// TLE tasukete

/**
 * date   : 2025-12-23 01:22:11
 * author : Nyaan
 */

#define NDEBUG

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tr2/dynamic_bitset>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility

namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;

template <typename T, typename U>
struct P : pair<T, U> {
  template <typename... Args>
  constexpr P(Args... args) : pair<T, U>(args...) {}

  using pair<T, U>::first;
  using pair<T, U>::second;

  P &operator+=(const P &r) {
    first += r.first;
    second += r.second;
    return *this;
  }
  P &operator-=(const P &r) {
    first -= r.first;
    second -= r.second;
    return *this;
  }
  P &operator*=(const P &r) {
    first *= r.first;
    second *= r.second;
    return *this;
  }
  template <typename S>
  P &operator*=(const S &r) {
    first *= r, second *= r;
    return *this;
  }
  P operator+(const P &r) const { return P(*this) += r; }
  P operator-(const P &r) const { return P(*this) -= r; }
  P operator*(const P &r) const { return P(*this) *= r; }
  template <typename S>
  P operator*(const S &r) const {
    return P(*this) *= r;
  }
  P operator-() const { return P{-first, -second}; }
};

using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T>
int sz(const T &t) {
  return t.size();
}

template <typename T, typename U>
inline bool amin(T &x, U y) {
  return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
  return (x < y) ? (x = y, true) : false;
}

template <typename T>
inline T Max(const vector<T> &v) {
  return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
  return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
  return accumulate(begin(v), end(v), 0LL);
}

template <typename T>
int lb(const vector<T> &v, const T &a) {
  return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
  return upper_bound(begin(v), end(v), a) - begin(v);
}

constexpr long long TEN(int n) {
  long long ret = 1, x = 10;
  for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
  return ret;
}

template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
  return make_pair(t, u);
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
  vector<T> ret(v.size() + 1);
  if (rev) {
    for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
  } else {
    for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
  }
  return ret;
};

template <typename T>
vector<T> mkuni(const vector<T> &v) {
  vector<T> ret(v);
  sort(ret.begin(), ret.end());
  ret.erase(unique(ret.begin(), ret.end()), ret.end());
  return ret;
}

template <typename F>
vector<int> mkord(int N, F f) {
  vector<int> ord(N);
  iota(begin(ord), end(ord), 0);
  sort(begin(ord), end(ord), f);
  return ord;
}

template <typename T>
vector<int> mkinv(vector<T> &v) {
  int max_val = *max_element(begin(v), end(v));
  vector<int> inv(max_val + 1, -1);
  for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
  return inv;
}

vector<int> mkiota(int n) {
  vector<int> ret(n);
  iota(begin(ret), end(ret), 0);
  return ret;
}

template <typename T>
T mkrev(const T &v) {
  T w{v};
  reverse(begin(w), end(w));
  return w;
}

template <typename T>
bool nxp(T &v) {
  return next_permutation(begin(v), end(v));
}

// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
  vector<vector<T>> ret;
  vector<T> v;
  auto dfs = [&](auto rc, int i) -> void {
    if (i == (int)a.size()) {
      ret.push_back(v);
      return;
    }
    for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
  };
  dfs(dfs, 0);
  return ret;
}

// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
  T res = I;
  for (; n; f(a = a * a), n >>= 1) {
    if (n & 1) f(res = res * a);
  }
  return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I = T{1}) {
  return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}

template <typename T>
T Rev(const T &v) {
  T res = v;
  reverse(begin(res), end(res));
  return res;
}

template <typename T>
vector<T> Transpose(const vector<T> &v) {
  using U = typename T::value_type;
  if(v.empty()) return {};
  int H = v.size(), W = v[0].size();
  vector res(W, T(H, U{}));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) {
      res[j][i] = v[i][j];
    }
  }
  return res;
}

template <typename T>
vector<T> Rotate(const vector<T> &v, int clockwise = true) {
  using U = typename T::value_type;
  int H = v.size(), W = v[0].size();
  vector res(W, T(H, U{}));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) {
      if (clockwise) {
        res[W - 1 - j][i] = v[i][j];
      } else {
        res[j][H - 1 - i] = v[i][j];
      }
    }
  }
  return res;
}

}  // namespace Nyaan


// bit operation

namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
  return __builtin_popcountll(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
  if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace Nyaan


// inout

namespace Nyaan {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
  os << p.first << " " << p.second;
  return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  int s = (int)v.size();
  for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
  return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (auto &x : v) is >> x;
  return is;
}

istream &operator>>(istream &is, __int128_t &x) {
  string S;
  is >> S;
  x = 0;
  int flag = 0;
  for (auto &c : S) {
    if (c == '-') {
      flag = true;
      continue;
    }
    x *= 10;
    x += c - '0';
  }
  if (flag) x = -x;
  return is;
}

istream &operator>>(istream &is, __uint128_t &x) {
  string S;
  is >> S;
  x = 0;
  for (auto &c : S) {
    x *= 10;
    x += c - '0';
  }
  return is;
}

ostream &operator<<(ostream &os, __int128_t x) {
  if (x == 0) return os << 0;
  if (x < 0) os << '-', x = -x;
  string S;
  while (x) S.push_back('0' + x % 10), x /= 10;
  reverse(begin(S), end(S));
  return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
  if (x == 0) return os << 0;
  string S;
  while (x) S.push_back('0' + x % 10), x /= 10;
  reverse(begin(S), end(S));
  return os << S;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
  cin >> t;
  in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
  cout << t;
  if (sizeof...(u)) cout << sep;
  out(u...);
}

struct IoSetupNya {
  IoSetupNya() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(15);
    cerr << fixed << setprecision(7);
  }
} iosetupnya;

}  // namespace Nyaan


// debug


#ifdef NyaanDebug
#define trc(...) (void(0))
#else
#define trc(...) (void(0))
#endif

#ifdef NyaanLocal
#define trc2(...) (void(0))
#else
#define trc2(...) (void(0))
#endif


// macro

#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...)   \
  int __VA_ARGS__; \
  in(__VA_ARGS__)
#define inl(...)         \
  long long __VA_ARGS__; \
  in(__VA_ARGS__)
#define ins(...)      \
  string __VA_ARGS__; \
  in(__VA_ARGS__)
#define in2(s, t)                           \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i]);                         \
  }
#define in3(s, t, u)                        \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i]);                   \
  }
#define in4(s, t, u, v)                     \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i], v[i]);             \
  }
#define die(...)             \
  do {                       \
    Nyaan::out(__VA_ARGS__); \
    return;                  \
  } while (0)


namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }


//



using namespace std;




using namespace std;

// x / y (x > 0, y > 0) を管理、デフォルトで 1 / 1
// 入力が互いに素でない場合は gcd を取って格納
// seq : (1, 1) から (x, y) へのパス。右の子が正/左の子が負
template <typename Int>
struct SternBrocotTreeNode {
  using Node = SternBrocotTreeNode;

  Int lx, ly, x, y, rx, ry;
  vector<Int> seq;

  SternBrocotTreeNode() : lx(0), ly(1), x(1), y(1), rx(1), ry(0) {}

  SternBrocotTreeNode(Int X, Int Y) : SternBrocotTreeNode() {
    assert(1 <= X && 1 <= Y);
    Int g = gcd(X, Y);
    X /= g, Y /= g;
    while (min(X, Y) > 0) {
      if (X > Y) {
        Int d = X / Y;
        X -= d * Y;
        go_right(d - (X == 0 ? 1 : 0));
      } else {
        Int d = Y / X;
        Y -= d * X;
        go_left(d - (Y == 0 ? 1 : 0));
      }
    }
  }
  SternBrocotTreeNode(const pair<Int, Int> &xy)
      : SternBrocotTreeNode(xy.first, xy.second) {}
  SternBrocotTreeNode(const vector<Int> &_seq) : SternBrocotTreeNode() {
    for (const Int &d : _seq) {
      assert(d != 0);
      if (d > 0) go_right(d);
      if (d < 0) go_left(d);
    }
    assert(seq == _seq);
  }

  // pair<Int, Int> 型で分数を get
  pair<Int, Int> get() const { return make_pair(x, y); }
  // 区間の下限
  pair<Int, Int> lower_bound() const { return make_pair(lx, ly); }
  // 区間の上限
  pair<Int, Int> upper_bound() const { return make_pair(rx, ry); }

  // 根からの深さ
  Int depth() const {
    Int res = 0;
    for (auto &s : seq) res += abs(s);
    return res;
  }
  // 左の子に d 進む
  void go_left(Int d = 1) {
    if (d <= 0) return;
    if (seq.empty() or seq.back() > 0) seq.push_back(0);
    seq.back() -= d;
    rx += lx * d, ry += ly * d;
    x = rx + lx, y = ry + ly;
  }
  // 右の子に d 進む
  void go_right(Int d = 1) {
    if (d <= 0) return;
    if (seq.empty() or seq.back() < 0) seq.push_back(0);
    seq.back() += d;
    lx += rx * d, ly += ry * d;
    x = rx + lx, y = ry + ly;
  }
  // 親の方向に d 進む
  // d 進めたら true, 進めなかったら false を返す
  bool go_parent(Int d = 1) {
    if (d <= 0) return true;
    while (d != 0) {
      if (seq.empty()) return false;
      Int d2 = min(d, seq.back() < 0 ? -seq.back() : seq.back());
      if (seq.back() > 0) {
        x -= rx * d2, y -= ry * d2;
        lx = x - rx, ly = y - ry;
        seq.back() -= d2;
      } else {
        x -= lx * d2, y -= ly * d2;
        rx = x - lx, ry = y - ly;
        seq.back() += d2;
      }
      d -= d2;
      if (seq.back() == 0) seq.pop_back();
      if (d2 == Int{0}) break;
    }
    return true;
  }
  // SBT 上の LCA
  static Node lca(const Node &lhs, const Node &rhs) {
    Node n;
    for (int i = 0; i < min<int>(lhs.seq.size(), rhs.seq.size()); i++) {
      Int val1 = lhs.seq[i], val2 = rhs.seq[i];
      if ((val1 < 0) != (val2 < 0)) break;
      if (val1 < 0) n.go_left(min(-val1, -val2));
      if (val1 > 0) n.go_right(min(val1, val2));
      if (val1 != val2) break;
    }
    return n;
  }
  friend ostream &operator<<(ostream &os, const Node &rhs) {
    os << "\n";
    os << "L : ( " << rhs.lx << ", " << rhs.ly << " )\n";
    os << "M : ( " << rhs.x << ", " << rhs.y << " )\n";
    os << "R : ( " << rhs.rx << ", " << rhs.ry << " )\n";
    os << "seq : {";
    for (auto &x : rhs.seq) os << x << ", ";
    os << "} \n";
    return os;
  }
  friend bool operator<(const Node &lhs, const Node &rhs) {
    return lhs.x * rhs.y < rhs.x * lhs.y;
  }
  friend bool operator==(const Node &lhs, const Node &rhs) {
    return lhs.x == rhs.x and lhs.y == rhs.y;
  }
};

/**
 *  @brief Stern-Brocot Tree
 */


// 極小から始まる下向き凸包の頂点列挙
// (xl, yl) 始点, x in [xl, xr]
// inside(x, y) : (x, y) が凸包内部か？
// candicate(x, y, c, d) : (x, y) が凸包外部にあるとする。
// 凸包内部の点 (x + sc, y + sd) が存在すればそのような s を返す
// (ただし s は誤差でズレてもいいように s ± 2 を探索する仕様)
// 存在しなければ任意の値 (-1 でもよい) を返す
template <typename Int>
vector<pair<Int, Int>> enumerate_convex(
    Int xl, Int yl, Int xr, const function<bool(Int, Int)>& inside,
    const function<Int(Int, Int, Int, Int)>& candicate) {
  assert(xl <= xr);

  // inside かつ x <= xr
  auto f = [&](Int x, Int y) { return x <= xr && inside(x, y); };

  // (a, b) から (c, d) 方向に進めるだけ進む
  auto go = [&](Int a, Int b, Int c, Int d) -> Int {
    assert(f(a, b));
    Int r = 1, s = 0;
    while (f(a + r * c, b + r * d)) r *= 2;
    while ((r /= 2) != 0) {
      if (f(a + r * c, b + r * d)) s += r, a += r * c, b += r * d;
    }
    return s;
  };

  // (a, b) が out, (a + c * k, b + d * k) が in とする
  // out の間進めるだけ進む
  auto go2 = [&](Int a, Int b, Int c, Int d, Int k) {
    assert(!f(a, b) and f(a + c * k, b + d * k));
    Int ok = 0, ng = k;
    while (ok + 1 < ng) {
      Int m = (ok + ng) / 2;
      (f(a + c * m, b + d * m) ? ng : ok) = m;
    }
    return ok;
  };

  vector<pair<Int, Int>> ps;
  Int x = xl, y = yl;
  assert(f(x, y) and go(x, y, 0, -1) == 0);
  ps.emplace_back(x, y);
  SternBrocotTreeNode<Int> sb;
  while (x < xr) {
    Int a, b;
    if (f(x + 1, y)) {
      a = 1, b = 0;
    } else {
      while (!f(x + sb.lx, y + sb.ly)) {
        assert(!sb.seq.empty());
        Int bc = sb.seq.back();
        sb.go_parent(bc < 0 ? -bc : bc);
      }
      while (true) {
        assert(f(x + sb.lx, y + sb.ly));
        assert(!f(x + sb.rx, y + sb.ry));
        if (f(x + sb.lx + sb.rx, y + sb.ly + sb.ry)) {
          Int s = go(x + sb.lx, y + sb.ly, sb.rx, sb.ry);
          assert(s > 0);
          sb.go_right(s);
        } else {
          Int c = candicate(x + sb.rx, y + sb.ry, sb.lx, sb.ly);
          if (sb.lx) c = min(c, (xr - x - sb.rx) / sb.lx);
          Int s = -1;
          // 念のため周囲を調べる(フェイルセーフ)
          for (Int d = -2; d <= 2; d++) {
            Int v = c + d;
            if (v > 0 && f(x + sb.lx * v + sb.rx, y + sb.ly * v + sb.ry)) {
              s = v;
              break;
            }
          }
          if (s == -1) {
            a = sb.lx, b = sb.ly;
            break;
          } else {
            Int t = go2(x + sb.rx, y + sb.ry, sb.lx, sb.ly, s);
            sb.go_left(t);
          }
        }
      }
    }
    Int s = go(x, y, a, b);
    x += a * s, y += b * s;
    ps.emplace_back(x, y);
  }
  return ps;
}




// sum_{0 <= i < N} (ai + b) // m
template <typename T>
T sum_of_floor(T n, T m, T a, T b) {
  T ret = 0;
  if (a >= m) ret += (n - 1) * n * (a / m) / 2, a %= m;
  if (b >= m) ret += n * (b / m), b %= m;
  T y = (a * n + b) / m;
  if (y == 0) return ret;
  T x = y * m - b;
  ret += (n - (x + a - 1) / a) * y;
  ret += sum_of_floor(y, a, m, (a - x % a) % a);
  return ret;
}

// verify www.codechef.com/viewsolution/36222026
// count x : ax + b mod m < yr, 0 <= x < xr
template <typename T>
T mod_affine_range_counting(T a, T b, T m, T xr, T yr) {
  assert(0 <= yr && yr <= m);
  return sum_of_floor(xr, m, a, b + m) - sum_of_floor(xr, m, a, b + m - yr);
}


using namespace Nyaan;


template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
  static_assert(r * mod == 1, "this code has bugs.");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  constexpr mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }
  constexpr mint operator+() const { return mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  constexpr mint inverse() const {
    int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
    while (y > 0) {
      t = x / y;
      x -= t * y, u -= t * v;
      tmp = x, x = y, y = tmp;
      tmp = u, u = v, v = tmp;
    }
    return mint{u};
  }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }

  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};





using namespace std;

// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
  vector<T> f, g, h;
  Binomial(int MAX = 0) {
    assert(T::get_mod() != 0 && "Binomial<mint>()");
    f.resize(1, T{1});
    g.resize(1, T{1});
    h.resize(1, T{1});
    if (MAX > 0) extend(MAX + 1);
  }

  void extend(int m = -1) {
    int n = f.size();
    if (m == -1) m = n * 2;
    m = min<int>(m, T::get_mod());
    if (n >= m) return;
    f.resize(m);
    g.resize(m);
    h.resize(m);
    for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
    g[m - 1] = f[m - 1].inverse();
    h[m - 1] = g[m - 1] * f[m - 2];
    for (int i = m - 2; i >= n; i--) {
      g[i] = g[i + 1] * T(i + 1);
      h[i] = g[i] * f[i - 1];
    }
  }

  T fac(int i) {
    if (i < 0) return T(0);
    while (i >= (int)f.size()) extend();
    return f[i];
  }

  T finv(int i) {
    if (i < 0) return T(0);
    while (i >= (int)g.size()) extend();
    return g[i];
  }

  T inv(int i) {
    if (i < 0) return -inv(-i);
    while (i >= (int)h.size()) extend();
    return h[i];
  }

  T C(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r) * finv(r);
  }

  inline T operator()(int n, int r) { return C(n, r); }

  template <typename I>
  T multinomial(const vector<I>& r) {
    static_assert(is_integral<I>::value == true);
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return T(0);
      n += x;
    }
    T res = fac(n);
    for (auto& x : r) res *= finv(x);
    return res;
  }

  template <typename I>
  T operator()(const vector<I>& r) {
    return multinomial(r);
  }

  T C_naive(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    T ret = T(1);
    r = min(r, n - r);
    for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
    return ret;
  }

  T P(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r);
  }

  // [x^r] 1 / (1-x)^n
  T H(long long n, long long r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};


//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;
// using mint = LazyMontgomeryModInt<1000000007>;
using vm = vector<mint>;
using vvm = vector<vm>;
Binomial<mint> C;

mint naive(ll N) {
  mint ans = 0;
  rep1(H, N) {
    ll x = (2 * N + 1) / (2 * H + 1);
    if (x < 3) break;
    ans += (x - 1) / 2;
  }
  return ans;
}

mint calc(ll N) {
  // (-2x+1)(2y+1)>=2N+2
  auto inside = [&](i128 x, i128 y) -> bool {
    return (-2 * x + 1) * (2 * y + 1) >= 2 * N + 2;
  };
  auto candicate = [&](i128 x, i128 y, i128 c, i128 d) -> i128 {
    if (c == 0) return infLL;
    i128 a = x * d - y * c;
    // (-2x+1)(2y+1)=2N+2
    // xd-yc=a
    // (-2x+1)(2yc+c)=c(2N+2)
    // (-2x+1)(2(xd-a)+c)=c(2N+2)
    // (-2x+1)(2dx+(c-2a))=const
    i128 A = -2 * 2 * d;
    i128 B = -2 * (c - 2 * a) + 2 * d;
    long double x_dst = -1.0l * B / (2 * A);
    i128 res = max<i128>(1, (x_dst - x) / c + 0.5);
    trc(x, y, c, d, res);
    return res;
  };

  auto cv = enumerate_convex<i128>(-(N + 1), 0, 0, inside, candicate);
  trc(cv);

  mint ans = 0;
  rep(i, sz(cv) - 1) {
    ll y = cv[i].se;
    ll dx = cv[i + 1].fi - cv[i].fi;
    ll dy = cv[i + 1].se - cv[i].se;

    mint cur = sum_of_floor<i128>(dx, dx, dy, dx - 1) - dx;
    cur += mint{dx} * y;
    ans += cur;
  }
  return ans + 1;
}

void q() {
  inl(N);
  out(calc(N));
}

void Nyaan::solve() {
  int t = 1;
  in(t);
  while (t--) q();
}