#line 1 "library/my_template.hpp"
#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using pi = pair<ll, ll>;
using vi = vector<ll>;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

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 vec(type, name, ...) vector<type> name(__VA_ARGS__)
#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 FOR4_R(i, a, b, c) for (ll i = (b)-1; i >= ll(a); i -= (c))
#define overload4(a, b, c, d, e, ...) e
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) \
  overload4(__VA_ARGS__, FOR4_R, 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

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

#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())

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); }
// (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, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}

template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}

template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

ll binary_search(function<bool(ll)> check, ll ok, ll ng) {
  assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    if (check(x))
      ok = x;
    else
      ng = x;
  }
  return ok;
}

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);
}

vi s_to_vi(const string &S, char first_char) {
  vi A(S.size());
  FOR(i, S.size()) { A[i] = S[i] - first_char; }
  return A;
}

template <typename T>
vector<T> cumsum(vector<T> &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;
}

template <typename CNT, typename T>
vc<CNT> bincount(const vc<T> &A, int size) {
  vc<CNT> C(size);
  for (auto &&x: A) { ++C[x]; }
  return C;
}

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

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  int n = len(A);
  assert(len(I) == n);
  vc<T> B(n);
  FOR(i, n) B[i] = A[I[i]];
  return B;
}
#line 1 "library/other/io.hpp"
// based on yosupo's fastio
#include <unistd.h>

namespace detail {
template <typename T, decltype(&T::is_modint) = &T::is_modint>
std::true_type check_value(int);
template <typename T>
std::false_type check_value(long);
} // namespace detail

template <typename T>
struct is_modint : decltype(detail::check_value<T>(0)) {};
template <typename T>
using is_modint_t = enable_if_t<is_modint<T>::value>;
template <typename T>
using is_not_modint_t = enable_if_t<!is_modint<T>::value>;

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <class T, is_modint_t<T> * = nullptr>
  bool read_single(T &ref) {
    long long val = 0;
    bool f = read_single(val);
    ref = T(val);
    return f;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <class A, class B, class C>
  bool read_single(tuple<A, B, C> &p) {
    return (read_single(get<0>(p)) && read_single(get<1>(p))
            && read_single(get<2>(p)));
  }
  template <class A, class B, class C, class D>
  bool read_single(tuple<A, B, C, D> &p) {
    return (read_single(get<0>(p)) && read_single(get<1>(p))
            && read_single(get<2>(p)) && read_single(get<3>(p)));
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char &val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string &s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double &x) {
    ostringstream oss;
    oss << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double &x) {
    ostringstream oss;
    oss << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <class T, is_modint_t<T> * = nullptr>
  void write(T &ref) {
    write(ref.val);
  }
  template <class T>
  void write(const vector<T> &val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> &val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <class A, class B, class C>
  void write(const tuple<A, B, C> &val) {
    auto &[a, b, c] = val;
    write(a), write(' '), write(b), write(' '), write(c);
  }
  template <class A, class B, class C, class D>
  void write(const tuple<A, B, C, D> &val) {
    auto &[a, b, c, d] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d);
  }
  template <class A, class B, class C, class D, class E>
  void write(const tuple<A, B, C, D, E> &val) {
    auto &[a, b, c, d, e] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e);
  }
  template <class A, class B, class C, class D, class E, class F>
  void write(const tuple<A, B, C, D, E, F> &val) {
    auto &[a, b, c, d, e, f] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e), write(' '), write(f);
  }
  template <class T, size_t S>
  void write(const array<T, S> &val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if(val < 0){
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if(negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};

Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);

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

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __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 1 "library/ds/mo.hpp"
struct Mo {
  vector<pair<int, int> > lr;

  void add(int l, int r) { /* [l, r) */
    lr.emplace_back(l, r);
  }

  template <typename AL, typename AR, typename EL, typename ER, typename O>
  void calc(const AL &add_left, const AR &add_right, const EL &erase_left,
            const ER &erase_right, const O &query) {
    int n = 1;
    for (auto &&[l, r]: lr) chmax(n, r);
    int q = (int)lr.size();
    int bs = n / min<int>(n, sqrt(q));
    vector<int> ord(q);
    iota(begin(ord), end(ord), 0);
    sort(begin(ord), end(ord), [&](int a, int b) {
      int ablock = lr[a].first / bs, bblock = lr[b].first / bs;
      if (ablock != bblock) return ablock < bblock;
      return (ablock & 1) ? lr[a].second > lr[b].second
                          : lr[a].second < lr[b].second;
    });
    int l = 0, r = 0;
    for (auto idx: ord) {
      while (l > lr[idx].first) add_left(--l);
      while (r < lr[idx].second) add_right(r++);
      while (l < lr[idx].first) erase_left(l++);
      while (r > lr[idx].second) erase_right(--r);
      query(idx);
    }
  }

  template <typename A, typename E, typename O>
  void calc(const A &add, const E &erase, const O &query) {
    calc(add, add, erase, erase, query);
  }
};
#line 1 "library/ds/fastset.hpp"
/* 64分木。
insert, erase
[]での存在判定
next, prev
*/
struct FastSet {
  using uint = unsigned;
  using ull = unsigned long long;

  int bsr(ull x) { return 63 - __builtin_clzll(x); }
  int bsf(ull x) { return __builtin_ctzll(x); }

  static constexpr uint B = 64;
  int n, lg;
  vc<vc<ull>> seg;
  FastSet(int _n) : n(_n) {
    do {
      seg.push_back(vc<ull>((_n + B - 1) / B));
      _n = (_n + B - 1) / B;
    } while (_n > 1);
    lg = int(seg.size());
  }
  bool operator[](int i) const { return (seg[0][i / B] >> (i % B) & 1) != 0; }
  void insert(int i) {
    for (int h = 0; h < lg; h++) {
      seg[h][i / B] |= 1ULL << (i % B);
      i /= B;
    }
  }
  void erase(int i) {
    for (int h = 0; h < lg; h++) {
      seg[h][i / B] &= ~(1ULL << (i % B));
      if (seg[h][i / B]) break;
      i /= B;
    }
  }

  // x以上最小の要素を返す。存在しなければ n。
  int next(int i) {
    for (int h = 0; h < lg; h++) {
      if (i / B == seg[h].size()) break;
      ull d = seg[h][i / B] >> (i % B);
      if (!d) {
        i = i / B + 1;
        continue;
      }
      // find
      i += bsf(d);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += bsf(seg[g][i / B]);
      }
      return i;
    }
    return n;
  }

  // x以下最大の要素を返す。存在しなければ -1。
  int prev(int i) {
    if (i < 0) return -1;
    chmin(i, n - 1);
    for (int h = 0; h < lg; h++) {
      if (i == -1) break;
      ull d = seg[h][i / B] << (63 - i % 64);
      if (!d) {
        i = i / B - 1;
        continue;
      }
      // find
      i += bsr(d) - (B - 1);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += bsr(seg[g][i / B]);
      }
      return i;
    }
    return -1;
  }

  // [l, r) 内の要素を全部集める
  vc<int> collect(int l, int r) {
    vc<int> res;
    int x = l - 1;
    while (1) {
      x = next(x + 1);
      if (x >= r) break;
      res.eb(x);
    }
    return res;
  }

  void debug() {
    string s;
    FOR(i, n) s += ((*this)[i] ? '1' : '0');
    print(s);
  }
};

// for mistype
using FaseSet = FastSet;
#line 2 "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 {
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  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:
    int l, r;
    const Graph* G;
  };

  bool is_prepared() { return prepared; }
  constexpr bool is_directed() { return directed; }

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

  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;
  }

  // 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(M) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }

  void read_parent(int off = 1) {
    FOR3(v, 1, N) {
      INT(p);
      p -= off;
      add(p, v);
    }
    build();
  }

  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(v, N) 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]};
  }

  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);
    }
  }
};
#line 3 "library/graph/range_to_range_graph.hpp"

template <typename T>
struct Range_to_Range_Graph {
  int n;
  int n_node;
  vc<tuple<int, int, T>> edges;

  Range_to_Range_Graph(int n) : n(n), n_node(n * 3) {
    FOR3(i, 2, n + n) { edges.eb(to_upper_idx(i / 2), to_upper_idx(i), 0); }
    FOR3(i, 2, n + n) { edges.eb(to_lower_idx(i), to_lower_idx(i / 2), 0); }
  }

  inline int to_upper_idx(const int& i) {
    if (i >= n) return i - n;
    return n + i;
  }

  inline int to_lower_idx(const int& i) {
    if (i >= n) return i - n;
    return n + n + i;
  }

  void add(int frm, int to, T wt) { edges.eb(frm, to, wt); }

  void add_frm_range(int frm_l, int frm_r, int to, T wt) {
    int l = frm_l + n, r = frm_r + n;
    while (l < r) {
      if (l & 1) add(to_lower_idx(l++), to, wt);
      if (r & 1) add(to_lower_idx(--r), to, wt);
      l >>= 1, r >>= 1;
    }
  }

  void add_to_range(int frm, int to_l, int to_r, T wt) {
    int l = to_l + n, r = to_r + n;
    while (l < r) {
      if (l & 1) add(frm, to_upper_idx(l++), wt);
      if (r & 1) add(frm, to_upper_idx(--r), wt);
      l >>= 1, r >>= 1;
    }
  }

  void add_range_to_range(int frm_l, int frm_r, int to_l, int to_r, T wt) {
    int new_node = n_node++;
    add_frm_range(frm_l, frm_r, new_node, wt);
    add_to_range(new_node, to_l, to_r, T(0));
  }

  Graph<T, 1> build() {
    Graph<T, 1> G(n_node);
    for (auto&& [a, b, c]: edges) G.add(a, b, c);
    G.build();
    return G;
  }
};
#line 1 "library/flow/maxflow.hpp"
template <typename Cap>
struct MaxFlowGraph {
  struct Edge {
    int to, rev;
    Cap cap;
  };

  const Cap INF;
  int N;
  vvc<Edge> G;
  vc<int> prog, level;
  Cap flow_ans;
  bool calculated;

  MaxFlowGraph(int N, Cap INF) : INF(INF), N(N), calculated(0) {}

  void add(int frm, int to, Cap cap) {
    assert(0 <= frm && frm < N);
    assert(0 <= to && to < N);
    assert(Cap(0) <= cap);
    if (len(G) < N) G.resize(N);
    G[frm].eb(Edge{to, (int)G[to].size(), cap});
    G[to].eb(Edge{frm, (int)G[frm].size() - 1, 0});
  }

  Cap flow(int source, int sink) {
    if (calculated) return flow_ans;
    calculated = true;
    chmax(N, source + 1);
    chmax(N, sink + 1);
    G.resize(N);
    flow_ans = 0;
    while (set_level(source, sink)) {
      fill(all(prog), 0);
      prog.assign(N, 0);
      while (1) {
        Cap x = flow_dfs(source, sink, INF);
        if (x == 0) break;
        flow_ans += x;
        chmin(flow_ans, INF);
        if (flow_ans == INF) return flow_ans;
      }
    }
    return flow_ans;
  }

  // 最小カットの値および、カットを表す 01 列を返す
  pair<Cap, vc<int>> cut(int source, int sink) {
    Cap f = flow(source, sink);
    vc<int> res(N);
    FOR(v, N) res[v] = (level[v] >= 0 ? 0 : 1);
    return {f, res};
  }

  // 残余グラフの辺
  vc<tuple<int, int, Cap>> get_edges() {
    vc<tuple<int, int, Cap>> edges;
    FOR(v, N) for (auto&& e: G[v]) { edges.eb(v, e.to, e.cap); }
    return edges;
  }

private:
  bool set_level(int source, int sink) {
    level.assign(N, -1);
    level[source] = 0;
    queue<int> que;
    que.push(source);
    while (!que.empty()) {
      int v = que.front();
      que.pop();
      for (auto&& e: G[v]) {
        if (e.cap > 0 && level[e.to] == -1) {
          level[e.to] = level[v] + 1;
          if (e.to == sink) return true;
          que.push(e.to);
        }
      }
    }
    return false;
  }

  Cap flow_dfs(int v, int sink, Cap lim) {
    if (v == sink) return lim;
    Cap res = 0;
    for (int& i = prog[v]; i < (int)G[v].size(); ++i) {
      auto& e = G[v][i];
      if (e.cap > 0 && level[e.to] == level[v] + 1) {
        Cap a = flow_dfs(e.to, sink, min(lim, e.cap));
        if (a > 0) {
          e.cap -= a;
          G[e.to][e.rev].cap += a;
          res += a;
          lim -= a;
          if (lim == 0) break;
        }
      }
    }
    return res;
  }
};
#line 1 "library/other/xor_range.hpp"
// lo <= a ^ x < hi となる x の区間 [li, ri) たちを返す
vc<pi> xor_range(ll a, ll lo, ll hi) {
  vc<pi> res;
  FOR(k, 64) {
    if (lo == hi) break;
    ll b = 1LL << k;
    if (lo & b) {
      res.eb((lo ^ a), (lo ^ a) + b);
      lo += b;
    }
    if (hi & b) {
      res.eb((hi - b) ^ a, ((hi - b) ^ a) + b);
      hi -= b;
    }
    if (a & b) a ^= b;
  }
  return res;
}
#line 8 "main.cpp"

void solve() {
  LL(N, LIM);
  VEC(ll, A, N);

  auto f = [&](auto& dfs, vi A, int k) -> ll {
    if (len(A) == 0) return 0;
    // A[i] < 2^{k+1} が分かっている
    // xor <= k となるようになるべくたくさん選ぶ
    if ((1 << (k + 1)) - 1 <= LIM) return len(A);
    if ((1 << k) > LIM) {
      // それぞれ解く
      vi B, C;
      for (auto&& x: A) {
        if (x & 1 << k)
          B.eb(x ^ (1 << k));
        else
          C.eb(x);
      }
      return max(dfs(dfs, B, k - 1), dfs(dfs, C, k - 1));
    }
    // 補グラフで最大独立集合。補グラフは二部グラフなので、最大マッチングで解ける。
    // a^b > LIM に辺を張って、最大マッチング。
    sort(all(A));
    Range_to_Range_Graph<int> RGG(len(A) + 2);
    int source = len(A), sink = len(A) + 1;
    FOR(i, len(A)) {
      ll a = A[i];
      if (a >= (1 << k)) break;
      for (auto&& [l, r]: xor_range(a, LIM + 1, 1 << (k + 1))) {
        l = LB(A, l);
        r = LB(A, r);
        RGG.add_to_range(i, l, r, 1);
      }
    }
    FOR(i, len(A)) {
      ll a = A[i];
      if (a < (1 << k))
        RGG.add(source, i, 1);
      else
        RGG.add(i, sink, 1);
    }
    auto tmp = RGG.build();
    int INF = 1 << 30;
    MaxFlowGraph<int> G(tmp.N, INF);
    for (auto&& e: tmp.edges) {
      if (e.cost == 0) {
        G.add(e.frm, e.to, INF);
      } else {
        G.add(e.frm, e.to, 1);
      }
    }
    ll match = G.flow(source, sink);
    return len(A) - match;
  };
  ll ANS = f(f, A, 20);
  print(ANS);
}

signed main() {
  cin.tie(nullptr);
  ios::sync_with_stdio(false);
  cout << setprecision(15);

  LL(T);
  // ll T = 1;
  FOR(T) solve();

  return 0;
}