l), hi(entry.r), chi{-1, -1}, has_lazy(false) {}
  };
  ll n;
  vc<Node> nodes;

  int make_node(const Entry &entry)
  {
    int i = nodes.size();
    nodes.eb(entry);
    return i;
  }

  void all_apply(int i, const F &f)
  {
    nodes[i].entry.unit = AM::mapping(f, nodes[i].entry.unit);
    nodes[i].entry.val = AM::mapping(f, nodes[i].entry.val);
    nodes[i].prod = AM::mapping(f, nodes[i].prod);
    if (nodes[i].has_lazy)
      nodes[i].lazy = AM::composition(f, nodes[i].lazy);
    else
    {
      nodes[i].lazy = f;
      nodes[i].has_lazy = true;
    }
  }

  void push(int i)
  {
    if (!nodes[i].has_lazy) return;
    F f = nodes[i].lazy;
    repi(d, 2)
    {
      int j = nodes[i].chi[d];
      if (j != -1) all_apply(j, f);
    }
    nodes[i].lazy = AM::id();
    nodes[i].has_lazy = false;
  }

  void pull(int i)
  {
    int l = nodes[i].chi[0], r = nodes[i].chi[1];
    S sml = l == -1 ? AM::e() : nodes[l].prod;
    S smr = r == -1 ? AM::e() : nodes[r].prod;
    nodes[i].prod = AM::op(AM::op(sml, nodes[i].entry.val), smr);
    nodes[i].lo = l == -1 ? nodes[i].entry.l : nodes[l].lo;
    nodes[i].hi = r == -1 ? nodes[i].entry.r : nodes[r].hi;
  }

  void insert(Entry x, ll a, ll b, int i)
  {
    push(i);
    ll c = a + (b - a) / 2;
    int d = x.l < c ? 0 : 1;
    if ((d == 0 && nodes[i].entry.l < x.l) || (d == 1 && x.l < nodes[i].entry.l))
      swap(x, nodes[i].entry);
    int j = nodes[i].chi[d];
    if (j == -1)
    {
      j = make_node(x);
      nodes[i].chi[d] = j;
    }
    else if (d == 0)
      insert(std::move(x), a, c, j);
    else
      insert(std::move(x), c, b, j);
    pull(i);
  }

  template <class G>
  void modify_entry(ll p, const G &g, int i)
  {
    push(i);
    if (nodes[i].entry.l == p)
      g(nodes[i].entry);
    else
      modify_entry(p, g, nodes[i].chi[p < nodes[i].entry.l ? 0 : 1]);
    pull(i);
  }

  void split(ll p)
  {
    if (p == 0 || p == n) return;
    int i = 0, prev = -1;
    while (i != -1)
    {
      push(i);
      if (nodes[i].entry.l == p) return;
      if (nodes[i].entry.l < p)
        prev = i, i = nodes[i].chi[1];
      else
        i = nodes[i].chi[0];
    }
    ll l = nodes[prev].entry.l, r = nodes[prev].entry.r;
    S unit = nodes[prev].entry.unit;
    modify_entry(l, [&](Entry &x)
    {
      x.r = p;
      x.val = repeat(x.unit, p - l);
    }, 0);
    insert(Entry(p, r, unit), 0, n, 0);
  }

  void apply_impl(ll l, ll r, const F &f, int i)
  {
    if (nodes[i].hi <= l || r <= nodes[i].lo) return;
    if (l <= nodes[i].lo && nodes[i].hi <= r)
    {
      all_apply(i, f);
      return;
    }
    push(i);
    ll p = nodes[i].entry.l;
    if (l <= p && p < r)
    {
      nodes[i].entry.unit = AM::mapping(f, nodes[i].entry.unit);
      nodes[i].entry.val = AM::mapping(f, nodes[i].entry.val);
    }
    repi(d, 2)
    {
      int j = nodes[i].chi[d];
      if (j != -1) apply_impl(l, r, f, j);
    }
    pull(i);
  }

public:
  
  SparseLazySegmentTree(ll n, const S &initial_value) : n(n)
  {
    assert(n >= 0);
    nodes.eb(Entry(0, n, initial_value));
  }

  void apply(ll l, ll r, const F &f)
  {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return;
    split(l), split(r);
    apply_impl(l, r, f, 0);
  }

  S all_prod() const { return nodes[0].prod; }

};

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

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

template <class T, class L = ll>
struct MonoidLenSum
{
  using UL = make_unsigned_ext_t<common_type_t<L, ull>>;
  struct S
  {
    L len;
    T val;
    CPP_DUMP_DEFINE_DATA(len, val);
  };
};

template <class T, auto infty = nullptr>
struct MonoidMinMax
{
  struct S
  {
    T mn = resolved_infty<T, infty>(), mx = -resolved_infty<T, infty>();
    CPP_DUMP_DEFINE_DATA(mn, mx);
  };
};

template <class T, auto infty = nullptr, class L = ll>
struct MonoidMinSum
{
  using UL = make_unsigned_ext_t<common_type_t<L, ull>>;
  struct S
  {
    T mn = resolved_infty<T, infty>(), sum = T{};
    L len = 0;
    CPP_DUMP_DEFINE_DATA(mn, sum, len);
  };
};

template <class T, auto infty = nullptr, class L = ll>
struct MonoidMaxSum
{
  using UL = make_unsigned_ext_t<common_type_t<L, ull>>;
  struct S
  {
    T mx = -resolved_infty<T, infty>(), sum = T{};
    L len = 0;
    CPP_DUMP_DEFINE_DATA(mx, sum, len);
  };
};

template <class T, auto infty = nullptr, class L = ll>
struct MonoidMinMaxSum
{
  using UL = make_unsigned_ext_t<common_type_t<L, ull>>;
  struct S
  {
    T mn = resolved_infty<T, infty>(), mx = -resolved_infty<T, infty>(), sum = T{};
    L len = 0;
    CPP_DUMP_DEFINE_DATA(mn, mx, sum, len);
  };
};

template <class T, auto infty = nullptr>
struct ActedMonoidAddMin
{
  using M = MonoidMin<T, infty>;
  using MF = MonoidAdd<T>;
  using S = typename M::S;
  using F = typename MF::S;
  static constexpr auto op = M::op;
  static constexpr auto e = M::e;
  static constexpr S mapping(F f, S x) { return f + x; }
  static constexpr auto composition = MF::op;
  static constexpr auto id = MF::e;
};

template <class T, auto infty = nullptr>
struct ActedMonoidAddMax
{
  using M = MonoidMax<T, infty>;
  using MF = MonoidAdd<T>;
  using S = typename M::S;
  using F = typename MF::S;
  static constexpr auto op = M::op;
  static constexpr auto e = M::e;
  static constexpr auto composition = MF::op;
  static constexpr auto id = MF::e;
};

template <class T, auto infty = nullptr>
struct ActedMonoidAddMinMax
{
  using M = MonoidMinMax<T, infty>;
  using MF = MonoidAdd<T>;
  using S = typename M::S;
  using F = typename MF::S;
  static constexpr auto op = M::op;
  static constexpr auto e = M::e;
  static constexpr auto composition = MF::op;
  static constexpr auto id = MF::e;
};

void main2()
{
  LL(N, L, K);
  VEC(ll, N, X);

  auto judge = [&](ll P)
  {
    if (P < 0)
      return false;
    dump(P);
    SparseLazySegmentTree<ActedMonoidAddMin<ll>> seg(L, 0);
    fe(x : X)
    {
      ll l = x - P, r = x + P;
      if (l < 0)
        seg.apply(safemod(l, L), L, 1);
      seg.apply(max(0LL, l), min(L, r), 1);
      if (r > L)
        seg.apply(0, r % L, 1);
      dump(seg.content());
    }
    return seg.all_prod() >= K;
  };
  ll ans = expsearch(judge, 0).first;
  PRINT(ans);
}

void test()
{

}

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

template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      
    };
  
    CERR("\n[FAST_IO]\n\n", "32");
    
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[MULTI_TESTCASE]\n\n", "33");
    local(while (true))
    {
      dump("T");
      IN(uint, T);
      while (T--)
      {
        dump("new testcase");
        main2();
      }
    }
    
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}


// submit-code: hash=0772ad2b7395 source=0772ad2b73958fcfb94e56a41fcc3dbf17ce171292c6bda089c9551570676df2 input=f07add052fc7a32b4d23160769431c530e06393172a49e7a2f44a192a4f67a1d body=f6692d740c0e68efc85da63164635588ad315dc9fbf669d3778390a23a911cfc