#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
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;
}
#endif
#line 1 "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;

#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 2 "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 "library/alg/monoid/max_idx.hpp"

template <typename T, bool tie_is_left = true>
struct Monoid_Max_Idx {
  using value_type = pair<T, int>;
  using X = value_type;
  static X op(X x, X y) {
    if (x.fi > y.fi) return x;
    if (x.fi < y.fi) return y;
    if (x.se > y.se) swap(x, y);
    return (tie_is_left ? x : y);
  }
  static constexpr X unit() { return {-infty<T>, -1}; }
  static constexpr bool commute = true;
};
#line 2 "library/ds/segtree/segtree_beats.hpp"

template <typename ActedMonoid>
struct SegTree_Beats {
  using AM = ActedMonoid;
  using MX = typename AM::Monoid_X;
  using MA = typename AM::Monoid_A;
  using X = typename MX::value_type;
  using A = typename MA::value_type;
  int n, log, size;
  vc<X> dat;
  vc<A> laz;

  SegTree_Beats() {}
  SegTree_Beats(int n) { build(n); }
  template <typename F>
  SegTree_Beats(int n, F f) {
    build(n, f);
  }
  SegTree_Beats(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());
    laz.assign(size, MA::unit());
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  void update(int k) { dat[k] = MX::op(dat[2 * k], dat[2 * k + 1]); }
  void set(int p, X x) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    dat[p] = x;
    for (int i = 1; i <= log; i++) update(p >> i);
  }

  X get(int p) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    return dat[p];
  }

  /*
  void all_apply(int k, A a) {
    dat[k] = ActedMonoid::act(dat[k], a);
    if (k < size) {
      laz[k] = MA::op(laz[k], a);
      if (dat[k].fail) push(k), update(k);
    }
  }
  */

  vc<X> get_all() {
    FOR(k, 1, size) { push(k); }
    return {dat.begin() + size, dat.begin() + size + n};
  }

  X prod(int l, int r) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return MX::unit();
    l += size, r += size;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) push(l >> i);
      if (((r >> i) << i) != r) push((r - 1) >> i);
    }
    X xl = MX::unit(), xr = MX::unit();
    while (l < r) {
      if (l & 1) xl = MX::op(xl, dat[l++]);
      if (r & 1) xr = MX::op(dat[--r], xr);
      l >>= 1, r >>= 1;
    }
    return MX::op(xl, xr);
  }

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

  void apply(int l, int r, A a) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return;
    l += size, r += size;
    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_at(l++, a);
      if (r & 1) apply_at(--r, a);
      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);
    }
  }

private:
  void apply_at(int k, A a) {
    int sz = 1 << (log - topbit(k));
    dat[k] = AM::act(dat[k], a, sz);
    if (k < size) {
      laz[k] = MA::op(laz[k], a);
      if (dat[k].fail) push(k), update(k);
    }
  }

  void push(int k) {
    if (laz[k] == MA::unit()) return;
    apply_at(2 * k, laz[k]), apply_at(2 * k + 1, laz[k]);
    laz[k] = MA::unit();
  }
};
#line 2 "library/ds/segtree/beats_summin_chmax.hpp"
template <typename T>
struct Beats_SumMin_Chmax {
  struct SumMin {
    struct X {
      T sum, min, minc, min2;
      bool fail;
    };
    using value_type = X;
    static X op(const X& x, const X& y) {
      if (x.min == infty<T>) return y;
      if (y.min == infty<T>) return x;
      X z;
      z.sum = x.sum + y.sum;

      z.min = min(x.min, y.min);
      z.minc = (x.min == z.min ? x.minc : 0) + (y.min == z.min ? y.minc : 0);

      z.min2 = infty<T>;
      if (z.min < x.min && x.min < z.min2) z.min2 = x.min;
      if (z.min < x.min2 && x.min2 < z.min2) z.min2 = x.min2;
      if (z.min < y.min && y.min < z.min2) z.min2 = y.min;
      if (z.min < y.min2 && y.min2 < z.min2) z.min2 = y.min2;

      z.fail = 0;
      return z;
    }
    static constexpr X unit() { return {0, infty<T>, 0, infty<T>, 0}; }
    bool commute = true;
  };
  struct AddChmax {
    using X = pair<T, T>;
    using value_type = X;
    static constexpr X op(const X& x, const X& y) {
      auto [a, c] = x;
      auto [d, f] = y;
      a += d, c += d, c = max(c, f);
      return {a, c};
    }
    static constexpr X unit() { return {0, -infty<T>}; }
    bool commute = false;
  };
  struct Beats {
    using Monoid_X = SumMin;
    using Monoid_A = AddChmax;
    using X = typename Monoid_X::value_type;
    using A = typename Monoid_A::value_type;
    static X act(X& x, const A& a, int cnt) {
      assert(!x.fail);
      if (x.min == infty<T>) return x;
      auto [add, ma] = a;
      x.sum += cnt * add, x.min += add, x.min2 += add;
      if (ma == -infty<T>) return x;

      T before_min = x.min;
      x.min = max(x.min, ma);
      if (x.minc == cnt) { x.min2 = x.min, x.sum = cnt * x.min; }
      elif (x.min2 > x.min) { x.sum += (x.min - before_min) * x.minc; }
      else {
        x.fail = 1;
      }
      return x;
    }
  };
  using X = typename SumMin::X;
  SegTree_Beats<Beats> seg;
  Beats_SumMin_Chmax(vc<T>& A) {
    seg.build(len(A), [&](int i) -> X { return from_element(A[i]); });
  }
  template <typename F>
  Beats_SumMin_Chmax(int n, F f) {
    seg.build(n, [&](int i) -> X { return from_element(f(i)); });
  }
  void set(int i, T x) { seg.set(i, from_element(x)); }

  // (sum, min)
  pair<T, T> prod(int l, int r) {
    auto e = seg.prod(l, r);
    return {e.sum, e.min};
  }
  static X from_element(T x) { return {x, x, 1, x}; }

  void chmax(int l, int r, T x) { seg.apply(l, r, {0, x}); }
  void add(int l, int r, T x) { seg.apply(l, r, {x, -infty<T>}); }
};
#line 7 "main.cpp"

void from_string(vvc<int>& A, int& p, vc<string> X) {
  ll N = len(X);
  map<char, vc<pair<int, int>>> MP;
  FOR(i, N) FOR(j, N) {
    char x = X[i][j];
    MP[x].eb(i, j);
  }

  for (auto& [x, pos]: MP) {
    if (x == '.') { continue; }
    for (auto& [i, j]: pos) { A[i][j] = p; }
    p++;
  }
}

void out(vvc<int> A) {
  ll N = len(A);
  ll ans = 0;
  FOR(i, N) FOR(j, N) chmax(ans, A[i][j]);
  vc<int> CNT(ans + 1);
  FOR(i, N) FOR(j, N) { CNT[A[i][j]]++; }
  FOR(x, 1, ans + 1) { assert(CNT[x] == 4); }
  print(ans);
  FOR(i, N) print(A[i]);
}

void put_24(vvc<int>& A, int& p, int i, int j) {
  A[i + 0][j + 0] = p + 0;
  A[i + 0][j + 1] = p + 0;
  A[i + 0][j + 2] = p + 1;
  A[i + 0][j + 3] = p + 1;
  A[i + 1][j + 0] = p + 0;
  A[i + 1][j + 1] = p + 1;
  A[i + 1][j + 2] = p + 0;
  A[i + 1][j + 3] = p + 1;
  p += 2;
}

void put_42(vvc<int>& A, int& p, int i, int j) {
  A[i + 0][j + 0] = p + 0;
  A[i + 1][j + 0] = p + 0;
  A[i + 2][j + 0] = p + 1;
  A[i + 3][j + 0] = p + 1;
  A[i + 0][j + 1] = p + 0;
  A[i + 1][j + 1] = p + 1;
  A[i + 2][j + 1] = p + 0;
  A[i + 3][j + 1] = p + 1;
  p += 2;
}

void fill_rect(vvc<int>& A, int& p, int x1, int x2, int y1, int y2) {
  if ((x2 - x1) % 4 == 0) {
    assert((y2 - y1) % 2 == 0);
    FOR(i, (x2 - x1) / 4) {
      FOR(j, (y2 - y1) / 2) { put_42(A, p, x1 + 4 * i, y1 + 2 * j); }
    }
  } else {
    assert((x2 - x1) % 2 == 0);
    assert((y2 - y1) % 4 == 0);
    FOR(i, (x2 - x1) / 2) {
      FOR(j, (y2 - y1) / 4) { put_24(A, p, x1 + 2 * i, y1 + 4 * j); }
    }
  }
}

void solve_4(ll N) {
  vv(int, A, N, N);
  int p = 1;
  fill_rect(A, p, 0, N, 0, N);
  out(A);
}

void solve_2(ll N) {
  vv(int, A, N, N);
  int p = 1;
  fill_rect(A, p, 0, N, 0, N - 2);
  fill_rect(A, p, 0, N - 2, N - 2, N);
  out(A);
}

vvc<int> solve_odd(ll N) {
  if (N == 1) {
    vv(int, A, N, N);
    return A;
  }
  if (N == 3) {
    vv(int, A, 3, 3);
    A[0] = {1, 2, 1};
    A[1] = {2, 1, 1};
    A[2] = {2, 2, 0};
    return A;
  }
  if (N == 5) {
    vv(int, A, N, N);
    int p = 1;
    vc<string> X = {"aabdb", "acbbd", "caedd", "cc.e.", "..ee."};
    from_string(A, p, X);
    return A;
  }
  if (N == 7) {
    vv(int, A, N, N);
    int p = 1;
    vc<string> X = {"ababcdc", "aabbccd", "eefhfdd", "egffhii",
                    "gejhhki", "gg.j.ik", "..jj.kk"};
    from_string(A, p, X);
    return A;
  }
  if (N == 9) {
    vv(int, A, N, N);
    int p = 1;
    vc<string> X
        = {"aacbcdede", "fabccddee", "afbbgghih", "ffjjgkhhi", "lljmmgkii",
           "nlnjmkkoo", "lnnmpqpro", "ststppqor", "sstt.qqrr"};
    from_string(A, p, X);
    return A;
  }
  if (N == 13) {
    vv(int, A, N, N);
    int p = 1;
    vc<string> X = {"!!ZZ##$%$%&'&", "(!)Z*#$$%%&&'", "!(Z)#*++,-,''",
                    "(())**+/,,-00", "1132344+/--50", "6123347//8805",
                    "16229947::855", "66;;9<77:8=>=", "??;@@9<AA:==>",
                    "B?B;@<<ACAC>>", "?BB@DDEEFCCGG", "HIHIDJEJEFKGK",
                    "HHII.DJJFFKKG"};
    from_string(A, p, X);
    return A;
  }
  if (N % 4 == 3) {
    auto A = solve_odd(N - 2);
    A.resize(N);
    FOR(i, N) A[i].resize(N);
    int p = (N - 2) * (N - 2) / 4 + 1;
    fill_rect(A, p, N - 2, N, 0, N - 3);
    fill_rect(A, p, 0, N - 3, N - 2, N);
    A[N - 3][N - 2] = p, A[N - 3][N - 1] = p;
    A[N - 2][N - 1] = p, A[N - 1][N - 2] = p;
    ++p;
    A[N - 2][N - 3] = p, A[N - 2][N - 2] = p;
    A[N - 1][N - 3] = p, A[N - 1][N - 1] = p;
    ++p;
    return A;
  }
  assert(N % 4 == 1 && N >= 17);

  vvc<int> B = solve_odd(N - 8);
  int p = (N - 8) * (N - 8) / 4 + 1;
  vv(int, A, N, N);
  FOR(i, N - 8) FOR(j, N - 8) { A[8 + i][8 + j] = B[i][j]; }
  fill_rect(A, p, 13, N, 0, 8);
  fill_rect(A, p, 0, 8, 13, N);

  vv(int, C, 13, 13);
  vc<string> X
      = {".....!!ZZ##$$", ".....%!&Z'#($", ".....!%Z&#'$(", ".....%%&&''((",
         ".....)*)*+,+,", "/-/00))**++,,", "-//0101232344", "--55116223374",
         "88599::6;<;47", "=8=59:66;;<77", "8==9>>:??<<@@", "ABABC>CD?DE@E",
         "AABB>CC?DDEE@"};
  from_string(C, p, X);
  reverse(all(C));
  FOR(i, 13) reverse(all(C[i]));
  FOR(i, 13) FOR(j, 13) {
    if (C[i][j] != 0) A[i][j] = C[i][j];
  }
  return A;
}

void solve() {
  LL(N);
  if (N % 4 == 0) return solve_4(N);
  if (N % 4 == 2) return solve_2(N);
  auto A = solve_odd(N);
  out(A);
}

signed main() {
  INT(T);
  FOR(T) solve();
  return 0;
}