/**
 * date   : 2024-12-21 01:57:29
 * 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(); }


//


template <typename T, typename E, T (*f)(T, T), T (*g)(T, E), E (*h)(E, E),
          T (*ti)(), E (*ei)()>
struct LazySegmentTreeBase {
  int n, log, s;
  vector<T> val;
  vector<E> laz;

  explicit LazySegmentTreeBase() {}
  explicit LazySegmentTreeBase(const vector<T>& vc) { init(vc); }

  void init(const vector<T>& vc) {
    n = 1, log = 0, s = vc.size();
    while (n < s) n <<= 1, log++;
    val.resize(2 * n, ti());
    laz.resize(n, ei());
    for (int i = 0; i < s; ++i) val[i + n] = vc[i];
    for (int i = n - 1; i; --i) _update(i);
  }

  void update(int l, int r, const E& x) {
    if (l == r) return;
    l += n, r += n;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) _push(l >> i);
      if (((r >> i) << i) != r) _push((r - 1) >> i);
    }
    {
      int l2 = l, r2 = r;
      while (l < r) {
        if (l & 1) _apply(l++, x);
        if (r & 1) _apply(--r, x);
        l >>= 1;
        r >>= 1;
      }
      l = l2;
      r = r2;
    }
    for (int i = 1; i <= log; i++) {
      if (((l >> i) << i) != l) _update(l >> i);
      if (((r >> i) << i) != r) _update((r - 1) >> i);
    }
  }

  T query(int l, int r) {
    if (l == r) return ti();
    l += n, r += n;
    T L = ti(), R = ti();
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) _push(l >> i);
      if (((r >> i) << i) != r) _push((r - 1) >> i);
    }
    while (l < r) {
      if (l & 1) L = f(L, val[l++]);
      if (r & 1) R = f(val[--r], R);
      l >>= 1;
      r >>= 1;
    }
    return f(L, R);
  }

  void set_val(int k, const T& x) {
    k += n;
    for (int i = log; i >= 1; i--) {
      if (((k >> i) << i) != k || (((k + 1) >> i) << i) != (k + 1))
        _push(k >> i);
    }
    val[k] = x;
    for (int i = 1; i <= log; i++) {
      if (((k >> i) << i) != k || (((k + 1) >> i) << i) != (k + 1))
        _update(k >> i);
    }
  }

  void update_val(int k, const E& x) {
    k += n;
    for (int i = log; i >= 1; i--) {
      if (((k >> i) << i) != k || (((k + 1) >> i) << i) != (k + 1))
        _push(k >> i);
    }
    val[k] = g(val[k], x);
    for (int i = 1; i <= log; i++) {
      if (((k >> i) << i) != k || (((k + 1) >> i) << i) != (k + 1))
        _update(k >> i);
    }
  }

  T get_val(int k) {
    k += n;
    for (int i = log; i >= 1; i--) {
      if (((k >> i) << i) != k || (((k + 1) >> i) << i) != (k + 1))
        _push(k >> i);
    }
    return val[k];
  }

  template <class G>
  int max_right(int l, G check) {
    assert(0 <= l && l <= s);
    assert(check(ti()));
    if (l == n) return n;
    l += n;
    for (int i = log; i >= 1; i--) _push(l >> i);
    T sm = ti();
    do {
      while (l % 2 == 0) l >>= 1;
      if (!check(f(sm, val[l]))) {
        while (l < n) {
          _push(l);
          l = (2 * l);
          if (check(f(sm, val[l]))) {
            sm = f(sm, val[l]);
            l++;
          }
        }
        return l - n;
      }
      sm = f(sm, val[l]);
      l++;
    } while ((l & -l) != l);
    return s;
  }

  template <class G>
  int min_left(int r, G check) {
    assert(0 <= r && r <= s);
    assert(check(ti()));
    if (r == 0) return 0;
    r += n;
    for (int i = log; i >= 1; i--) _push((r - 1) >> i);
    T sm = ti();
    do {
      r--;
      while (r > 1 && (r % 2)) r >>= 1;
      if (!check(f(val[r], sm))) {
        while (r < n) {
          _push(r);
          r = (2 * r + 1);
          if (check(f(val[r], sm))) {
            sm = f(val[r], sm);
            r--;
          }
        }
        return r + 1 - n;
      }
      sm = f(val[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

 private:
  void _push(int i) {
    if (laz[i] != ei()) {
      val[2 * i + 0] = g(val[2 * i + 0], laz[i]);
      val[2 * i + 1] = g(val[2 * i + 1], laz[i]);
      if (2 * i < n) {
        compose(laz[2 * i + 0], laz[i]);
        compose(laz[2 * i + 1], laz[i]);
      }
      laz[i] = ei();
    }
  }
  inline void _update(int i) { val[i] = f(val[2 * i + 0], val[2 * i + 1]); }
  inline void _apply(int i, const E& x) {
    if (x != ei()) {
      val[i] = g(val[i], x);
      if (i < n) compose(laz[i], x);
    }
  }
  inline void compose(E& a, const E& b) { a = a == ei() ? b : h(a, b); }
};

namespace SegmentTreeUtil {

template <typename T>
struct Pair {
  T first, second;
  Pair() = default;
  Pair(const T& f, const T& s) : first(f), second(s) {}
  operator T() const { return first; }
  friend ostream& operator<<(ostream& os, const Pair<T>& p) {
    os << T(p.first);
    return os;
  }
};

template <typename T>
T Mx(T a, T b) {
  return max(a, b);
}
template <typename T>
T Mn(T a, T b) {
  return min(a, b);
}
template <typename T>
T Update(T, T b) {
  return b;
}
template <typename T>
T Add(T a, T b) {
  return a + b;
}
template <typename T>
Pair<T> Psum(Pair<T> a, Pair<T> b) {
  return Pair<T>(a.first + b.first, a.second + b.second);
}
template <typename T>
Pair<T> Padd(Pair<T> a, T b) {
  return Pair<T>(a.first + a.second * b, a.second);
}
template <typename T>
Pair<T> PUpdate(Pair<T> a, T b) {
  return Pair<T>(a.second * b, a.second);
}
template <typename T>
Pair<T> Pid() {
  return Pair<T>(T{}, T{});
}
template <typename T>
T Zero() {
  return T{};
}
template <typename T, T val>
T Const() {
  return val;
}

template <typename T, T MINF>
struct AddMax_LazySegmentTree : LazySegmentTreeBase<T, T, Mx<T>, Add<T>, Add<T>,
                                                    Const<T, MINF>, Zero<T>> {
  using base =
      LazySegmentTreeBase<T, T, Mx<T>, Add<T>, Add<T>, Const<T, MINF>, Zero<T>>;
  AddMax_LazySegmentTree(const vector<T>& v) : base(v) {}
};

template <typename T, T INF>
struct AddMin_LazySegmentTree
    : LazySegmentTreeBase<T, T, Mn<T>, Add<T>, Add<T>, Const<T, INF>, Zero<T>> {
  using base =
      LazySegmentTreeBase<T, T, Mn<T>, Add<T>, Add<T>, Const<T, INF>, Zero<T>>;
  AddMin_LazySegmentTree(const vector<T>& v) : base(v) {}
};

template <typename T>
struct AddSum_LazySegmentTree
    : LazySegmentTreeBase<Pair<T>, T, Psum<T>, Padd<T>, Add<T>, Pid<T>,
                          Zero<T>> {
  using base = LazySegmentTreeBase<Pair<T>, T, Psum<T>, Padd<T>, Add<T>, Pid<T>,
                                   Zero<T>>;
  AddSum_LazySegmentTree(const vector<T>& v) {
    vector<Pair<T>> w(v.size());
    for (int i = 0; i < (int)v.size(); i++) w[i] = Pair<T>(v[i], 1);
    base::init(w);
  }
};

template <typename T, T MINF>
struct UpdateMax_LazySegmentTree
    : LazySegmentTreeBase<T, T, Mx<T>, Update<T>, Update<T>, Const<T, MINF>,
                          Const<T, MINF>> {
  using base = LazySegmentTreeBase<T, T, Mx<T>, Update<T>, Update<T>,
                                   Const<T, MINF>, Const<T, MINF>>;
  UpdateMax_LazySegmentTree(const vector<T>& v) : base(v) {}
};

template <typename T, T INF>
struct UpdateMin_LazySegmentTree
    : LazySegmentTreeBase<T, T, Mn<T>, Update<T>, Update<T>, Const<T, INF>,
                          Const<T, INF>> {
  using base = LazySegmentTreeBase<T, T, Mn<T>, Update<T>, Update<T>,
                                   Const<T, INF>, Const<T, INF>>;
  UpdateMin_LazySegmentTree(const vector<T>& v) : base(v) {}
};

template <typename T, T UNUSED_VALUE>
struct UpdateSum_LazySegmentTree
    : LazySegmentTreeBase<Pair<T>, T, Psum<T>, PUpdate<T>, Update<T>, Pid<T>,
                          Const<T, UNUSED_VALUE>> {
  using base = LazySegmentTreeBase<Pair<T>, T, Psum<T>, PUpdate<T>, Update<T>,
                                   Pid<T>, Const<T, UNUSED_VALUE>>;
  UpdateSum_LazySegmentTree(const vector<T>& v) {
    vector<Pair<T>> w(v.size());
    for (int i = 0; i < (int)v.size(); i++) w[i] = Pair<T>(v[i], 1);
    base::init(w);
  }
};

}  // namespace SegmentTreeUtil
using SegmentTreeUtil::AddMax_LazySegmentTree;
using SegmentTreeUtil::AddMin_LazySegmentTree;
using SegmentTreeUtil::AddSum_LazySegmentTree;
using SegmentTreeUtil::UpdateMax_LazySegmentTree;
using SegmentTreeUtil::UpdateMin_LazySegmentTree;
using SegmentTreeUtil::UpdateSum_LazySegmentTree;

/**
 * @brief 使用頻度の高い遅延セグメント木
 * @docs docs/segment-tree/lazy-segment-tree-utility.md
 */




// LazySegmentTree
template <typename T, typename E, typename F, typename G, typename H>
struct LazySegmentTree {
  int n, height;
  F f;
  G g;
  H h;
  T ti;
  E ei;
  vector<T> dat;
  vector<E> laz;
  LazySegmentTree(int _n, F _f, G _g, H _h, T _ti, E _ei)
      : f(_f), g(_g), h(_h), ti(_ti), ei(_ei) {
    init(_n);
  }
  LazySegmentTree(const vector<T> &v, F _f, G _g, H _h, T _ti, E _ei)
      : f(_f), g(_g), h(_h), ti(_ti), ei(_ei) {
    init((int)v.size());
    build(v);
  }
  void init(int _n) {
    n = 1;
    height = 0;
    while (n < _n) n <<= 1, height++;
    dat.assign(2 * n, ti);
    laz.assign(2 * n, ei);
  }
  void build(const vector<T> &v) {
    int _n = v.size();
    init(_n);
    for (int i = 0; i < _n; i++) dat[n + i] = v[i];
    for (int i = n - 1; i; i--)
      dat[i] = f(dat[(i << 1) | 0], dat[(i << 1) | 1]);
  }
  inline T reflect(int k) { return laz[k] == ei ? dat[k] : g(dat[k], laz[k]); }
  inline void eval(int k) {
    if (laz[k] == ei) return;
    laz[(k << 1) | 0] = h(laz[(k << 1) | 0], laz[k]);
    laz[(k << 1) | 1] = h(laz[(k << 1) | 1], laz[k]);
    dat[k] = reflect(k);
    laz[k] = ei;
  }
  inline void thrust(int k) {
    for (int i = height; i; i--) eval(k >> i);
  }
  inline void recalc(int k) {
    while (k >>= 1) dat[k] = f(reflect((k << 1) | 0), reflect((k << 1) | 1));
  }
  void update(int a, int b, E x) {
    if (a >= b) return;
    thrust(a += n);
    thrust(b += n - 1);
    for (int l = a, r = b + 1; l < r; l >>= 1, r >>= 1) {
      if (l & 1) laz[l] = h(laz[l], x), l++;
      if (r & 1) --r, laz[r] = h(laz[r], x);
    }
    recalc(a);
    recalc(b);
  }
  void set_val(int a, T x) {
    thrust(a += n);
    dat[a] = x;
    laz[a] = ei;
    recalc(a);
  }
  T get_val(int a) {
    thrust(a += n);
    return reflect(a);
  }
  T query(int a, int b) {
    if (a >= b) return ti;
    thrust(a += n);
    thrust(b += n - 1);
    T vl = ti, vr = ti;
    for (int l = a, r = b + 1; l < r; l >>= 1, r >>= 1) {
      if (l & 1) vl = f(vl, reflect(l++));
      if (r & 1) vr = f(reflect(--r), vr);
    }
    return f(vl, vr);
  }
};

//





using namespace std;

namespace internal {
unsigned long long non_deterministic_seed() {
  unsigned long long m =
      chrono::duration_cast<chrono::nanoseconds>(
          chrono::high_resolution_clock::now().time_since_epoch())
          .count();
  m ^= 9845834732710364265uLL;
  m ^= m << 24, m ^= m >> 31, m ^= m << 35;
  return m;
}
unsigned long long deterministic_seed() { return 88172645463325252UL; }

// 64 bit の seed 値を生成 (手元では seed 固定)
// 連続で呼び出すと同じ値が何度も返ってくるので注意
// #define RANDOMIZED_SEED するとシードがランダムになる
unsigned long long seed() {
#if defined(DETERMINISTIC_SEED)
  return deterministic_seed();
#elif defined(NyaanLocal) && !defined(RANDOMIZED_SEED)
  return deterministic_seed();
#else
  return non_deterministic_seed();
#endif
}

}  // namespace internal


namespace my_rand {
using i64 = long long;
using u64 = unsigned long long;

// [0, 2^64 - 1)
u64 rng() {
  static u64 _x = internal::seed();
  return _x ^= _x << 7, _x ^= _x >> 9;
}

// [l, r]
i64 rng(i64 l, i64 r) {
  assert(l <= r);
  return l + rng() % u64(r - l + 1);
}

// [l, r)
i64 randint(i64 l, i64 r) {
  assert(l < r);
  return l + rng() % u64(r - l);
}

// choose n numbers from [l, r) without overlapping
vector<i64> randset(i64 l, i64 r, i64 n) {
  assert(l <= r && n <= r - l);
  unordered_set<i64> s;
  for (i64 i = n; i; --i) {
    i64 m = randint(l, r + 1 - i);
    if (s.find(m) != s.end()) m = r - i;
    s.insert(m);
  }
  vector<i64> ret;
  for (auto& x : s) ret.push_back(x);
  sort(begin(ret), end(ret));
  return ret;
}

// [0.0, 1.0)
double rnd() { return rng() * 5.42101086242752217004e-20; }
// [l, r)
double rnd(double l, double r) {
  assert(l < r);
  return l + rnd() * (r - l);
}

template <typename T>
void randshf(vector<T>& v) {
  int n = v.size();
  for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]);
}

}  // namespace my_rand

using my_rand::randint;
using my_rand::randset;
using my_rand::randshf;
using my_rand::rnd;
using my_rand::rng;





using namespace std;

struct Timer {
  chrono::high_resolution_clock::time_point st;

  Timer() { reset(); }
  void reset() { st = chrono::high_resolution_clock::now(); }

  long long elapsed() {
    auto ed = chrono::high_resolution_clock::now();
    return chrono::duration_cast<chrono::milliseconds>(ed - st).count();
  }
  long long operator()() { return elapsed(); }
};





using namespace std;

struct modint_2_61m1 {
  using M = modint_2_61m1;
  using u64 = unsigned long long;
  using u128 = __uint128_t;

  static constexpr u64 mod = (1uLL << 61) - 1;
  u64 x;

  static constexpr u64 modulo(u128 y) {
    u64 val = (y >> 61) + (y & mod);
    return val >= mod ? val - mod : val;
  }

  modint_2_61m1() : x(0) {}
  modint_2_61m1(long long _x) {
    long long y = _x % (long long)mod;
    if (y < 0) y += mod;
    x = y;
  }

  static M raw(u64 y) {
    M res;
    res.x = y;
    return res;
  }
  u64 get() const { return x; }
  static u64 get_mod() { return mod; }

  friend M operator+(const M& l, const M& r) {
    u64 y = l.x + r.x;
    if (y >= mod) y -= mod;
    return raw(y);
  }
  friend M operator-(const M& l, const M& r) {
    u64 y = l.x - r.x;
    if (y >= mod) y += mod;
    return raw(y);
  }
  friend M operator*(const M& l, const M& r) {
    return raw(modulo(u128(l.x) * r.x));
  }
  friend M operator/(const M& l, const M& r) { return l * r.inverse(); }

  M& operator+=(const M& r) { return *this = *this + r; }
  M& operator-=(const M& r) { return *this = *this - r; }
  M& operator*=(const M& r) { return *this = *this * r; }
  M& operator/=(const M& r) { return *this = *this / r; }
  M operator-() const { return raw(x ? mod - x : u64{0}); }
  M operator+() const { return *this; }

  M pow(u64 e) const {
    M res{1}, a{*this};
    while (e) {
      if (e & 1) res = res * a;
      a = a * a;
      e >>= 1;
    }
    return res;
  }
  M inverse() const {
    assert(x != 0);
    return this->pow(mod - 2);
  }

  friend bool operator==(const M& l, const M& r) { return l.x == r.x; }
  friend bool operator!=(const M& l, const M& r) { return l.x != r.x; }
  friend ostream& operator<<(ostream& os, const M& r) { return os << r.x; }
};




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 Nyaan;
using mint = LazyMontgomeryModInt<1000000021>;

using namespace Nyaan;

constexpr int B = 3;
mint base[B];
struct BaseInit {
  BaseInit() { rep(i, B) base[i] = rng(1, mint::get_mod() - 1); }
} baseinit;

struct Data {
  mint a[B];
  Data() { rep(i, B) a[i] = 0; }
  Data(ll x) {
    if (x >= 0) {
      rep(i, B) a[i] = base[i].pow(x);
    } else {
      rep(i, B) a[i] = base[i].pow(-x).inverse();
    }
  }
  friend Data merge(const Data& a, const Data& b) {
    Data res;
    rep(i, B) res.a[i] = a.a[i] + b.a[i];
    return res;
  }
  friend Data merge2(const Data& a, const Data& b) {
    Data res;
    rep(i, B) res.a[i] = a.a[i] * b.a[i];
    return res;
  }
  bool operator==(const Data& rhs) {
    rep(i, B) if (a[i] != rhs.a[i]) return false;
    return true;
  }
};

void q() {
  inl(N, Q);
  vl A(N);
  in(A);

  AddSum_LazySegmentTree<ll> segsum(A);

  auto f = [&](const Data& a, const Data& b) { return merge(a, b); };
  auto g = [&](const Data& a, const Data& b) { return merge2(a, b); };
  auto h = [&](const Data& a, const Data& b) { return merge2(a, b); };
  Data ti = Data{};
  Data ei = Data{};
  rep(i, B) ei.a[i] = 1;

  V<Data> init, initrev;
  rep(i, N) {
    init.push_back(Data{A[i]});
    initrev.push_back(Data{-A[i]});
  }
  LazySegmentTree seg(init, f, g, h, ti, ei);
  LazySegmentTree segrev(initrev, f, g, h, ti, ei);

  rep(q, Q) {
    ini(cmd, L, R);
    --L;
    if (cmd == 1) {
      inl(v);
      segsum.update(L, R, v);
      seg.update(L, R, Data{v});
      segrev.update(L, R, Data{-v});
    } else {
      ll s = segsum.query(L, R);
      if (s % ((R - L) / 2)) {
        out("NO");
      } else {
        ll x = s / ((R - L) / 2);
        Data h1 = merge2(seg.query(L, R), Data{-x});
        Data h2 = segrev.query(L, R);
        int ok = 1;
        rep(i, B) if (h1.a[i] != h2.a[i]) ok = 0;
        out(ok ? "YES" : "NO");
      }
    }
  }
}

void Nyaan::solve() {
  Timer timer;
  int t = 1;
  // in(t);
  while (t--) q();
  trc2(timer());
}