#line 1 "/home/maspy/compro/library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else

// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
// いまの CF だとこれ入れると動かない？
// #pragma GCC target("avx2,popcnt")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
  vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}

template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &... others) {
  vc<T> &res = first;
  (res.insert(res.end(), others.begin(), others.end()), ...);
}
#endif
#line 1 "/home/maspy/compro/library/other/io.hpp"
#define FASTIO
#include <unistd.h>

// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
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, SZ - pir + pil, stdin);
  pil = 0;
  if (pir < SZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}

void rd(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 rd_real(T &x) {
  string s;
  rd(s);
  x = stod(s);
}

template <typename T>
void rd_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++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}

void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }

template <class T, class U>
void rd(pair<T, U> &p) {
  return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd(x);
    rd_tuple<N + 1>(t);
  }
}
template <class... T>
void rd(tuple<T...> &tpl) {
  rd_tuple(tpl);
}

template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
  for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
  for (auto &d: x) rd(d);
}

void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd(h), read(t...);
}

void wt(const char c) {
  if (por == SZ) flush();
  obuf[por++] = c;
}
void wt(const string s) {
  for (char c: s) wt(c);
}
void wt(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt(s[i]);
}

template <typename T>
void wt_integer(T x) {
  if (por > SZ - 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 wt_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt(s);
}

void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }

template <class T, class U>
void wt(const pair<T, U> val) {
  wt(val.first);
  wt(' ');
  wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt(' '); }
    const auto x = std::get<N>(t);
    wt(x);
    wt_tuple<N + 1>(t);
  }
}
template <class... T>
void wt(tuple<T...> tpl) {
  wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}
template <class T>
void wt(const vector<T> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}

void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt(head);
  if (sizeof...(Tail)) wt(' ');
  print(forward<Tail>(tail)...);
}

// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;

#if defined(LOCAL)
#define SHOW(...) SHOW_IMPL(__VA_ARGS__, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, SHOW1)(__VA_ARGS__)
#define SHOW_IMPL(_1, _2, _3, _4, _5, _6, NAME, ...) NAME
#define SHOW1(x) print(#x, "=", (x)), flush()
#define SHOW2(x, y) print(#x, "=", (x), #y, "=", (y)), flush()
#define SHOW3(x, y, z) print(#x, "=", (x), #y, "=", (y), #z, "=", (z)), flush()
#define SHOW4(x, y, z, w) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush()
#define SHOW5(x, y, z, w, v) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v)), flush()
#define SHOW6(x, y, z, w, v, u) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v), #u, "=", (u)), flush()
#else
#define SHOW(...)
#endif

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define U32(...)   \
  u32 __VA_ARGS__; \
  read(__VA_ARGS__)
#define U64(...)   \
  u64 __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

#line 1 "/home/maspy/compro/library/ds/bit_vector.hpp"
struct Bit_Vector {
  int n;
  bool prepared = 0;
  vc<pair<u64, u32>> dat;
  Bit_Vector(int n) : n(n) { dat.assign((n + 127) >> 6, {0, 0}); }
  void set(int i) {
    assert(!prepared);
    dat[i >> 6].fi |= u64(1) << (i & 63);
  }
  void reset() {
    fill(all(dat), pair<u64, u32>{0, 0});
    prepared = 0;
  }
  void build() {
    prepared = 1;
    FOR(i, len(dat) - 1) dat[i + 1].se = dat[i].se + popcnt(dat[i].fi);
  }
  // [0, k) 内の 1 の個数
  bool operator[](int i) { return dat[i >> 6].fi >> (i & 63) & 1; }
  int count_prefix(int k, bool f = true) {
    assert(prepared);
    auto [a, b] = dat[k >> 6];
    int ret = b + popcnt(a & ((u64(1) << (k & 63)) - 1));
    return (f ? ret : k - ret);
  }
  int count(int L, int R, bool f = true) { return count_prefix(R, f) - count_prefix(L, f); }
  string to_string() {
    string ans;
    FOR(i, n) ans += '0' + (dat[i / 64].fi >> (i % 64) & 1);
    return ans;
  }
};
#line 1 "/home/maspy/compro/library/ds/index_compression.hpp"
template <typename T>
struct Index_Compression_DISTINCT_SMALL {
  static_assert(is_same_v<T, int>);
  int mi, ma;
  vc<int> dat;
  vc<int> build(vc<int> X) {
    mi = 0, ma = -1;
    if (!X.empty()) mi = MIN(X), ma = MAX(X);
    dat.assign(ma - mi + 2, 0);
    for (auto& x: X) dat[x - mi + 1]++;
    FOR(i, len(dat) - 1) dat[i + 1] += dat[i];
    for (auto& x: X) { x = dat[x - mi]++; }
    FOR_R(i, 1, len(dat)) dat[i] = dat[i - 1];
    dat[0] = 0;
    return X;
  }
  int operator()(ll x) { return dat[clamp<ll>(x - mi, 0, ma - mi + 1)]; }
};

template <typename T>
struct Index_Compression_SAME_SMALL {
  static_assert(is_same_v<T, int>);
  int mi, ma;
  vc<int> dat;
  vc<int> build(vc<int> X) {
    mi = 0, ma = -1;
    if (!X.empty()) mi = MIN(X), ma = MAX(X);
    dat.assign(ma - mi + 2, 0);
    for (auto& x: X) dat[x - mi + 1] = 1;
    FOR(i, len(dat) - 1) dat[i + 1] += dat[i];
    for (auto& x: X) { x = dat[x - mi]; }
    return X;
  }
  int operator()(ll x) { return dat[clamp<ll>(x - mi, 0, ma - mi + 1)]; }
};

template <typename T>
struct Index_Compression_SAME_LARGE {
  vc<T> dat;
  vc<int> build(vc<T> X) {
    vc<int> I = argsort(X);
    vc<int> res(len(X));
    for (auto& i: I) {
      if (!dat.empty() && dat.back() == X[i]) {
        res[i] = len(dat) - 1;
      } else {
        res[i] = len(dat);
        dat.eb(X[i]);
      }
    }
    dat.shrink_to_fit();
    return res;
  }
  int operator()(T x) { return LB(dat, x); }
};

template <typename T>
struct Index_Compression_DISTINCT_LARGE {
  vc<T> dat;
  vc<int> build(vc<T> X) {
    vc<int> I = argsort(X);
    vc<int> res(len(X));
    for (auto& i: I) { res[i] = len(dat), dat.eb(X[i]); }
    dat.shrink_to_fit();
    return res;
  }
  int operator()(T x) { return LB(dat, x); }
};

template <typename T, bool SMALL>
using Index_Compression_DISTINCT =
    typename std::conditional<SMALL, Index_Compression_DISTINCT_SMALL<T>,
                              Index_Compression_DISTINCT_LARGE<T>>::type;
template <typename T, bool SMALL>
using Index_Compression_SAME =
    typename std::conditional<SMALL, Index_Compression_SAME_SMALL<T>,
                              Index_Compression_SAME_LARGE<T>>::type;

// SAME: [2,3,2] -> [0,1,0]
// DISTINCT: [2,2,3] -> [0,2,1]
// (x): lower_bound(X,x) をかえす
template <typename T, bool SAME, bool SMALL>
using Index_Compression =
    typename std::conditional<SAME, Index_Compression_SAME<T, SMALL>,
                              Index_Compression_DISTINCT<T, SMALL>>::type;
#line 2 "/home/maspy/compro/library/alg/monoid/add.hpp"

template <typename E>
struct Monoid_Add {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
  static constexpr X inverse(const X &x) noexcept { return -x; }
  static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
  static constexpr X unit() { return X(0); }
  static constexpr bool commute = true;
};
#line 4 "/home/maspy/compro/library/ds/wavelet_matrix/wavelet_matrix.hpp"

// 静的メソッドinverseの存在をチェックするテンプレート
template <typename, typename = std::void_t<>>
struct has_inverse : std::false_type {};

template <typename T>
struct has_inverse<T, std::void_t<decltype(T::inverse(std::declval<typename T::value_type>()))>> : std::true_type {};

struct Dummy_Data_Structure {
  using MX = Monoid_Add<bool>;
  void build(const vc<bool>& A) {}
};

template <typename Y, bool SMALL_Y, typename SEGTREE = Dummy_Data_Structure>
struct Wavelet_Matrix {
  using Mono = typename SEGTREE::MX;
  using T = typename Mono::value_type;
  static_assert(Mono::commute);

  int n, log, K;
  Index_Compression<Y, true, SMALL_Y> IDX;
  vc<Y> ItoY;
  vc<int> mid;
  vc<Bit_Vector> bv;
  vc<SEGTREE> seg;

  Wavelet_Matrix() {}
  Wavelet_Matrix(const vc<Y>& A) { build(A); }
  Wavelet_Matrix(const vc<Y>& A, vc<T>& SUM_Data) { build(A, SUM_Data); }
  template <typename F>
  Wavelet_Matrix(int n, F f) {
    build(n, f);
  }

  template <typename F>
  void build(int m, F f) {
    vc<Y> A(m);
    vc<T> S(m);
    for (int i = 0; i < m; ++i) tie(A[i], S[i]) = f(i);
    build(A, S);
  }

  void build(const vc<Y>& A) { build(A, vc<T>(len(A), Mono::unit())); }
  void build(const vc<Y>& A, vc<T> S) {
    n = len(A);
    vc<int> B = IDX.build(A);
    K = 0;
    for (auto& x: B) chmax(K, x + 1);
    ItoY.resize(K);
    FOR(i, n) ItoY[B[i]] = A[i];
    log = 0;
    while ((1 << log) < K) ++log;
    mid.resize(log), bv.assign(log, Bit_Vector(n));
    vc<int> B0(n), B1(n);
    vc<T> S0(n), S1(n);
    seg.resize(log + 1);
    seg[log].build(S);
    for (int d = log - 1; d >= 0; --d) {
      int p0 = 0, p1 = 0;
      for (int i = 0; i < n; ++i) {
        bool f = (B[i] >> d & 1);
        if (!f) { B0[p0] = B[i], S0[p0] = S[i], p0++; }
        if (f) { bv[d].set(i), B1[p1] = B[i], S1[p1] = S[i], p1++; }
      }
      swap(B, B0), swap(S, S0);
      move(B1.begin(), B1.begin() + p1, B.begin() + p0);
      move(S1.begin(), S1.begin() + p1, S.begin() + p0);
      mid[d] = p0, bv[d].build(), seg[d].build(S);
    }
  }

  // [L,R) x [0,y)
  int prefix_count(int L, int R, Y y) {
    int p = IDX(y);
    if (L == R || p == 0) return 0;
    if (p == K) return R - L;
    int cnt = 0;
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (p >> d & 1) cnt += r0 - l0, L = l1, R = r1;
      if (!(p >> d & 1)) L = l0, R = r0;
    }
    return cnt;
  }

  // [L,R) x [y1,y2)
  int count(int L, int R, Y y1, Y y2) { return prefix_count(L, R, y2) - prefix_count(L, R, y1); }

  // [L,R) x [0,y)
  pair<int, T> prefix_count_and_prod(int L, int R, Y y) {
    int p = IDX(y);
    if (p == 0) return {0, Mono::unit()};
    if (p == K) return {R - L, seg[log].prod(L, R)};
    int cnt = 0;
    T t = Mono::unit();
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (p >> d & 1) { cnt += r0 - l0, t = Mono::op(t, seg[d].prod(l0, r0)), L = l1, R = r1; }
      if (!(p >> d & 1)) L = l0, R = r0;
    }
    return {cnt, t};
  }

  // [L,R) x [y1,y2)
  pair<int, T> count_and_prod(int L, int R, Y y1, Y y2) {
    if constexpr (has_inverse<Mono>::value) {
      auto [c1, t1] = prefix_count_and_prod(L, R, y1);
      auto [c2, t2] = prefix_count_and_prod(L, R, y2);
      return {c2 - c1, Mono::op(Mono::inverse(t1), t2)};
    }
    int lo = IDX(y1), hi = IDX(y2), cnt = 0;
    T t = Mono::unit();
    auto dfs = [&](auto& dfs, int d, int L, int R, int a, int b) -> void {
      assert(b - a == (1 << d));
      if (hi <= a || b <= lo) return;
      if (lo <= a && b <= hi) {
        cnt += R - L, t = Mono::op(t, seg[d].prod(L, R));
        return;
      }
      --d;
      int c = (a + b) / 2;
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      dfs(dfs, d, l0, r0, a, c), dfs(dfs, d, l1, r1, c, b);
    };
    dfs(dfs, log, L, R, 0, 1 << log);
    return {cnt, t};
  }

  // [L,R) x [y1,y2)
  T prefix_prod(int L, int R, Y y) { return prefix_count_and_prod(L, R, y).se; }
  // [L,R) x [y1,y2)
  T prod(int L, int R, Y y1, Y y2) { return count_and_prod(L, R, y1, y2).se; }
  T prod_all(int L, int R) { return seg[log].prod(L, R); }

  Y kth(int L, int R, int k) {
    assert(0 <= k && k < R - L);
    int p = 0;
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (k < r0 - l0) {
        L = l0, R = r0;
      } else {
        k -= r0 - l0, L = l1, R = r1, p |= 1 << d;
      }
    }
    return ItoY[p];
  }

  // y 以上最小 OR infty<Y>
  Y next(int L, int R, Y y) {
    int k = IDX(y);
    int p = K;

    auto dfs = [&](auto& dfs, int d, int L, int R, int a, int b) -> void {
      if (p <= a || L == R || b <= k) return;
      if (d == 0) {
        chmin(p, a);
        return;
      }
      --d;
      int c = (a + b) / 2;
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      dfs(dfs, d, l0, r0, a, c), dfs(dfs, d, l1, r1, c, b);
    };
    dfs(dfs, log, L, R, 0, 1 << log);
    return (p == K ? infty<Y> : ItoY[p]);
  }

  // y 以下最大 OR -infty<T>
  Y prev(int L, int R, Y y) {
    int k = IDX(y + 1);
    int p = -1;
    auto dfs = [&](auto& dfs, int d, int L, int R, int a, int b) -> void {
      if (b - 1 <= p || L == R || k <= a) return;
      if (d == 0) {
        chmax(p, a);
        return;
      }
      --d;
      int c = (a + b) / 2;
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      dfs(dfs, d, l1, r1, c, b), dfs(dfs, d, l0, r0, a, c);
    };
    dfs(dfs, log, L, R, 0, 1 << log);
    return (p == -1 ? -infty<Y> : ItoY[p]);
  }

  Y median(bool UPPER, int L, int R) {
    assert(0 <= L && L < R && R <= n);
    int k = (UPPER ? (R - L) / 2 : (R - L - 1) / 2);
    return kth(L, R, k);
  }

  pair<Y, T> kth_value_and_prod(int L, int R, int k) {
    assert(0 <= k && k <= R - L);
    if (k == R - L) return {infty<Y>, seg[log].prod(L, R)};
    int p = 0;
    T t = Mono::unit();
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (k < r0 - l0) {
        L = l0, R = r0;
      } else {
        t = Mono::op(t, seg[d].prod(l0, r0)), k -= r0 - l0, L = l1, R = r1, p |= 1 << d;
      }
    }
    t = Mono::op(t, seg[0].prod(L, L + k));
    return {ItoY[p], t};
  }

  T prod_index_range(int L, int R, int k1, int k2) {
    static_assert(has_inverse<Mono>::value);
    T t1 = kth_value_and_prod(L, R, k1).se;
    T t2 = kth_value_and_prod(L, R, k2).se;
    return Mono::op(Mono::inverse(t1), t2);
  }

  // [L,R) x [0,y) での check(cnt, prod) が true となる最大の (cnt,prod)
  template <typename F>
  pair<int, T> max_right(F check, int L, int R) {
    int cnt = 0;
    T t = Mono::unit();
    assert(check(0, Mono::unit()));
    if (check(R - L, seg[log].prod(L, R))) { return {R - L, seg[log].prod(L, R)}; }
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      int cnt1 = cnt + r0 - l0;
      T t1 = Mono::op(t, seg[d].prod(l0, r0));
      if (check(cnt1, t1)) {
        cnt = cnt1, t = t1, L = l1, R = r1;
      } else {
        L = l0, R = r0;
      }
    }
    return {cnt, t};
  }

  void set(int i, T t) {
    assert(0 <= i && i < n);
    int L = i, R = i + 1;
    seg[log].set(L, t);
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (l0 < r0) L = l0, R = r0;
      if (l0 == r0) L = l1, R = r1;
      seg[d].set(L, t);
    }
  }
  void multiply(int i, T t) {
    assert(0 <= i && i < n);
    int L = i, R = i + 1;
    seg[log].multiply(L, t);
    for (int d = log - 1; d >= 0; --d) {
      int l0 = bv[d].count_prefix(L, 0), r0 = bv[d].count_prefix(R, 0);
      int l1 = L + mid[d] - l0, r1 = R + mid[d] - r0;
      if (l0 < r0) L = l0, R = r0;
      if (l0 == r0) L = l1, R = r1;
      seg[d].multiply(L, t);
    }
  }
  void add(int i, T t) { multiply(i, t); }
};
#line 2 "/home/maspy/compro/library/ds/segtree/segtree.hpp"

template <class Monoid>
struct SegTree {
  using MX = Monoid;
  using X = typename MX::value_type;
  using value_type = X;
  vc<X> dat;
  int n, log, size;

  SegTree() {}
  SegTree(int n) { build(n); }
  template <typename F>
  SegTree(int n, F f) {
    build(n, f);
  }
  SegTree(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::unit(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    dat.assign(size << 1, MX::unit());
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  X get(int i) { return dat[size + i]; }
  vc<X> get_all() { return {dat.begin() + size, dat.begin() + size + n}; }

  void update(int i) { dat[i] = Monoid::op(dat[2 * i], dat[2 * i + 1]); }
  void set(int i, const X& x) {
    assert(i < n);
    dat[i += size] = x;
    while (i >>= 1) update(i);
  }

  void multiply(int i, const X& x) {
    assert(i < n);
    i += size;
    dat[i] = Monoid::op(dat[i], x);
    while (i >>= 1) update(i);
  }

  X prod(int L, int R) {
    assert(0 <= L && L <= R && R <= n);
    X vl = Monoid::unit(), vr = Monoid::unit();
    L += size, R += size;
    while (L < R) {
      if (L & 1) vl = Monoid::op(vl, dat[L++]);
      if (R & 1) vr = Monoid::op(dat[--R], vr);
      L >>= 1, R >>= 1;
    }
    return Monoid::op(vl, vr);
  }

  X prod_all() { return dat[1]; }

  template <class F>
  int max_right(F check, int L) {
    assert(0 <= L && L <= n && check(Monoid::unit()));
    if (L == n) return n;
    L += size;
    X sm = Monoid::unit();
    do {
      while (L % 2 == 0) L >>= 1;
      if (!check(Monoid::op(sm, dat[L]))) {
        while (L < size) {
          L = 2 * L;
          if (check(Monoid::op(sm, dat[L]))) { sm = Monoid::op(sm, dat[L++]); }
        }
        return L - size;
      }
      sm = Monoid::op(sm, dat[L++]);
    } while ((L & -L) != L);
    return n;
  }

  template <class F>
  int min_left(F check, int R) {
    assert(0 <= R && R <= n && check(Monoid::unit()));
    if (R == 0) return 0;
    R += size;
    X sm = Monoid::unit();
    do {
      --R;
      while (R > 1 && (R % 2)) R >>= 1;
      if (!check(Monoid::op(dat[R], sm))) {
        while (R < size) {
          R = 2 * R + 1;
          if (check(Monoid::op(dat[R], sm))) { sm = Monoid::op(dat[R--], sm); }
        }
        return R + 1 - size;
      }
      sm = Monoid::op(dat[R], sm);
    } while ((R & -R) != R);
    return 0;
  }

  // prod_{l<=i<r} A[i xor x]
  X xor_prod(int l, int r, int xor_val) {
    static_assert(Monoid::commute);
    X x = Monoid::unit();
    for (int k = 0; k < log + 1; ++k) {
      if (l >= r) break;
      if (l & 1) { x = Monoid::op(x, dat[(size >> k) + ((l++) ^ xor_val)]); }
      if (r & 1) { x = Monoid::op(x, dat[(size >> k) + ((--r) ^ xor_val)]); }
      l /= 2, r /= 2, xor_val /= 2;
    }
    return x;
  }
};
#line 2 "/home/maspy/compro/library/string/suffix_array.hpp"

#line 2 "/home/maspy/compro/library/alg/monoid/min.hpp"

template <typename E>
struct Monoid_Min {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return min(x, y); }
  static constexpr X unit() { return infty<E>; }
  static constexpr bool commute = true;
};
#line 2 "/home/maspy/compro/library/ds/sparse_table/sparse_table.hpp"

// 冪等なモノイドであることを仮定。disjoint sparse table より x 倍高速
template <class Monoid>
struct Sparse_Table {
  using MX = Monoid;
  using X = typename MX::value_type;
  int n, log;
  vvc<X> dat;

  Sparse_Table() {}
  Sparse_Table(int n) { build(n); }
  template <typename F>
  Sparse_Table(int n, F f) {
    build(n, f);
  }
  Sparse_Table(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::unit(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 1;
    while ((1 << log) < n) ++log;
    dat.resize(log);
    dat[0].resize(n);
    FOR(i, n) dat[0][i] = f(i);

    FOR(i, log - 1) {
      dat[i + 1].resize(len(dat[i]) - (1 << i));
      FOR(j, len(dat[i]) - (1 << i)) {
        dat[i + 1][j] = MX::op(dat[i][j], dat[i][j + (1 << i)]);
      }
    }
  }

  X prod(int L, int R) {
    if (L == R) return MX::unit();
    if (R == L + 1) return dat[0][L];
    int k = topbit(R - L - 1);
    return MX::op(dat[k][L], dat[k][R - (1 << k)]);
  }

  template <class F>
  int max_right(const F check, int L) {
    assert(0 <= L && L <= n && check(MX::unit()));
    if (L == n) return n;
    int ok = L, ng = n + 1;
    while (ok + 1 < ng) {
      int k = (ok + ng) / 2;
      bool bl = check(prod(L, k));
      if (bl) ok = k;
      if (!bl) ng = k;
    }
    return ok;
  }

  template <class F>
  int min_left(const F check, int R) {
    assert(0 <= R && R <= n && check(MX::unit()));
    if (R == 0) return 0;
    int ok = R, ng = -1;
    while (ng + 1 < ok) {
      int k = (ok + ng) / 2;
      bool bl = check(prod(k, R));
      if (bl) ok = k;
      if (!bl) ng = k;
    }
    return ok;
  }
};
#line 5 "/home/maspy/compro/library/string/suffix_array.hpp"

// 辞書順 i 番目の suffix が j 文字目始まりであるとき、
// SA[i] = j, ISA[j] = i
// |S|>0 を前提（そうでない場合 dummy 文字を追加して利用せよ）
struct Suffix_Array {
  vc<int> SA;
  vc<int> ISA;
  vc<int> LCP;
  Sparse_Table<Monoid_Min<int>> seg;
  bool build_seg;

  Suffix_Array(string& s) {
    build_seg = 0;
    assert(len(s) > 0);
    char first = 127, last = 0;
    for (auto&& c: s) {
      chmin(first, c);
      chmax(last, c);
    }
    SA = calc_suffix_array(s, first, last);
    calc_LCP(s);
  }

  Suffix_Array(vc<int>& s) {
    build_seg = 0;
    assert(len(s) > 0);
    SA = calc_suffix_array(s);
    calc_LCP(s);
  }

  // lcp(S[i:], S[j:])
  int lcp(int i, int j) {
    if (!build_seg) {
      build_seg = true;
      seg.build(LCP);
    }
    int n = len(SA);
    if (i == n || j == n) return 0;
    if (i == j) return n - i;
    i = ISA[i], j = ISA[j];
    if (i > j) swap(i, j);
    return seg.prod(i, j);
  }

  // S[i:] との lcp が n 以上であるような半開区間
  pair<int, int> lcp_range(int i, int n) {
    if (!build_seg) {
      build_seg = true;
      seg.build(LCP);
    }
    i = ISA[i];
    int a = seg.min_left([&](auto e) -> bool { return e >= n; }, i);
    int b = seg.max_right([&](auto e) -> bool { return e >= n; }, i);
    return {a, b + 1};
  }

  // -1: S[L1:R1) < S[L2, R2)
  //  0: S[L1:R1) = S[L2, R2)
  // +1: S[L1:R1) > S[L2, R2)
  int compare(int L1, int R1, int L2, int R2) {
    int n1 = R1 - L1, n2 = R2 - L2;
    int n = lcp(L1, L2);
    if (n == n1 && n == n2) return 0;
    if (n == n1) return -1;
    if (n == n2) return 1;
    return (ISA[L1 + n] > ISA[L2 + n] ? 1 : -1);
  }

private:
  void induced_sort(const vc<int>& vect, int val_range, vc<int>& SA,
                    const vc<bool>& sl, const vc<int>& lms_idx) {
    vc<int> l(val_range, 0), r(val_range, 0);
    for (int c: vect) {
      if (c + 1 < val_range) ++l[c + 1];
      ++r[c];
    }
    partial_sum(l.begin(), l.end(), l.begin());
    partial_sum(r.begin(), r.end(), r.begin());
    fill(SA.begin(), SA.end(), -1);
    for (int i = (int)lms_idx.size() - 1; i >= 0; --i)
      SA[--r[vect[lms_idx[i]]]] = lms_idx[i];
    for (int i: SA)
      if (i >= 1 && sl[i - 1]) SA[l[vect[i - 1]]++] = i - 1;
    fill(r.begin(), r.end(), 0);
    for (int c: vect) ++r[c];
    partial_sum(r.begin(), r.end(), r.begin());
    for (int k = (int)SA.size() - 1, i = SA[k]; k >= 1; --k, i = SA[k])
      if (i >= 1 && !sl[i - 1]) { SA[--r[vect[i - 1]]] = i - 1; }
  }

  vc<int> SA_IS(const vc<int>& vect, int val_range) {
    const int n = vect.size();
    vc<int> SA(n), lms_idx;
    vc<bool> sl(n);
    sl[n - 1] = false;
    for (int i = n - 2; i >= 0; --i) {
      sl[i] = (vect[i] > vect[i + 1] || (vect[i] == vect[i + 1] && sl[i + 1]));
      if (sl[i] && !sl[i + 1]) lms_idx.push_back(i + 1);
    }
    reverse(lms_idx.begin(), lms_idx.end());
    induced_sort(vect, val_range, SA, sl, lms_idx);
    vc<int> new_lms_idx(lms_idx.size()), lms_vec(lms_idx.size());
    for (int i = 0, k = 0; i < n; ++i)
      if (!sl[SA[i]] && SA[i] >= 1 && sl[SA[i] - 1]) {
        new_lms_idx[k++] = SA[i];
      }
    int cur = 0;
    SA[n - 1] = cur;
    for (size_t k = 1; k < new_lms_idx.size(); ++k) {
      int i = new_lms_idx[k - 1], j = new_lms_idx[k];
      if (vect[i] != vect[j]) {
        SA[j] = ++cur;
        continue;
      }
      bool flag = false;
      for (int a = i + 1, b = j + 1;; ++a, ++b) {
        if (vect[a] != vect[b]) {
          flag = true;
          break;
        }
        if ((!sl[a] && sl[a - 1]) || (!sl[b] && sl[b - 1])) {
          flag = !((!sl[a] && sl[a - 1]) && (!sl[b] && sl[b - 1]));
          break;
        }
      }
      SA[j] = (flag ? ++cur : cur);
    }
    for (size_t i = 0; i < lms_idx.size(); ++i) lms_vec[i] = SA[lms_idx[i]];
    if (cur + 1 < (int)lms_idx.size()) {
      auto lms_SA = SA_IS(lms_vec, cur + 1);
      for (size_t i = 0; i < lms_idx.size(); ++i) {
        new_lms_idx[i] = lms_idx[lms_SA[i]];
      }
    }
    induced_sort(vect, val_range, SA, sl, new_lms_idx);
    return SA;
  }

  vc<int> calc_suffix_array(const string& s, const char first = 'a',
                            const char last = 'z') {
    vc<int> vect(s.size() + 1);
    copy(begin(s), end(s), begin(vect));
    for (auto& x: vect) x -= (int)first - 1;
    vect.back() = 0;
    auto ret = SA_IS(vect, (int)last - (int)first + 2);
    ret.erase(ret.begin());
    return ret;
  }

  vc<int> calc_suffix_array(const vc<int>& s) {
    vc<int> ss = s;
    UNIQUE(ss);

    vc<int> vect(s.size() + 1);
    copy(all(s), vect.begin());
    for (auto& x: vect) x = LB(ss, x) + 1;
    vect.back() = 0;
    auto ret = SA_IS(vect, MAX(vect) + 2);
    ret.erase(ret.begin());
    return ret;
  }

  template <typename STRING>
  void calc_LCP(const STRING& s) {
    int n = s.size(), k = 0;
    ISA.resize(n);
    LCP.resize(n);
    for (int i = 0; i < n; i++) ISA[SA[i]] = i;
    for (int i = 0; i < n; i++, k ? k-- : 0) {
      if (ISA[i] == n - 1) {
        k = 0;
        continue;
      }
      int j = SA[ISA[i] + 1];
      while (i + k < n && j + k < n && s[i + k] == s[j + k]) k++;
      LCP[ISA[i]] = k;
    }
    LCP.resize(n - 1);
  }
};
#line 2 "/home/maspy/compro/library/ds/fastset.hpp"

// 64-ary tree
// space: (N/63) * u64
struct FastSet {
  static constexpr u32 B = 64;
  int n, log;
  vvc<u64> seg;

  FastSet() {}
  FastSet(int n) { build(n); }

  int size() { return n; }

  template <typename F>
  FastSet(int n, F f) {
    build(n, f);
  }

  void build(int m) {
    seg.clear();
    n = m;
    do {
      seg.push_back(vc<u64>((m + B - 1) / B));
      m = (m + B - 1) / B;
    } while (m > 1);
    log = len(seg);
  }
  template <typename F>
  void build(int n, F f) {
    build(n);
    FOR(i, n) { seg[0][i / B] |= u64(f(i)) << (i % B); }
    FOR(h, log - 1) {
      FOR(i, len(seg[h])) {
        seg[h + 1][i / B] |= u64(bool(seg[h][i])) << (i % B);
      }
    }
  }

  bool operator[](int i) const { return seg[0][i / B] >> (i % B) & 1; }
  void insert(int i) {
    for (int h = 0; h < log; h++) {
      seg[h][i / B] |= u64(1) << (i % B), i /= B;
    }
  }
  void add(int i) { insert(i); }
  void erase(int i) {
    u64 x = 0;
    for (int h = 0; h < log; h++) {
      seg[h][i / B] &= ~(u64(1) << (i % B));
      seg[h][i / B] |= x << (i % B);
      x = bool(seg[h][i / B]);
      i /= B;
    }
  }
  void remove(int i) { erase(i); }

  // min[x,n) or n
  int next(int i) {
    assert(i <= n);
    chmax(i, 0);
    for (int h = 0; h < log; h++) {
      if (i / B == seg[h].size()) break;
      u64 d = seg[h][i / B] >> (i % B);
      if (!d) {
        i = i / B + 1;
        continue;
      }
      i += lowbit(d);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += lowbit(seg[g][i / B]);
      }
      return i;
    }
    return n;
  }

  // max [0,x], or -1
  int prev(int i) {
    assert(i >= -1);
    if (i >= n) i = n - 1;
    for (int h = 0; h < log; h++) {
      if (i == -1) break;
      u64 d = seg[h][i / B] << (63 - i % B);
      if (!d) {
        i = i / B - 1;
        continue;
      }
      i -= __builtin_clzll(d);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += topbit(seg[g][i / B]);
      }
      return i;
    }
    return -1;
  }

  bool any(int l, int r) { return next(l) < r; }

  // [l, r)
  template <typename F>
  void enumerate(int l, int r, F f) {
    for (int x = next(l); x < r; x = next(x + 1)) f(x);
  }

  string to_string() {
    string s(n, '?');
    for (int i = 0; i < n; ++i) s[i] = ((*this)[i] ? '1' : '0');
    return s;
  }
};
#line 2 "/home/maspy/compro/library/alg/monoid/minmax.hpp"

template <class X>
struct Monoid_MinMax {
  using P = pair<X, X>;
  using value_type = P;
  static constexpr P op(const P x, const P y) noexcept {
    return {min(x.fi, y.fi), max(x.se, y.se)};
  }
  static constexpr P from_element(const X x) { return {x, x}; }
  static constexpr P unit() { return {infty<X>, -infty<X>}; }
  static constexpr bool commute = true;
};
#line 1 "/home/maspy/compro/library/string/suffix_tree.hpp"

#line 1 "/home/maspy/compro/library/seq/cartesian_tree.hpp"
/*
辞書順で高さを unique して、木にしている。
極大長方形アルゴリズムで線形時間構築。
*/
template <typename T, bool IS_MIN>
struct CartesianTree {
  int n;
  vc<T>& A;
  vc<pair<int, int>> range;
  vc<int> lch, rch, par;
  int root;

  CartesianTree(vc<T>& A) : n(len(A)), A(A) {
    range.assign(n, {-1, -1});
    lch.assign(n, -1);
    rch.assign(n, -1);
    par.assign(n, -1);
    if (n == 1) {
      range[0] = {0, 1};
      root = 0;
      return;
    }
    auto is_sm = [&](int i, int j) -> bool {
      if (IS_MIN) return (A[i] < A[j]) || (A[i] == A[j] && i < j);
      return (A[i] > A[j]) || (A[i] == A[j] && i < j);
    };
    vc<int> st;
    FOR(i, n) {
      while (!st.empty() && is_sm(i, st.back())) {
        lch[i] = st.back();
        st.pop_back();
      }
      range[i].fi = (st.empty() ? 0 : st.back() + 1);
      st.eb(i);
    }
    st.clear();
    FOR_R(i, n) {
      while (!st.empty() && is_sm(i, st.back())) {
        rch[i] = st.back();
        st.pop_back();
      }
      range[i].se = (st.empty() ? n : st.back());
      st.eb(i);
    }
    FOR(i, n) if (lch[i] != -1) par[lch[i]] = i;
    FOR(i, n) if (rch[i] != -1) par[rch[i]] = i;
    FOR(i, n) if (par[i] == -1) root = i;
  }

  // (l, r, h)
  tuple<int, int, T> maximum_rectangle(int i) {
    auto [l, r] = range[i];
    return {l, r, A[i]};
  }

  // (l, r, h)
  T max_rectangle_area() {
    assert(IS_MIN);
    T res = 0;
    FOR(i, n) {
      auto [l, r, h] = maximum_rectangle(i);
      chmax(res, (r - l) * h);
    }
    return res;
  }

  ll count_subrectangle(bool baseline) {
    assert(IS_MIN);
    ll res = 0;
    FOR(i, n) {
      auto [l, r, h] = maximum_rectangle(i);
      ll x = (baseline ? h : h * (h + 1) / 2);
      res += x * (i - l + 1) * (r - i);
    }
    return res;
  }
};
#line 2 "/home/maspy/compro/library/graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (len(used_e) <= e.id) used_e.resize(e.id + 1);
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 5 "/home/maspy/compro/library/string/suffix_tree.hpp"

// https://twitter.com/maspy_stars/status/1565901414236205057?s=20&t=S2Tu6ayozHcakxai8dmh4g
// 各ノードは、suffix array での長方形領域と見なして、
// グラフおよび、領域データを作る。
// sample: test/my_test/suffix_tree.test.cpp
template <typename SUFFIX>
pair<Graph<int, 1>, vc<tuple<int, int, int, int>>> suffix_tree(SUFFIX& X) {
  auto SA = X.SA;
  auto ISA = X.ISA;
  auto LCP = X.LCP;

  vc<tuple<int, int, int, int>> dat;
  vc<pair<int, int>> edges;

  int N = len(SA);
  if (N == 1) {
    Graph<int, 1> G(2);
    G.add(0, 1);
    G.build();
    dat.eb(0, 1, 1, 1), dat.eb(0, 1, 1, 2);
    return {G, dat};
  }

  dat.eb(0, N, 0, 0);
  CartesianTree<int, true> CT(LCP);

  auto dfs = [&](auto& dfs, int p, int idx, int h) -> void {
    int L = CT.range[idx].fi;
    int R = CT.range[idx].se + 1;
    int hh = LCP[idx];
    if (h < hh) {
      edges.eb(p, len(dat));
      p = len(dat);
      dat.eb(L, R, h, hh);
    }
    if (CT.lch[idx] == -1) {
      if (hh < N - SA[idx]) {
        edges.eb(p, len(dat));
        dat.eb(idx, idx + 1, hh, N - SA[idx]);
      }
    } else {
      dfs(dfs, p, CT.lch[idx], hh);
    }
    if (CT.rch[idx] == -1) {
      if (hh < N - SA[idx + 1]) {
        edges.eb(p, len(dat));
        dat.eb(idx + 1, idx + 2, hh, N - SA[idx + 1]);
      }
    } else {
      dfs(dfs, p, CT.rch[idx], hh);
    }
  };
  int r = CT.root;
  if (LCP[r] > 0) {
    edges.eb(0, 1);
    dat.eb(0, N, 0, LCP[r]);
    dfs(dfs, 1, r, LCP[r]);
  } else {
    dfs(dfs, 0, r, 0);
  }
  for (auto& [a, b, c, d]: dat) ++c, ++d;

  Graph<int, 1> G(len(dat));
  for (auto&& [a, b]: edges) G.add(a, b);
  G.build();
  return {G, dat};
}
#line 11 "main.cpp"

void solve() {
  STR(S);

  ll N = len(S);
  Suffix_Array X(S);
  Wavelet_Matrix<int, 1> WM(X.SA);

  auto [G, dat] = suffix_tree(X);

  SegTree<Monoid_MinMax<int>> seg(N, [&](int i) -> pair<int, int> { return {X.SA[i], X.SA[i]}; });

  vi ANS(N + 2);

  for (auto& [L, R, a, b]: dat) {
    if (a == 0) continue;
    auto [i, k] = seg.prod(L, R);
    auto check = [&](int n) -> bool {
      if (i + n > k - n) return 0;
      return WM.count(L, R, i + n, k - n + 1) >= 1;
    };
    int n = binary_search(check, a - 1, b, 0);
    // FOR(i, a, n + 1) ANS[i]++;
    ANS[a]++, ANS[n + 1]--;
  }

  ANS = cumsum<ll>(ANS, 0);
  int n = N / 3;
  ANS = {ANS.begin() + 1, ANS.begin() + n + 1};
  print(ANS);
}

signed main() { solve(); }