#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11
#include <bits/stdc++.h>
using namespace std;
using ld = decltype(EPS);
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;
#define vc vector
template <class T>
using vvc = vc<vc<T>>;
using vpll = vc<pll>;
using vstr = vc<string>;
using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  i128 res = 0;
  if (s.front() == '-')
  {
    for (int i = 1; i < (int)s.size(); i++)
      res = 10 * res + s[i] - '0';
    res = -res;
  }
  else
  {
    for (auto &&c : s)
      res = 10 * res + c - '0';
  }
  return res;
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  if (x < 0)
    x = -x, sign = "-";
  while (x > 0)
  {
    res += '0' + x % 10;
    x /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  is >> s;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}
#define cauto const auto
#define overload4(_1,_2,_3,_4,name,...) name
#define repi1(i,n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i,l,r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)
#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
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>;
template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divround(U a, V b) { return divfloor<T>(2 * T(a) + T(b), 2 * T(b)); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }
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, INF); }
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, INF); }
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)
  {
    if constexpr (sizeof(T) > sizeof(ull))
    {
      if ((u128)a < ((u128)1 << 120))
        return sqrtl(a);
    }
    else
      return sqrtl(a);
  }
  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 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;
}
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, 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;
}
#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) ([&](auto x) { return fx; })
template <class F>
auto gen_vec(int n, const F &f)
{
  vc<decltype(f(0))> res(n);
  repi(i, n) res[i] = f(i);
  return res;
}
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 vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}
template <class T = ll>
T ctol(const char &c, const string &s)
{
  repi(i, SZ<int>(s)) if (s[i] == c) return i;
  return -1;
}
template <class T = ll>
vc<T> stov(const string &s, const string &t)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return ctol(s[i], t); });
}
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 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, 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 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;
}
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()); }
template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V>
void sortunique(V &v)
{
  sort(ALL(v));
  unique(v);
}
template <class V, class U>
void rotate(V &v, U k)
{
  const U n = v.size();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}
template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;
template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};
template <class T, const T infty = INF>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr S e() { return infty; }
};
template <class T, const T infty = INF>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b) { return max(a, b); }
  static constexpr S e() { return -infty; }
};
template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0)
{
  const int n = v.size();
  vc<typename M::S> res(n + 1);
  res[0] = M::e();
  repi(i, n) res[i + 1] = M::op(res[i], v[i]);
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}
template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0)
{ return reversed(cuml<M>(reversed(v), right_index)); }
const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};
template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;
template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::lower_bound(v, val, comp, proj) - v.begin(); }
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
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); }
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>, typename V::const_iterator>
{
  auto it = v.lower_bound(val);
  return it == v.begin() ? v.end() : 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>, typename V::const_iterator>
{
  auto it = v.upper_bound(val);
  return it == v.begin() ? v.end() : 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>, typename V::const_iterator>
{ return v.upper_bound(val); }
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>, typename V::const_iterator>
{ return v.lower_bound(val); }
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); }
#define dump(...)
#define oj(...) __VA_ARGS__
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(); }
namespace fastio {
static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;
inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}
inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}
void rd1(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}
void rd1(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}
template <typename T>
void rd1_real(T &x) {
  string s;
  rd1(s);
  x = stod(s);
}
template <typename T>
void rd1_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  pil--;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus) x = -x;
  }
}
void rd1(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }
template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <size_t N = 0, typename T>
void rd1_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd1(x);
    rd1_tuple<N + 1>(t);
  }
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  rd1_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd1(array<T, N> &x) {
  for (auto &d: x) rd1(d);
}
template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd1(h), read(t...);
}
void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(const string s) {
  for (char c: s) wt1(c);
}
void wt1(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt1(s[i]);
}
template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}
template <typename T>
void wt1_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt1(s);
}
void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }
template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <size_t N = 0, typename T>
void wt1_tuple(const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt1(' '); }
    const auto x = std::get<N>(t);
    wt1(x);
    wt1_tuple<N + 1>(t);
  }
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  wt1_tuple(tpl);
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
void write() {}
template <class Head, class... Tail>
void write(Head &&head, Tail &&... tail) {
  wt1(head);
  write(std::forward<Tail>(tail)...);
}
void print() { wt1('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt1(head);
  if (sizeof...(Tail)) wt1(' ');
  print(std::forward<Tail>(tail)...);
}
}
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
namespace internal
{
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }
template <class T>
void READVECnodump(int n, vc<T> &v)
{
  v.resize(n);
  READnodump(v);
}
template <class T, class... Ts>
void READVECnodump(int n, vc<T> &v, vc<Ts> &...vs)
{ READVECnodump(n, v), READVECnodump(n, vs...); }
template <class T>
void READVEC2nodump(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  READnodump(v);
}
template <class T, class... Ts>
void READVEC2nodump(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ READVEC2nodump(n, m, v), READVEC2nodump(n, m, vs...); }
template <class T>
void READJAGnodump(int n, vvc<T> &v)
{
  v.resize(n);
  repi(i, n)
  {
    int k;
    READnodump(k);
    READVECnodump(k, v[i]);
  }
}
template <class T, class... Ts>
void READJAGnodump(int n, vvc<T> &v, vvc<Ts> &...vs)
{ READJAGnodump(n, v), READJAGnodump(n, vs...); }
};
#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 READVEC2(...) internal::READVEC2nodump(__VA_ARGS__); dump(__VA_ARGS__)
#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define READJAG(...) internal::READJAGnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define PRINT fastio::print
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, 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; }
namespace internal
{
template <size_t... I, class A, class B>
auto tuple_add_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}
};
template <class... Ts, class Tp>
tuple<Ts...> operator+=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> &a, const Tp &b) { return a += b; }
template <class T, class Add>
void offset(vvc<T> &v, const Add &add) { for (auto &vi : v) for (auto &vij : vi) vij += add; }
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)
        { ((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;
}
mt19937_64 mt;
template <class T = ll, class U1, class U2>
T randint(U1 l, U2 r)
{
  assert(T(l) <= T(r));
  return T(l) + mt() % (T(r) - T(l) + 1);
}
template <class T>
struct larger_int
{
  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>
using larger_int_t = typename larger_int<T>::type;
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 = 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);
  }
};
};
namespace internal
{
struct barrett32
{
  uint m;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};
template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static void set_mod(mod_type m) { 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); }
};
};
namespace internal
{
inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}
struct montgomery64odd
{
  ull m, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};
struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b;
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx);
    u128 sq = -u128(mx) % mx;
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    ull p = x & MASK(b);
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};
template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;
  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static void set_mod(mod_type m) { 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{(1LL << 61) - 1};
  static void set_mod(mod_type m) { 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); }
};
};
template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    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>
  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() { return Policy::mod(); }
    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m)) { return T::set_mod(m); }
    static mint raw(V v)
    {
      mint x;
      x._v = v;
      return x;
    }
    modint_impl() : _v(0) {}
    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(Policy::umod());
        if (x < 0)
          x += Policy::umod();
        rem = x;
      }
      else
        rem = V(v % Policy::umod());
      _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);
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }
    mint inv() const
    {
      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 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>;
template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;
template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;
template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};
template <class mint>
struct 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)
  {
    reserve(n);
    return pw[n];
  }
};
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 inv(int n)
  {
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }
  static T C(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    reserve(n);
    return fac_[n] * finv_[k] * finv_[n - k];
  }
};
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 ((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);
    T k = divround<U>((U)a1 * b1 + (U)a2 * b2, (U)a1 * a1 + (U)a2 * a2);
    b1 -= k * a1, b2 -= k * a2;
  }
  return {a1, a2};
}
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};
}
namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };
  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;
        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };
  template <typename T>
    requires(!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
  constexpr auto operator|(T &&t, const rat_closure &c)
  {
    return c(std::forward<T>(t));
  }
}
using mint = modint998244353;
using bi = Binomial<mint>;
void init()
{
  oj(mt.seed(random_device()()));
}
struct use_std_nth_element {
    template <class RandomIt, class Compare>
    void operator()(RandomIt first, RandomIt nth, RandomIt last,
                    Compare comp) const {
        std::nth_element(first, nth, last, comp);
    }
};
struct use_median_of_medians {
private:
    template <class RandomIt, class Compare>
    static void select_impl(RandomIt first, RandomIt nth, RandomIt last,
                            Compare comp) {
        using difference_type =
            typename std::iterator_traits<RandomIt>::difference_type;
        using value_type =
            typename std::iterator_traits<RandomIt>::value_type;
        constexpr difference_type small_limit = 32;
        while (last - first > small_limit) {
            RandomIt medians_end = first;
            for (RandomIt group_first = first; group_first < last;
                 group_first += 5) {
                const difference_type remaining = last - group_first;
                RandomIt group_last =
                    group_first + std::min<difference_type>(5, remaining);
                std::sort(group_first, group_last, comp);
                RandomIt median = group_first + (group_last - group_first) / 2;
                std::iter_swap(medians_end, median);
                ++medians_end;
            }
            RandomIt pivot_pos = first + (medians_end - first) / 2;
            select_impl(first, pivot_pos, medians_end, comp);
            const value_type pivot = *pivot_pos;
            RandomIt equal_first = std::partition(
                first, last,
                [&](const value_type& x) { return comp(x, pivot); });
            RandomIt equal_last = std::partition(
                equal_first, last,
                [&](const value_type& x) { return !comp(pivot, x); });
            if (nth < equal_first) {
                last = equal_first;
            } else if (nth >= equal_last) {
                first = equal_last;
            } else {
                return;
            }
        }
        std::sort(first, last, comp);
    }
public:
    template <class RandomIt, class Compare>
    void operator()(RandomIt first, RandomIt nth, RandomIt last,
                    Compare comp) const {
        assert(first <= nth && nth < last);
        select_impl(first, nth, last, comp);
    }
};
namespace sorted_matrix_selection_detail {
using rank_type = std::uint64_t;
enum class answer_kind : unsigned char { upper, lower, middle };
struct answer_location {
    answer_kind kind;
    rank_type middle_rank = 0;
};
template <class Accessor, class Compare, class LocalSelector>
class selector {
public:
    using value_type = std::decay_t<
        std::invoke_result_t<Accessor&, std::size_t, std::size_t>>;
    selector(std::size_t n, Accessor accessor, Compare compare,
             LocalSelector local_selector)
        : original_n_(n),
          accessor_(std::move(accessor)),
          compare_(std::move(compare)),
          local_selector_(std::move(local_selector)) {}
    value_type run(rank_type k) {
        if (original_n_ == 0) {
            throw std::invalid_argument(
                "sorted_matrix_select: n must be positive");
        }
        const rank_type rank_n = static_cast<rank_type>(original_n_);
        if (static_cast<std::size_t>(rank_n) != original_n_ ||
            rank_n > std::numeric_limits<rank_type>::max() / rank_n) {
            throw std::overflow_error(
                "sorted_matrix_select: n*n does not fit in uint64_t");
        }
        const rank_type total = rank_n * rank_n;
        if (k >= total) {
            throw std::out_of_range(
                "sorted_matrix_select: k is outside [0, n*n)");
        }
        std::vector<std::size_t> positions(original_n_);
        std::iota(positions.begin(), positions.end(), std::size_t{0});
        return biselect(positions, k + 1, k + 1).first;
    }
private:
    value_type at(const std::vector<std::size_t>& positions,
                  std::size_t i, std::size_t j) {
        return std::invoke(accessor_, positions[i],
                           original_n_ - 1 - positions[j]);
    }
    bool less(const value_type& x, const value_type& y) {
        return std::invoke(compare_, x, y);
    }
    static rank_type ceil_div4(rank_type x) {
        return x / 4 + static_cast<rank_type>(x % 4 != 0);
    }
    static rank_type ceil_div4_sum(rank_type x, rank_type y) {
        return x / 4 + y / 4 +
               static_cast<rank_type>((x % 4 + y % 4 + 3) / 4);
    }
    std::pair<value_type, value_type>
    biselect(const std::vector<std::size_t>& positions,
             rank_type k1, rank_type k2) {
        const std::size_t n = positions.size();
        const rank_type total = static_cast<rank_type>(n) * n;
        assert(n > 0);
        assert(1 <= k2 && k2 <= k1 && k1 <= total);
        if (n <= 2) {
            std::vector<value_type> values;
            values.reserve(n * n);
            for (std::size_t i = 0; i < n; ++i) {
                for (std::size_t j = 0; j < n; ++j) {
                    values.push_back(at(positions, i, j));
                }
            }
            std::sort(values.begin(), values.end(), std::ref(compare_));
            return {values[static_cast<std::size_t>(k1 - 1)],
                    values[static_cast<std::size_t>(k2 - 1)]};
        }
        const std::size_t sub_n = (n + 2) / 2;
        std::vector<std::size_t> sub_positions(sub_n);
        for (std::size_t i = 0; i < sub_n; ++i) {
            sub_positions[i] = positions[std::min(2 * i, n - 1)];
        }
        rank_type sub_k1;
        if ((n & 1U) == 0) {
            sub_k1 = static_cast<rank_type>(n) + 1 + ceil_div4(k1);
        } else {
            sub_k1 = ceil_div4_sum(
                k1, 2 * static_cast<rank_type>(n) + 1);
        }
        const rank_type sub_k2 = ceil_div4(k2);
        const rank_type sub_total =
            static_cast<rank_type>(sub_n) * sub_n;
        assert(1 <= sub_k2 && sub_k2 <= sub_k1 &&
               sub_k1 <= sub_total);
        auto [upper, lower] =
            biselect(sub_positions, sub_k1, sub_k2);
        assert(!less(upper, lower));
        rank_type rank_less_upper = 0;
        rank_type rank_greater_lower = 0;
        std::size_t first_less_upper = 0;
        std::size_t first_not_greater_lower = 0;
        for (std::size_t i = 0; i < n; ++i) {
            while (first_less_upper < n) {
                value_type x = at(positions, i, first_less_upper);
                if (less(x, upper)) break;
                ++first_less_upper;
            }
            rank_less_upper += n - first_less_upper;
            while (first_not_greater_lower < n) {
                value_type x =
                    at(positions, i, first_not_greater_lower);
                if (!less(lower, x)) break;
                ++first_not_greater_lower;
            }
            rank_greater_lower += first_not_greater_lower;
        }
        const auto locate = [&](rank_type k) -> answer_location {
            if (rank_less_upper <= k - 1) {
                return {answer_kind::upper, 0};
            }
            const rank_type count_not_greater_lower =
                total - rank_greater_lower;
            if (k <= count_not_greater_lower) {
                return {answer_kind::lower, 0};
            }
            return {answer_kind::middle,
                    k - count_not_greater_lower};
        };
        const answer_location where1 = locate(k1);
        const answer_location where2 = locate(k2);
        if (where1.kind != answer_kind::middle &&
            where2.kind != answer_kind::middle) {
            return {where1.kind == answer_kind::upper ? upper : lower,
                    where2.kind == answer_kind::upper ? upper : lower};
        }
        assert(less(lower, upper));
        assert(rank_less_upper + rank_greater_lower >= total);
        const rank_type middle_count =
            rank_less_upper - (total - rank_greater_lower);
        assert(middle_count <=
               static_cast<rank_type>(
                   std::numeric_limits<std::size_t>::max()));
        std::vector<value_type> middle;
        middle.reserve(static_cast<std::size_t>(middle_count));
        first_less_upper = 0;
        first_not_greater_lower = 0;
        for (std::size_t i = 0; i < n; ++i) {
            while (first_less_upper < n) {
                value_type x = at(positions, i, first_less_upper);
                if (less(x, upper)) break;
                ++first_less_upper;
            }
            while (first_not_greater_lower < n) {
                value_type x =
                    at(positions, i, first_not_greater_lower);
                if (!less(lower, x)) break;
                ++first_not_greater_lower;
            }
            for (std::size_t j = first_less_upper;
                 j < first_not_greater_lower; ++j) {
                middle.push_back(at(positions, i, j));
            }
        }
        assert(middle.size() == middle_count);
        const auto pick = [&](rank_type r) -> value_type {
            assert(1 <= r && r <= middle.size());
            auto nth = middle.begin() + static_cast<std::ptrdiff_t>(r - 1);
            std::invoke(local_selector_, middle.begin(), nth, middle.end(),
                        std::ref(compare_));
            return *nth;
        };
        value_type answer1 =
            where1.kind == answer_kind::upper
                ? upper
                : where1.kind == answer_kind::lower
                      ? lower
                      : pick(where1.middle_rank);
        value_type answer2 =
            where2.kind == answer_kind::upper
                ? upper
                : where2.kind == answer_kind::lower
                      ? lower
                      : pick(where2.middle_rank);
        return {std::move(answer1), std::move(answer2)};
    }
    std::size_t original_n_;
    Accessor accessor_;
    Compare compare_;
    LocalSelector local_selector_;
};
}
template <class Accessor, class Compare = std::less<>>
auto sorted_matrix_select(std::size_t n, std::uint64_t k,
                          Accessor accessor, Compare compare = {}) {
    using impl = sorted_matrix_selection_detail::selector<
        Accessor, Compare, use_std_nth_element>;
    return impl(n, std::move(accessor), std::move(compare),
                use_std_nth_element{})
        .run(k);
}
template <class T>
inline constexpr bool is_rational_ordered_v = is_integral_ext<T> || is_floating_point_v<T>;
template <class T>
struct Rational
{
  using value_type = T;
  T num, den;
private:
  void normalize_sign()
  {
    if constexpr (is_signed_ext<T> || is_floating_point_v<T>)
      if (den < T(0))
        num = -num, den = -den;
  }
  static T gcd_abs(T a, T b)
  {
    if constexpr (is_signed_ext<T>)
    {
      if (a < 0) a = -a;
      if (b < 0) b = -b;
    }
    while (b != T(0))
    {
      T r = a % b;
      a = b, b = r;
    }
    return a;
  }
public:
  Rational() : num(0), den(1) {}
  Rational(const T &num) : num(num), den(1) {}
  Rational(const T &num, const T &den) : num(num), den(den)
  {
    assert(den != T(0));
    normalize_sign();
  }
  pair<T, T> reduced() const
  {
    if constexpr (is_integral_ext<T>)
    {
      T g = gcd_abs(num, den);
      return g == T(0) ? pair<T, T>{num, den} : pair<T, T>{num / g, den / g};
    }
    else
      return {num, den};
  }
  Rational operator-() const { return {-num, den}; }
  Rational operator+() const { return *this; }
  Rational &operator+=(const Rational &rhs)
  {
    T rhs_num = rhs.num, rhs_den = rhs.den;
    num = num * rhs_den + rhs_num * den;
    den *= rhs_den;
    normalize_sign();
    return *this;
  }
  Rational &operator-=(const Rational &rhs)
  {
    T rhs_num = rhs.num, rhs_den = rhs.den;
    num = num * rhs_den - rhs_num * den;
    den *= rhs_den;
    normalize_sign();
    return *this;
  }
  Rational &operator*=(const Rational &rhs)
  {
    T rhs_num = rhs.num, rhs_den = rhs.den;
    num *= rhs_num;
    den *= rhs_den;
    normalize_sign();
    return *this;
  }
  Rational &operator/=(const Rational &rhs)
  {
    assert(rhs.num != T(0));
    T rhs_num = rhs.num, rhs_den = rhs.den;
    num *= rhs_den;
    den *= rhs_num;
    normalize_sign();
    return *this;
  }
  friend Rational operator+(Rational lhs, const Rational &rhs) { return lhs += rhs; }
  friend Rational operator-(Rational lhs, const Rational &rhs) { return lhs -= rhs; }
  friend Rational operator*(Rational lhs, const Rational &rhs) { return lhs *= rhs; }
  friend Rational operator/(Rational lhs, const Rational &rhs) { return lhs /= rhs; }
  friend bool operator==(const Rational &lhs, const Rational &rhs)
  {
    using C = larger_int_t<T>;
    return C(lhs.num) * C(rhs.den) == C(rhs.num) * C(lhs.den);
  }
  friend bool operator!=(const Rational &lhs, const Rational &rhs) { return !(lhs == rhs); }
  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator<(const Rational &lhs, const Rational &rhs)
  {
    using C = larger_int_t<T>;
    return C(lhs.num) * C(rhs.den) < C(rhs.num) * C(lhs.den);
  }
  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator>(const Rational &lhs, const Rational &rhs) { return rhs < lhs; }
  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator<=(const Rational &lhs, const Rational &rhs) { return !(rhs < lhs); }
  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator>=(const Rational &lhs, const Rational &rhs) { return !(lhs < rhs); }
  friend ostream &operator<<(ostream &os, const Rational &x)
  {
    auto [num, den] = x.reduced();
    return os << num << '/' << den;
  }
  friend void wt1(const Rational &x)
  {
    auto [num, den] = x.reduced();
    using fastio::wt1;
    wt1(num), wt1('/'), wt1(den);
  }
};
template <class T>
struct is_rational : false_type
{};
template <class T>
struct is_rational<Rational<T>> : true_type
{};
template <class T>
inline constexpr bool is_rational_v = is_rational<T>::value;
namespace std
{
template <class T>
struct numeric_limits<::Rational<T>> : numeric_limits<T>
{
  static constexpr bool is_specialized = numeric_limits<T>::is_specialized;
  static ::Rational<T> min() noexcept { return ::Rational<T>(numeric_limits<T>::min()); }
  static ::Rational<T> lowest() noexcept { return ::Rational<T>(numeric_limits<T>::lowest()); }
  static ::Rational<T> max() noexcept { return ::Rational<T>(numeric_limits<T>::max()); }
  static constexpr bool is_signed = numeric_limits<T>::is_signed;
  static constexpr bool is_integer = false;
  static constexpr bool is_exact = numeric_limits<T>::is_exact;
};
}
void main2()
{
  LL(N);
  VEC(ll, N, A);
  sort(ALL(A));
  auto ans = sorted_matrix_select(N, N * (N - 1) / 2, [&](ll i, ll j)
                                  { return Rational<int>(A[i], A[j]); });
  PRINT(ans.reduced());
}
void test()
{
}
template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
    };
    CERR("\n[FAST_IO]\n\n", "32");
    cout << fixed << setprecision(20);
    init();
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}