#define YRSD

#include <iostream>
#include <algorithm>
#include <array>
#include <bitset>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <string>
#include <tuple>
#include <bit>
#include <chrono>
#include <functional>
#include <iomanip>
#include <utility>
#include <type_traits>
#include <cassert>
#include <cctype>
#include <cmath>
#include <cstring>
#include <ctime>
#include <limits>
#include <ranges>
#include <concepts>

#define TE template <typename T>
#define TES template <typename T, typename ...S>
#define Z auto
#define ep emplace_back
#define eb emplace
#define fi first
#define se second
#define bg begin
#define ed end
#define all(x) bg(x), ed(x)

#define ov(a, b, c, d, e, ...) e
#define FO1(a) for (int _ = 0; _ < (a); ++_)
#define FO2(i, a) for (int i = 0; i < (a); ++i)
#define FO3(i, a, b) for (int i = (a); i < (b); ++i)
#define FO4(i, a, b, c) for (int i = (a); i < (b); i += (c))
#define FOR(...) ov(__VA_ARGS__, FO4, FO3, FO2, FO1)(__VA_ARGS__)
#define FF1(a) for (int _ = (a) - 1; _ >= 0; --_)
#define FF2(i, a) for (int i = (a) - 1; i >= 0; --i)
#define FF3(i, a, b) for (int i = (b) - 1; i >= (a); --i)
#define FF4(i, a, b, c) for (int i = (b) - 1; i >= (a); i -= (c))
#define FOR_R(...) ov(__VA_ARGS__, FF4, FF3, FF2, FF1)(__VA_ARGS__)
#define FOR_subset(t, s) for (int t = (s); t > -1; t = (t == 0 ? -1 : (t - 1) & s))

#define sort ranges::sort

using namespace std;

TE using vc = vector<T>;
TE using vvc = vc<vc<T>>;
TE using T1 = tuple<T>;
TE using T2 = tuple<T, T>;
TE using T3 = tuple<T, T, T>;
TE using T4 = tuple<T, T, T, T>;
TE using max_heap = priority_queue<T>;
TE using min_heap = priority_queue<T, vc<T>, greater<T>>;
using u8 = unsigned char; using uint = unsigned int; using ll = long long;      using ull = unsigned long long;
using ld = long double;   using i128 = __int128;     using u128 = __uint128_t;  using f128 = __float128;
using u16 = uint16_t;
using PII = pair<int, int>;   using PLL = pair<ll, ll>;

#ifdef YRSD
constexpr bool dbg = 1;
#else
constexpr bool dbg = 0;
#endif

istream &operator>>(istream &I, i128 &x) {
  static string s;
  I >> s;
  int f = s[0] == '-';
  x = 0;
  const int N = (int)s.size();
  FOR(i, f, N) x = x * 10 + s[i] - '0';
  if (f) x = -x;
  return I;
}
ostream &operator<<(ostream &O, i128 x) {
  static string s;
  s.clear();
  bool f = x < 0;
  if (f) x = -x;
  while (x) s += '0' + x % 10, x /= 10;
  if (s.empty()) s += '0';
  if (f) s += '-';
  reverse(all(s));
  return O << s;
}
istream &operator>>(istream &I, f128 &x) {
  static string s;
  I >> s, x = stold(s);
  return I;
}
ostream &operator<<(ostream &O, const f128 x) { return O << ld(x); }
template <typename... S>
istream &operator>>(istream &I, tuple<S...> &t) {
  return apply([&I](Z &...s) { ((I >> s), ...); }, t), I;
}
template <typename T, typename U>
istream &operator>>(istream &I, pair<T, U> &x) {
  return I >> x.fi >> x.se;
}
template <typename T, typename U>
ostream &operator<<(ostream &O, const pair<T, U> &x) {
  return O << x.fi << ' ' << x.se;
}
TE requires requires(T &c) { begin(c); end(c); } and 
                          (not is_same_v<decay_t<T>, string>)
istream &operator>>(istream &I, T &c) {
  for (Z &e : c) I >> e;
  return I;
}
TE requires requires(const T &c) { begin(c); end(c); } and 
  (not is_same_v<decay_t<T>, const char*>) and 
  (not is_same_v<decay_t<T>, string>) and 
  (not is_array_v<remove_reference_t<T>> or 
   not is_same_v<remove_extent_t<remove_reference_t<T>>, char>)
ostream &operator<<(ostream &O, const T &a) {
  if (a.empty()) return O;
  Z i = a.begin();
  O << *i++;
  for (; i != a.end(); ++i) O << ' ' << *i;
  return O;
}
void IN() {}
TE void IN(T &x, Z &...s) { cin >> x, IN(s...); }
void print() { cout << '\n'; }
TES void print(T &&x, S &&...y) {
  cout << x;
  if constexpr (sizeof...(S)) cout << ' ';
  print(forward<S>(y)...);
}
void put() {}
TES void put(T &&x, S &&...y) {
  cout << x;
  put(forward<S>(y)...);
}

#define INT(...)  int    __VA_ARGS__; IN(__VA_ARGS__)
#define UINT(...) uint   __VA_ARGS__; IN(__VA_ARGS__)
#define LL(...)   ll     __VA_ARGS__; IN(__VA_ARGS__)
#define ULL(...)  ull    __VA_ARGS__; IN(__VA_ARGS__)
#define I128(...) i128   __VA_ARGS__; IN(__VA_ARGS__)
#define STR(...)  string __VA_ARGS__; IN(__VA_ARGS__)
#define CH(...)   char   __VA_ARGS__; IN(__VA_ARGS__)
#define REAL(...) re     __VA_ARGS__; IN(__VA_ARGS__)
#define VEC(T, a, n) vc<T> a(n); IN(a)

void YES(bool o = 1) { print(o ? "YES" : "NO"); }
void Yes(bool o = 1) { print(o ? "Yes" : "No"); }
void yes(bool o = 1) { print(o ? "yes" : "no"); }
void NO(bool o = 1) { YES(not o); }
void No(bool o = 1) { Yes(not o); }
void no(bool o = 1) { yes(not o); }
void ALICE(bool o = 1) { print(o ? "ALICE" : "BOB"); }
void Alice(bool o = 1) { print(o ? "Alice" : "Bob"); }
void alice(bool o = 1) { print(o ? "alice" : "bob"); }
void BOB(bool o = 1) { ALICE(not o); }
void Bob(bool o = 1) { Alice(not o); }
void bob(bool o = 1) { alice(not o); }
void POSSIBLE(bool o = 1) { print(o ? "POSSIBLE" : "IMPOSSIBLE"); }
void Possible(bool o = 1) { print(o ? "Possible" : "Impossible"); }
void possible(bool o = 1) { print(o ? "possible" : "impossible"); }
void IMPOSSIBLE(bool o = 1) { POSSIBLE(not o); }
void Impossible(bool o = 1) { Possible(not o); }
void impossible(bool o = 1) { possible(not o); }
void TAK(bool o = 1) { print(o ? "TAK" : "NIE"); }
void NIE(bool o = 1) { TAK(not o); }

#if (__cplusplus >= 202002L)
#include <numbers>
constexpr ld pi = numbers::pi_v<ld>;
#endif
TE constexpr T inf = numeric_limits<T>::max();
template <> constexpr i128 inf<i128> = i128(inf<ll>) * 2'000'000'000'000'000'000;
template <typename T, typename U>
constexpr pair<T, U> inf<pair<T, U>> = {inf<T>, inf<U>};

TE constexpr static inline int pc(T x) { return popcount(make_unsigned_t<T>(x)); }
constexpr static inline ll len(const Z &a) { return a.size(); }

void reverse(Z &a) { reverse(all(a)); }

void unique(Z &a) {
  sort(a);
  a.erase(unique(all(a)), a.end());
}
TE vc<int> inverse(const vc<T> &a) {
  int N = len(a);
  vc<int> b(N, -1);
  FOR(i, N) if (a[i] != -1) b[a[i]] = i;
  return b;
}

Z QMAX(const Z &a) { return *max_element(all(a)); }
Z QMIN(const Z &a) { return *min_element(all(a)); }
TE Z QMAX(T l, T r) { return *max_element(l, r); }
TE Z QMIN(T l, T r) { return *min_element(l, r); }
constexpr bool chmax(Z &a, const Z &b) { return (a < b ? a = b, 1 : 0); }
constexpr bool chmin(Z &a, const Z &b) { return (a > b ? a = b, 1 : 0); }

vc<int> argsort(const Z &a) {
  vc<int> I(len(a));
  iota(all(I), 0);
  sort(I, [&](int i, int k) { return a[i] < a[k] or (a[i] == a[k] and i < k); });
  return I;
}
TE vc<T> rearrange(const vc<T> &a, const vc<int> &I) {
  int N = len(I);
  vc<T> b(N);
  FOR(i, N) b[i] = a[I[i]];
  return b;
}
template <int of = 1, typename T> 
vc<T> pre_sum(const vc<T> &a) {
  int N = len(a);
  vc<T> c(N + 1);
  FOR(i, N) c[i + 1] = c[i] + a[i];
  if (of == 0) c.erase(c.begin());
  return c;
}

TE constexpr static int topbit(T x) {
  if (x == 0) return - 1;
  if constexpr (sizeof(T) <= 4) return 31 - __builtin_clz(x);
  else return 63 - __builtin_clzll(x);
}
TE constexpr static int lowbit(T x) {
  if (x == 0) return -1;
  if constexpr (sizeof(T) <= 4) return __builtin_ctz(x);
  else return __builtin_ctzll(x);
}

TE constexpr T floor(T x, T y) { return x / y - (x % y and (x ^ y) < 0); }
TE constexpr T ceil(T x, T y) { return floor(x + y - 1, y); }
TE constexpr T bmod(T x, T y) { return x - floor(x, y) * y; }
TE constexpr pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return pair{q, x - q * y};
}
TE T SUM(const Z &v) { return accumulate(all(v), T()); }
TE T SUM(Z l, Z r) { return accumulate(l, r, T()); }
int lb(const Z &a, Z x) { return lower_bound(all(a), x) - a.begin(); }
TE int lb(T l, T r, Z x) { return lower_bound(l, r, x) - l; }
int ub(const Z &a, Z x) { return upper_bound(all(a), x) - a.begin(); }
TE int ub(T l, T r, Z x) { return upper_bound(l, r, x) - l; }

template <bool ck = 1>
ll bina(Z f, ll l, ll r) {
  if (ck) assert(f(l));
  while (abs(l - r) > 1) {
    ll x = (r + l) >> 1;
    (f(x) ? l : r) = x;
  }
  return l;
}
TE T bina_real(Z f, T l, T r, int c = 100) {
  while (c--) {
    T x = (l + r) / 2;
    (f(x) ? l : r) = x;
  }
  return (l + r) / 2;
}

Z pop(Z &s) {
  if constexpr (requires { s.pop_back(); }) {
    Z x = s.back();
    return s.pop_back(), x;
  } else if constexpr (requires { s.top(); }) {
    Z x = s.top();
    return s.pop(), x;
  } else {
    Z x = s.front();
    return s.pop(), x;
  }
}
void setp(int x) { cout << fixed << setprecision(x); }

TE inline void sh(vc<T> &a, int N, T b = {}) {
  a.resize(N, b);
}
#ifdef YRSD
void DBG() { cerr << "]" << endl; }
TES void DBG(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  DBG(forward<S>(y)...);
}
#define debug(...) cerr << "[" << __LINE__ << "]: [" #__VA_ARGS__ "] = [", DBG(__VA_ARGS__)
void ERR() { cerr << endl; }
TES void ERR(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  ERR(forward<S>(y)...);
}
#define err(...) cerr << "[" << __LINE__ << "]: ", ERR(__VA_ARGS__)
#else
#define debug(...) void(0721)
#define err(...)   void(0721)
#endif

template <typename T, int sz = 1 << 12>
struct node_pool {
  using np = T*;
  union U {
    U *nx;
    alignas(T) u8 nd[sizeof(T)];
  };
  vc<U*> a, b;
  U *l = 0, *r = 0, *h = 0;

  node_pool() { add(); }
  ~node_pool() { re(); } 

  void reserve(int x) { 
    int t = (x + sz - 1) / sz, n = len(b) + len(a);
    FOR(i, t, n) a.ep(new U[sz]);
  }

  template <typename... S>
  np gen(S&&...x) {
    U* s = ac();
    return ::new(&s->nd) T{std::forward<S>(x)...};
  }

  np cl(np x) { 
    if (not x) return x;
    return gen(*x);
  }

  void del(np x) { 
    if (not x) return ;
    x->~T();
    U* s = reinterpret_cast<U*>(x);
    s->nx = h, h = s;
  }

  void reset() { h = 0, re(), add(); } 

  U* ac() {
    if (h) {
      U* s = h;
      return h = h->nx, s;
    }
    if (l == r) add();
    return l++;
  }

  void add() {
    U* x;
    if (not a.empty()) x = pop(a);
    else x = new U[sz];
    b.ep(x), l = x, r = x + sz;
  }

  void re() { 
    if (not b.empty()) {
      for (Z x : b) delete[] x;
      b.clear();
    }
    if (not a.empty()) {
      for (Z x : a) delete[] x;
      a.clear();
    }
    l = r = 0;
  }
};

template <typename T, int sz = 1 << 12>
struct node_pool_sim {
  using np = T*;
  union U {
    U *nx;
    alignas(T) u8 nd[sizeof(T)];
  };

  vc<U*> a;
  U *l = 0, *r = 0;

  node_pool_sim() { add(); }

  ~node_pool_sim() {
    for (Z x : a) delete[] x;
  }

  template <typename ...S>
  np gen(S &&...x) {
    U *s = ac();
    return ::new(&s->nd) T(std::forward<S>(x)...);
  }

  U *ac() {
    if (l == r) add();
    return l++;
  }

  void add() {
    U* x = new U[sz];
    a.ep(x), l = x, r = x + sz;
  }
};

template <typename am, bool persistent = 0, typename ll = int, int sz = 1 << 18>
struct segdl_t {
  using AM = am;
  using MX = AM::MX;
  using MA = AM::MA;
  using X = MX::X;
  using A = MA::X;

  struct node {
    node *l, *r;
    X x;
    A f;
    void upd() {
      x = l ? l->x : MX::unit();
      if (r) x = MX::op(x, r->x);
    }
  };
  using np = node*;
  node_pool<node, sz> a;
  ll L, R;

  segdl_t(ll l, ll r) : L(l), R(r) {}
  ~segdl_t() { a.reset(); }

  np newnode() { return newnode(MX::unit()); }
  np newnode(X x) { return a.gen(nullptr, nullptr, x, MA::unit()); }
  np newnode(np l, np r) {
    np t = a.gen(l, r, MX::unit(), MA::unit());
    return t->upd(), t;
  }
  np newnode(const vc<X> &a) {
    Z f = [&](Z &f, int l, int r) -> np {
      if (l == r) return 0;
      if (l + 1 == r) return newnode(a[l]);
      int m = (l + r) >> 1;
      return newnode(f(f, l, m), f(f, m, r));
    };
    return f(f, 0, len(a));
  }

  np cl(np x) {
    if (not persistent or not x) return x;
    return a.cl(x);
  }

  void push(np x, ll l, ll r) {
    if (x->f == MA::unit()) return;
    ll m = (l + r) >> 1;
    x->l = x->l ? cl(x->l) : newnode();
    x->l->x = AM::act(x->l->x, x->f, m - l);
    x->l->f = MA::op(x->l->f, x->f);
    x->r = x->r ? cl(x->r) : newnode();
    x->r->x = AM::act(x->r->x, x->f, r - m);
    x->r->f = MA::op(x->r->f, x->f);
    x->f = MA::unit();
  }

  vc<X> get_all(np x) {
    vc<X> s;
    Z f = [&](Z &f, np x, ll l, ll r, A a) -> void {
      if (not x) return;
      if (l + 1 == r) return s.ep(AM::act(x->x, a, l, r)), void();
      ll m = (l + r) >> 1;
      a = MA::op(x->f, a);
      f(f, x->l, l, m, a), f(f, x->r, m, r, a);
    };
    return f(f, x, L, R, MA::unit()), s;
  }

  void reset() { a.reset(); }

  X prod(np x, ll l, ll r) {
    if (l == r or not x) return MX::unit();
    return _p(x, L, R, l, r, MA::unit());
  }

  X _p(np x, ll l, ll r, ll ql, ll qr, A f) {
    chmax(ql, l), chmin(qr, r);
    if (ql >= qr) return MX::unit();
    if (not x) return AM::act(MX::unit(), f, qr - ql);
    if (l == ql and r == qr) return AM::act(x->x, f, r - l);
    ll m = (l + r) >> 1;
    f = MA::op(x->f, f);
    return MX::op(_p(x->l, l, m, ql, qr, f), _p(x->r, m, r, ql, qr, f));
  }

  np set(np x, ll i, const X &w) {
    if (not x) x = newnode();
    return _s(x, L, R, i, w);
  }

  np _s(np x, ll l, ll r, ll i, const X &w, bool cp = 1) {
    if (l + 1 == r) {
      if (cp) {
        if (cp) x = cl(x);
        return x->x = w, x->f = MA::unit(), x;
      }
    }
    push(x, l, r);
    ll m = (l + r) >> 1;
    bool f = 1;
    x = cl(x);
    if (i < m) {
      if (not x->l) x->l = newnode(), f = 0;
      x->l = _s(x->l, l, m, i, w, f);
    } else {
      if (not x->r) x->r = newnode(), f = 0;
      x->r = _s(x->r, m, r, i, w, f);
    }
    return x->upd(), x;
  }

  np multiply(np x, ll i, const X &w) {
    if (not x) x = newnode();
    return _m(x, L, R, i, w);
  }

  np _m(np x, ll l, ll r, ll i, const X &w, bool cp = 1) {
    if (l + 1 == r) {
      if (cp) x = cl(x);
      return x->x = MX::op(x->x, w), x->f = MA::unit(), x;
    }
    push(x, l, r);
    ll m = (l + r) >> 1;
    if (cp) x = cl(x);
    bool f = 1;
    if (i < m) {
      if (not x->l) x->l = newnode(), f = 0;
      x->l = _m(x->l, l, m, i, w, f);
    } else {
      if (not x->r) x->r = newnode(), f = 0;
      x->r = _m(x->r, m, r, i, w, f);
    }
    return x->upd(), x;
  }

  np apply(np x, ll l, ll r, const A &f) {
    if (l == r) return x;
    return _a(x, L, R, l, r, f);
  }

  np _a(np x, ll l, ll r, ll ql, ll qr, const A &f) {
    if (not x) x = newnode();
    chmax(ql, l), chmin(qr, r);
    if (ql >= qr) return x;
    if (l == ql and r == qr)
      return x = cl(x), x->x = AM::act(x->x, f, r - l), 
             x->f = MA::op(x->f, f), x;
    push(x, l, r);
    ll m = (l + r) >> 1;
    x = cl(x);
    x->l = _a(x->l, l, m, ql, qr, f), x->r = _a(x->r, m, r, ql, qr, f);
    return x->upd(), x;
  }

  ll max_right(np x, const Z &f, ll l) {
    X w = MX::unit();
    return _max_r(x, f, L, R, l, w);
  }

  ll _max_r(np x, const Z &f, ll l, ll r, ll ql, X &w) {
    if (r <= ql) return r;
    X pr = x ? x->x : MX::unit();
    if (ql <= l and f(MX::op(w, pr))) return w = MX::op(w, pr), R;
    if (l + 1 == r) return l;
    push(x, l, r);
    ll m = (l + r) >> 1, k = _max_r(x ? x->l : x, f, l, m, ql, w);
    if (k != R) return k;
    return _max_r(x ? x->r : x, f, m, r, ql, w);
  }

  ll min_left(np x, const Z &f, ll l) {
    X w = MX::unit();
    return _min_l(x, f, L, R, l, w);
  }

  ll _min_l(np x, const Z &f, ll l, ll r, ll qr, X &w) {
    if (qr <= l) return L;
    X pr = x ? x->x : MX::unit();
    if (r <= qr and f(MX::op(pr, w))) return w = MX::op(pr, w), L;
    if (l + 1 == r) return r;
    push(x, l, r);
    ll m = (l + 1) >> 1, k = _min_l(x ? x->r : x, f, m, r, qr, w);
    if (k != L) return k;
    return _min_l(x ? x->l : x, l, m, qr, w);
  }

  
  np copy(np x, np y, ll l, ll r, A f = MA::unit()) {
    if (x == y) return x;
    x = cl(x);
    _copy(x, y, L, R, l, r, f);
    return x;
  }

  void _copy(np x, np y, ll l, ll r, ll ql, ll qr, A f) {
    chmax(ql, l), chmin(qr, r);
    if (ql >= qr) return;
    if (l == ql and r == qr) {
      if (y) *x = {y->l, y->r, AM::act(y->x, f, r - l), MA::op(y->f, f)};
      else *x = {0, 0, AM::act(MX::unit(), f, r - l), f};
      return;
    }
    push(x, l, r);
    if (y) f = MA::op(y->f, f);
    ll m = (l + r) >> 1;
    x->l = x->l ? cl(x->l) : newnode();
    x->r = x->r ? cl(x->r) : newnode();
    _copy(x->l, y ? y->l : y, l, m, ql, qr, f);
    _copy(x->r, y ? y->r : y, m, r, ql, qr, f);
    x->upd();
  }

  
  np merge_to(np x, np y, const Z &op, A f = MA::unit()) {
    return _merge_to(x, y, L, R, op, f);
  }

  np _merge_to(np x, np y, ll l, ll r, const Z &op, A f) {
    if (not x) return y;
    if (not y) return x;
    np re = x;
    if (l + 1 == r) return re->x = op(x->x, y->x), re->f = MA::unit(), re;
    push(x, l, r);
    ll m = (l + r) >> 1;
    f = MA::op(y->f, f);
    re->l = _merge_to(x->l, y->l, l, m, op, f);
    re->r = _merge_to(x->r, y->r, m, r, op, f);
    return re->upd(), a.del(y), re;
  }

  np merge_new(np x, np y, const Z &op, A f = MA::unit()) {
    return _merge_new(x, y, L, R, op, f);
  }

  np _merge_new(np x, np y, ll l, ll r, const Z &op, A f) {
    if (not x) return y;
    if (not y) return x;
    np re = newnode();
    if (l + 1 == r) return re->x = op(x->x, y->x), re->f = MA::unit(), re;
    ll m = (l + r) >> 1;
    f = MA::op(x->f, f);
    re->l = _merge_new(x->l, y->l, l, m, op, f);
    re->r = _merge_new(x->r, y->r, m, r, op, f);
    return re->upd(), re;
  }

  
  np spl(np x, np &y, ll l, ll r, A f = MA::unit()) {
    if (not x and y) x = newnode();
    return _spl(x, y, L, R, l, r, f), x;
  }
  
  void _spl(np &x, np &y, ll l, ll r, ll ql, ll qr, A f = MA::unit()) {
    if (not y) return;
    chmax(ql, l), chmin(qr, r);
    if (ql >= qr) return;
    if (l == ql and r == qr) {
      x = y;
      
      y = 0;
      return;
    }
    if (not x) x = newnode();
    ll m = (l + r) >> 1;
    f = MA::op(y->f, f);
    _spl(x->l, y->l, l, m, ql, qr, f);
    _spl(x->r, y->r, m, r, ql, qr, f);
    x->upd();
    if (not y) return;
    if (not y->l and not y->r) y = 0;
    else y->upd();
  }
};

TE struct monoid_add {
  using X = T;
  static constexpr inline X op(const X &x, const X &y) { return x + y; }
  static constexpr inline X inv(const X &x) { return -x; }
  static constexpr inline X pow(const X &x, ll n) { return X(n) * x; }
  static constexpr inline X unit() { return X(0); }
  static constexpr bool commute = 1;
};

TE struct monoid_affine {
  struct X {
    T a, b;
    T eval(T x) const { return a * x + b; }
    bool operator==(const X &p) const { return a == p.a and b == p.b; }
    bool operator!=(const X &p) const { return a != p.a or b != p.b; }
  };
  static constexpr X op(const X &f, const X &g) {
    return {f.a * g.a, f.b * g.a + g.b};
  }
  static constexpr X inv(const X &x) {
    Z [a, b] = x;
    a = T(1) / a;
    return {a, a * -b};
  }
  static constexpr X unit() { return {T(1), T(0)}; }
  static constexpr X make_add(T x) { return {T(1), x}; }
  static constexpr X make_mul(T x) { return {T(x), T(0)}; }
  static constexpr bool commute = 0;
};

TE struct a_monoid_sum_affine {
  using MX = monoid_add<T>;
  using MA = monoid_affine<T>;
  using X = MX::X;
  using A = MA::X;
  static constexpr X act(const X &x, const A &a, ll sz) {
    return x * a.a + T(sz) * a.b;
  }
};

using AM = a_monoid_sum_affine<ull>;
using MX = AM::MX;
using MA = AM::MA;
using DS = segdl_t<AM, 0, ll>;
using np = DS::np;
constexpr ull mod = 1'000'000'000'000'000'000ull + 9;
void Yorisou() {
  LL(N, Q);
  DS seg(0, N);
  np t[5]{};
  array<ull, 5> s{};
  FOR(Q) {
    LL(x, l, r);
    ++r;
    if (x == 0) {
      ull g[5]{};
      FOR(i, 5) g[i] = seg.prod(t[i], l, r);
      ull mx = QMAX(g, g + 5);
      int c = 0, id = -1;
      FOR(i, 5) if (mx == g[i]) {
        ++c, id = i;
      }
      if (c == 1) {
        s[id] += mx;
        if (s[id] >= mod) s[id] -= mod;
      }
    } else {
      --x;
      FOR(i, 5) {
        if (i == x) {
          t[i] = seg.apply(t[i], l, r, MA::make_add(1));
        } else {
          t[i] = seg.apply(t[i], l, r, MA::make_mul(0));
        }
      }
    }
  }
  FOR(i, 5) s[i] += seg.prod(t[i], 0, N);
  print(s);
}
constexpr int tests = 0, fl = 0, DB = 10;
int main() {
  cin.tie(0)->sync_with_stdio(0);
  int T = 1;
  if (fl) cerr.tie(0);
  if (tests and not fl) IN(T);
  for (int i = 0; i < T or fl; ++i) {
    Yorisou();
    if (fl and i % DB == 0) cerr << "Case: " << i << '\n';
  }
  return 0;
}