#define SINGLE_TESTCASE
// #define MULTI_TESTCASE
// #define AOJ_TESTCASE

#define FAST_IO
// #define FAST_CIO
// #define INTERACTIVE

#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11



/**
 * @brief テンプレート（型）
 * @docs docs/template/template_types.md
 */

#include <bits/stdc++.h>
using namespace std;

#ifndef EPS
#define EPS 1e-11
#endif
using ld = decltype(EPS);

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;

#define vc vector
template <class T>
using vvc = vc<vc<T>>;
template <class T>
using vvvc = vc<vc<vc<T>>>;

using vb = vc<bool>;
using vl = vc<ll>;
using vpll = vc<pll>;
using vtlll = vc<tlll>;
using vtllll = vc<tllll>;
using vstr = vc<string>;
using vvb = vvc<bool>;
using vvl = vvc<ll>;

template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;
template <class T>
using pqg = priority_queue<T>;

#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  i128 res = 0;
  if (s.front() == '-')
  {
    for (int i = 1; i < (int)s.size(); i++)
      res = 10 * res + s[i] - '0';
    res = -res;
  }
  else
  {
    for (auto &&c : s)
      res = 10 * res + c - '0';
  }
  return res;
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  if (x < 0)
    x = -x, sign = "-";
  while (x > 0)
  {
    res += '0' + x % 10;
    x /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  is >> s;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}
#endif

#define cauto const auto


/**
 * @brief テンプレート（rep）
 * @docs docs/template/template_rep.md
 */

// https://trap.jp/post/1224/

#define overload4(_1, _2, _3, _4, name, ...) name
#define rep1(i, n) for (ll i = 0, nnnnn = ll(n); i < nnnnn; i++)
#define rep2(i, l, r) for (ll i = ll(l), rrrrr = ll(r); i < rrrrr; i++)
#define rep3(i, l, r, d) for (ll i = ll(l), rrrrr = ll(r), ddddd = ll(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i, n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i, l, r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi3(i, l, r, d) for (int i = int(l), rrrrr = int(r), ddddd = int(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)

#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif
#ifndef EPS
#define EPS 1e-11
#endif


/**
 * @brief テンプレート（演算）
 * @docs docs/template/template_math.md
 */

template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T, class U>
inline bool chmax(T &a, U b) { return a < b ? a = b, true : false; }

template <class T = ll, class U, class V>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V>
inline constexpr T divceil(U a, V b) { return T(a) / T(b) + (T(a) % T(b) && (T(a) ^ T(b)) >= 0); }
template <class T = ll, class U, class V>
inline constexpr T divround(U a, V b) { return divfloor<T>(2 * T(a) + T(b), 2 * T(b)); }
template <class T = ll, class U, class V>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;

  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B, class M>
T mul_limited(A a, B b, M m = INF)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (b == 0)
    return 0;
  return T(a) > T(m) / T(b) ? T(m) : T(a) * T(b);
}
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m = INF)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (a <= 1 || b == 0)
    return min(ipow<T>(a, b), T(m));
  
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
    {
      if (res > T(m) / tmp)
        return m;
      res *= tmp;
    }
    b >>= 1;
    if (b == 0)
      break;
    if (tmp > T(m) / tmp)
      return m;
    tmp *= tmp;
  }
  return res;
}

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;
  if (k == 2)
  {
    if constexpr (sizeof(T) > sizeof(ull))
    {
      if ((u128)a < ((u128)1 << 120))
        return sqrtl(a);
    }
    else
      return sqrtl(a);
  }

  auto isok = [&](T x) -> bool
  {
    if (x == 0)
      return true;
    T res = 1, k2 = k;
    while (true)
    {
      if (k2 & 1)
      {
        if (res > T(a) / x)
          return false;
        res *= x;
      }
      k2 >>= 1;
      if (k2 == 0)
        break;
      if (x > T(a) / x)
        return false;
      x *= x;
    }
    return res <= T(a);
  };

  T x = pow(a, 1.0 / k);
  bool up = true;
  while (!isok(x))
    up = false, x--;
  if (up)
  {
    while (x < numeric_limits<T>::max() && isok(x + 1))
      x++;
  }
  return x;
}

// https://misawa.github.io/others/avoid_errors/techniques_to_avoid_errors.html
template <class D = decltype(EPS), class A>
int SGN(A a, const D &eps = EPS) { return int(a > eps) - int(a < -eps); }

// 位取り記数法と同じ順番（下位桁が後ろ）
// 0 に対しては {0} が返る
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base)
{
  assert(val >= 0);
  assert(base >= 2);
  if (val == 0)
    return {0};
  vc<T> a;
  while (val > 0)
  {
    a.emplace_back(val % base);
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
// 位取り記数法と同じ順番（下位桁が後ろ）
template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base, int n)
{
  assert(val >= 0);
  assert(base >= 2);
  assert(n >= 0);
  vc<T> a(n);
  repi(i, n)
  {
    a[i] = val % base;
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  auto a = base_repr(val, base);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base, int n)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  assert(n >= 0);
  auto a = base_repr(val, base, n);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}


/**
 * @brief テンプレート（vector）
 * @docs docs/template/template_vector.md
 */

#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back

template <class F>
auto gen_vec(const int &n, const F &f)
{
  vc<decltype(f(0))> res(n);
  repi(i, n) res[i] = f(i);
  return res;
}

// https://qiita.com/Chippppp/items/13150f5e0ea99f444d97#%E5%A4%9A%E6%AC%A1%E5%85%83vector%E7%94%9F%E6%88%90%E9%96%A2%E6%95%B0
template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}

template <class T = ll>
T ctol(const char &c, const string &s)
{
  repi(i, SZ<int>(s)) if (s[i] == c) return i;
  return -1;
}
template <class T = ll>
vc<T> stov(const string &s, const char &first)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return s[i] - first; });
}
template <class T = ll>
vc<T> stov(const string &s, const string &t)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return ctol(s[i], t); });
}

template <class T>
vc<T> concat(const vvc<T> &vs)
{
  vc<T> res;
  for (cauto &v : vs)
    res.insert(res.end(), ALL(v));
  return res;
}
template <class T>
vc<T> concat(const vc<T> &v) { return v; }
template <class T, class... Ts>
vc<T> concat(vc<T> v, const vc<Ts> &...vs)
{
  (v.insert(v.end(), ALL(vs)), ...);
  return v;
}

template <class T, class I>
T vecget(const vc<T> &v, I i, const T &dflt_negative = -INF, const T &dflt_positive = INF)
{
  if (i < 0)
    return dflt_negative;
  if (i >= SZ<int>(v))
    return dflt_positive;
  return v[i];
}

#ifndef INF
#define INF 4'000'000'000'000'000'037LL
#endif


/**
 * @brief テンプレート（アルゴリズム）
 * @docs docs/template/template_algo.md
 */

template <class V>
auto SUM(const V &v)
{
  typename V::value_type s{};
  fec(vi : v) s += vi;
  return s;
}
template <class T, class V>
T SUM(const V &v)
{
  T s{};
  fec(vi : v) s += vi;
  return s;
}
template <class V>
auto MAX(const V &v) { return *max_element(ALL(v)); }
template <class V>
auto MIN(const V &v) { return *min_element(ALL(v)); }
template <class I = ll, class V>
I ARGMAX(const V &v) { return max_element(ALL(v)) - v.begin(); }
template <class I = ll, class V>
I ARGMIN(const V &v) { return min_element(ALL(v)) - v.begin(); }

template<class T = ll, class V>
T mex(const V &a)
{
  int n = a.size();
  vector<bool> exists(n, false);
  repi(i, n) if (0 <= a[i] && a[i] < n) exists[a[i]] = true;
  repi(x, n) if (!exists[x]) return x;
  return n;
}

template <class T = ll>
vc<T> permid(const int &n, const int &base_index = 0)
{
  vc<T> p(n);
  repi(i, n) p[i] = i + base_index;
  return p;
}
template <class T>
vc<T> perminv(const vc<T> &p)
{
  if (p.empty())
    return {};
  const int n = p.size();
  vc<T> q(MAX(p) + 1);
  repi(i, n) if (p[i] >= 0) q[p[i]] = i;
  return q;
}
// a[p[i]] for all i
template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p)
{
  const int n = p.size();
  vc<T> res(n);
  repi(i, n)
  {
    assert(0 <= p[i] && p[i] < U(a.size()));
    res[i] = a[p[i]];
  }
  return res;
}

template <class V>
V reversed(const V &v) { return V(v.rbegin(), v.rend()); }

#if __cplusplus < 202002L
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  sort(ALL(v), forward<Args>(args)...);
  return v;
}
#else
template <class V, class... Args>
V sorted(V v, Args&&... args)
{
  ranges::sort(v, forward<Args>(args)...);
  return v;
}
#endif

template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V>
V uniqued(V v) { unique(v); return v; }

template <class V>
void sortunique(V &v)
{
  sort(ALL(v));
  unique(v);
}
template <class V>
V sortuniqued(V v) { sortunique(v); return v; }

// 01234 -> 12340
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
// 01234 -> 12340
template <class V, class U>
V rotated(V v, U k) { rotate(v, k); return v; }

template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}

// 12
// 34 -> 246
// 56    135
// (反時計回り)
template <class VV, class U = ll>
VV rot90(const VV &a, U k = 1)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  k = (k % 4 + 4) % 4;
  if (k == 0)
    return a;
  else if (k == 1)
  {
    VV b(m);
    repi(j, m) b[j].resize(n);
    repi(i, n)
    {
      assert(SZ<int>(a[i]) == m);
      repi(j, m) b[m - 1 - j][i] = a[i][j];
    }
    return b;
  }
  else if (k == 2)
  {
    VV b(n);
    repi(i, n) b[i].resize(m);
    repi(i, n)
    {
      assert(SZ<int>(a[i]) == m);
      repi(j, m) b[n - 1 - i][m - 1 - j] = a[i][j];
    }
    return b;
  }
  else
  {
    VV b(m);
    repi(j, m) b[j].resize(n);
    repi(i, n)
    {
      assert(SZ<int>(a[i]) == m);
      repi(j, m) b[j][n - 1 - i] = a[i][j];
    }
    return b;
  }
}

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e() { return 0; }
};
template <class T, const T infty = INF>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr S e() { return infty; }
};
template <class T, const T infty = INF>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b) { return max(a, b); }
  static constexpr S e() { return -infty; }
};

// left_index が 0 なら、長さ n+1 で a.front() が e()
// left_index が 1 なら、長さ n で e() がない
template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0)
{
  const int n = v.size();
  vc<typename M::S> res(n + 1);
  res[0] = M::e();
  repi(i, n) res[i + 1] = M::op(res[i], v[i]);
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}
// right_index が 0 なら、長さ n+1 で a.back() が e()
// right_index が 1 なら、長さ n で e() がない
template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0)
{ return reversed(cuml<M>(reversed(v), right_index)); }
template <class T>
vc<T> cumlsum(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidAdd<T>>(v, left_index); }
template <class T>
vc<T> cumrsum(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidAdd<T>>(v, right_index); }
template <class T>
vc<T> cumlmin(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMin<T>>(v, left_index); }
template <class T>
vc<T> cumrmin(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMin<T>>(v, right_index); }
template <class T>
vc<T> cumlmax(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidMax<T>>(v, left_index); }
template <class T>
vc<T> cumrmax(const vc<T> &v, int right_index = 0)
{ return cumr<MonoidMax<T>>(v, right_index); }

// デフォルトでは長さ n+1
// left_index, right_index をそれぞれ 1 にすると、左右が削除される
template <class T>
vc<T> adjd(const vc<T> &v, int left_index = 0, int right_index = 0)
{
  int n = v.size();
  vc<T> res(n + 1);
  res[0] = v[0];
  repi(i, 1, n) res[i] = v[i] - v[i - 1];
  res[n] = -v[n - 1];
  res.erase(res.end() - right_index, res.end());
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}

const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};


/**
 * @brief テンプレート（二分探索）
 * @docs docs/template/template_binsearch.md
 */

template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;

// --- LB, UB ---

#if __cplusplus < 202002L
struct identity
{
  template <class T>
  constexpr T &&operator()(T &&t) const noexcept
  { return forward<T>(t); }
};
namespace internal
{
  template <class T = ll, class V, class Judge>
  inline T bound_helper(const V &v, Judge judge)
  {
    int l = -1, r = v.size();
    while (r - l > 1)
    {
      int m = (l + r) / 2;
      if (judge(m))
        l = m;
      else
        r = m;
    }
    return r;
  }
};
// val <= v[i] となる最小の i (val 未満の値の個数)
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return comp(proj(*(v.begin() + i)), val); });
}
// val < v[i] となる最小の i (val 以下の値の個数)
template <class T = ll, class V, class Value, class Comp = less<>, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{
  return internal::bound_helper(v, [&](int i) -> bool
                                { return !comp(val, proj(*(v.begin() + i))); });
}
#define DEFAULT_COMP less<>
#else
// val <= v[i] となる最小の i (val 未満の値の個数)
template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::lower_bound(v, val, comp, proj) - v.begin(); }
// val < v[i] となる最小の i (val 以下の値の個数)
template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::upper_bound(v, val, comp, proj) - v.begin(); }
#define DEFAULT_COMP ranges::less
#endif

// --- vector 等の lt, leq, gt, geq ---

// v[i] < val となる最大の i (なければ -1)
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj) - 1; }
// v[i] <= val となる最大の i (なければ -1)
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_max(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj) - 1; }
// val < v[i] となる最小の i (なければ n)
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj); }
// val <= v[i] となる最小の i (なければ n)
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_min(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj); }
// v[i] < val となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj); }
// v[i] <= val となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto leq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return UB<T>(v, val, comp, proj); }
// val < v[i] となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto gt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - UB<T>(v, val, comp, proj); }
// val <= v[i] となる i の個数
template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto geq_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return SZ<T>(v) - LB<T>(v, val, comp, proj); }
// l <= v[i] < r となる i の個数
template <class T = ll, class V, class L, class R, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto in_cnt(const V &v, L l, R r, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{
  if (l > r)
    return 0;
  return lt_cnt<T>(v, r, comp, proj) - lt_cnt<T>(v, l, comp, proj);
}

// --- set 等の lt, leq, gt, geq ---

// *it < val となる最大の it (なければ end())
template <class V, class Value>
inline auto lt_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.lower_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}
// *it <= val となる最大の it (なければ end())
template <class V, class Value>
inline auto leq_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.upper_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}
// val < *it となる最小の it (なければ end())
template <class V, class Value>
inline auto gt_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.upper_bound(val); }
// val <= *it となる最小の it (なければ end())
template <class V, class Value>
inline auto geq_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.lower_bound(val); }

// --- 自作二分探索 ---

// (ok, ng)
template <class T = ll, class Judge, class InitOk, class InitNg>
pair<T, T> binsearch(const Judge &judge, const InitOk &init_ok, const InitNg &init_ng)
{
  T ok(init_ok), ng(init_ng);
  assert(judge(ok));
  assert(!judge(ng));
  while (ok - ng != 1 && ng - ok != 1)
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}
template <class T = ld, class Judge, class InitOk, class InitNg>
T binsearch_real(const Judge &judge, const InitOk &init_ok, const InitNg &init_ng, int iteration_count = 100)
{
  T ok(init_ok), ng(init_ng);
  assert(judge(ok));
  assert(!judge(ng));
  repi(_, iteration_count)
  {
    T mid = (ok + ng) / 2;
    (judge(mid) ? ok : ng) = mid;
  }
  return ok;
}
// (ok, ng)
template <class T = ll, class Judge, class InitVal>
pair<T, T> expsearch(const Judge &judge, const InitVal &init_val, bool positive = true)
{
  T ok, ng;
  if (judge(init_val))
  {
    ok = init_val, ng = init_val + (positive ? 1 : -1);
    for (int i = 1; judge(ng); i++)
      ok = ng, ng = init_val + (positive ? 1 : -1) * (T(1) << i);
  }
  else
  {
    ng = init_val, ok = init_val + (positive ? 1 : -1);
    for (int i = 1; !judge(ok); i++)
      ng = ok, ok = init_val + (positive ? 1 : -1) * (T(1) << i);
  }
  while (ok - ng != 1 && ng - ok != 1)
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}


/**
 * @brief テンプレート（ビット演算）
 * @docs docs/template/template_bit.md
 */

template <class T>
inline constexpr ull pow2(T k) { return 1ULL << k; }
template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

#if __cplusplus < 202002L
// x == 0 ならば 0、そうでなければ 1 + floor(log2(x))
// 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... 
inline constexpr ull bit_width(ull x) { return x == 0 ? 0 : 64 - __builtin_clzll(x); }
// 0, 1, 2, 2, 4, 4, 4, 4, 8, 8, ...
inline constexpr ull bit_floor(ull x) { return x == 0 ? 0ULL : 1ULL << (bit_width(x) - 1); }
// 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, ...
inline constexpr ull bit_ceil(ull x) { return x == 0 ? 1ULL : 1ULL << bit_width(x - 1); }
inline constexpr ull countr_zero(ull x) { assert(x != 0); return __builtin_ctzll(x); }
inline constexpr ull popcount(ull x) { return __builtin_popcountll(x); }
inline constexpr bool has_single_bit(ull x) { return popcount(x) == 1; }
#else
// 0, 1, 2, 2, 3, 3, 3, 3, 4, 4, ... 
inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }
// 0, 1, 2, 2, 4, 4, 4, 4, 8, 8, ...
inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }
// 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, ...
inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }
#endif

inline constexpr ull lsb_pos(ull x) { assert(x != 0); return countr_zero(x); }
inline constexpr ull msb_pos(ull x) { assert(x != 0); return bit_width(x) - 1; }
inline constexpr ull lsb_mask(ull x) { assert(x != 0); return x & -x; }
inline constexpr ull msb_mask(ull x) { assert(x != 0); return bit_floor(x); }

inline constexpr bool btest(ull x, uint k) { return (x >> k) & 1; }
template <class T>
inline void bset(T &x, uint k, bool b = 1) { b ? x |= (1ULL << k) : x &= ~(1ULL << k); }
template <class T>
inline void bflip(T &x, uint k) { x ^= (1ULL << k); }
inline constexpr bool bsubset(ull x, ull y) { return (x & y) == x; }
inline constexpr bool bsupset(ull x, ull y) { return (x & y) == y; }
inline constexpr ull bsetminus(ull x, ull y) { return x & ~y; }



/**
 * @brief テンプレート（dump）
 * @docs docs/template/template_dump.md
 */

#ifdef LOCAL
#include <cpp-dump.hpp> // https://github.com/philip82148/cpp-dump
namespace cpp_dump::_detail
{
  inline string export_var(
      const i128 &x, const string &indent, size_t last_line_length,
      size_t current_depth, bool fail_on_newline, const export_command &command
  ) {
    return export_var(i128tos(x), indent, last_line_length, current_depth, fail_on_newline, command);
  }
} // namespace cpp_dump::_detail
#define dump(...) cpp_dump(__VA_ARGS__)
namespace cp = cpp_dump;
CPP_DUMP_SET_OPTION_GLOBAL(log_label_func, cp::log_label::line());
CPP_DUMP_SET_OPTION_GLOBAL(max_iteration_count, 10000);
#define local(...) __VA_ARGS__
#else
#define dump(...)
#define local(...)
#endif

/**
 * @brief テンプレート（入出力）
 * @docs docs/template/template_inout.md
 */

// https://judge.yosupo.jp/submission/170706 (maspy さん)
// https://judge.yosupo.jp/submission/21623  (Nyaan さん)
#if defined FAST_IO and not defined LOCAL
namespace fastio {
static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
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, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}

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

void rd1(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
}

void rd1(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 rd1_real(T &x) {
  string s;
  rd1(s);
  x = stod(s);
}

template <typename T>
void rd1_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 rd1(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }
// void rd1(f128 &x) { rd1_real(x); }

template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <size_t N = 0, typename T>
void rd1_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd1(x);
    rd1_tuple<N + 1>(t);
  }
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  rd1_tuple(tpl);
}

template <size_t N = 0, typename T>
void rd1(array<T, N> &x) {
  for (auto &d: x) rd1(d);
}
template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}

void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd1(h), read(t...);
}

void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(const string s) {
  for (char c: s) wt1(c);
}
void wt1(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt1(s[i]);
}

template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 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 wt1_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt1(s);
}

template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }
// void wt1(f128 x) { wt1_real(x); }

template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <size_t N = 0, typename T>
void wt1_tuple(const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt1(' '); }
    const auto x = std::get<N>(t);
    wt1(x);
    wt1_tuple<N + 1>(t);
  }
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  wt1_tuple(tpl);
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}

void write() {}
template <class Head, class... Tail>
void write(Head &&head, Tail &&... tail) {
  wt1(head);
  write(forward<Tail>(tail)...);
}

void print() { wt1('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt1(head);
  if (sizeof...(Tail)) wt1(' ');
  print(forward<Tail>(tail)...);
}

} // namespace fastio

#endif

#if defined FAST_IO and not defined LOCAL
struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;
#endif

// https://trap.jp/post/1224/

// ---- 入力 ----
#if defined LOCAL or not defined FAST_IO
template <class T, class U>
istream &operator>>(istream &is, pair<T, U> &p)
{
  is >> p.first >> p.second;
  return is;
}
template <class... Ts>
istream &operator>>(istream &is, tuple<Ts...> &t)
{
  apply([&](auto &...a)
        { (is >> ... >> a); }, t);
  return is;
}
template <class T, size_t n>
istream &operator>>(istream &is, array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
template <class T>
istream &operator>>(istream &is, vc<T> &a)
{
  const size_t n = a.size();
  for (size_t i = 0; i < n; i++)
    is >> a[i];
  return is;
}
#endif

namespace internal
{

#if defined LOCAL or not defined FAST_IO
template <class... Ts>
void CIN(Ts &...a) { (cin >> ... >> a); }
#endif

#if defined FAST_IO and not defined LOCAL
template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }
#else
template <class... Ts>
void READnodump(Ts &...a) { CIN(a...); }
#endif

template <class T>
void READVECnodump(int n, vc<T> &v)
{
  v.resize(n);
  READnodump(v);
}
template <class T, class... Ts>
void READVECnodump(int n, vc<T> &v, vc<Ts> &...vs)
{ READVECnodump(n, v), READVECnodump(n, vs...); }

template <class T>
void READVEC2nodump(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  READnodump(v);
}
template <class T, class... Ts>
void READVEC2nodump(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ READVEC2nodump(n, m, v), READVEC2nodump(n, m, vs...); }

template <class T>
void READJAGnodump(int n, vvc<T> &v)
{
  v.resize(n);
  repi(i, n)
  {
    int k;
    READnodump(k);
    READVECnodump(k, v[i]);
  }
}
template <class T, class... Ts>
void READJAGnodump(int n, vvc<T> &v, vvc<Ts> &...vs)
{ READJAGnodump(n, v), READJAGnodump(n, vs...); }

}; // namespace internal

#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define IN(T, ...) T __VA_ARGS__; READ(__VA_ARGS__)

#define CHAR(...) IN(char, __VA_ARGS__)
#define INT(...) IN(int, __VA_ARGS__)
#define LL(...) IN(ll, __VA_ARGS__)
#define STR(...) IN(string, __VA_ARGS__)
#define ARR(T, n, ...) array<T, n> __VA_ARGS__; READ(__VA_ARGS__)

#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define READVEC2(...) internal::READVEC2nodump(__VA_ARGS__); dump(__VA_ARGS__)

#define VEC(T, n, ...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define VEC2(T, n, m, ...) vvc<T> __VA_ARGS__; READVEC2(n, m, __VA_ARGS__)

#define READJAG(...) internal::READJAGnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define JAG(T, n, ...) vvc<T> __VA_ARGS__; READJAG(n, __VA_ARGS__)

// ----------

// ----- 出力 -----
#ifdef INTERACTIVE
#define ENDL endl
#else
#define ENDL '\n'
#endif

#if defined LOCAL or not defined FAST_IO
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p)
{
  os << p.first << ' ' << p.second;
  return os;
}

namespace internal
{

template <size_t N = 0, typename T>
void cout_tuple(ostream &os, const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { os << ' '; }
    const auto x = std::get<N>(t);
    os << x;
    cout_tuple<N + 1>(os, t);
  }
}

}; // namespace internal

template <class... Ts>
ostream &operator<<(ostream &os, const tuple<Ts...> &t)
{
  internal::cout_tuple(os, t);
  return os;
}
template <class T, size_t n>
ostream &operator<<(ostream &os, const array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << a[i];
  }
  return os;
}
template <class T>
ostream &operator<<(ostream &os, const vc<T> &v)
{
  const size_t n = v.size();
  for (size_t i = 0; i < n; i++)
  {
    if (i)
      os << ' ';
    os << v[i];
  }
  return os;
}

namespace internal
{

template <class T>
void COUTW() {}
template <class... Ts>
void COUTW(const Ts &...a) { (cout << ... << a); }

template <class T>
void COUTP() { cout << ENDL; }
template <class T>
void COUTP(const T &a) { cout << a << ENDL; }
template <class T, class... Ts>
void COUTP(const T &a, const Ts &...b)
{
  cout << a;
  (cout << ... << (cout << ' ', b));
  cout << ENDL;
}

}; // namespace internal
#endif

#if defined FAST_IO and not defined LOCAL
#define WRITE fastio::write
#define PRINT fastio::print
#else
#define WRITE internal::COUTW
#define PRINT internal::COUTP
#endif
#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

template <class T>
void PRINTV(const vc<T> &v) { for (auto &vi : v) PRINT(vi); }
#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)
// ----------

// ----- 基準ずらし -----
template <class T, class U, class P>
pair<T, U> operator+=(pair<T, U> &a, const P &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> &a, const P &b) { return a += b; }

template <class T, size_t n, class A>
array<T, n> operator+=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> &a, const A &b) { return a += b; }

namespace internal
{

template <size_t... I, class A, class B>
auto tuple_add_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}

}; // namespace internal

template <class... Ts, class Tp>
tuple<Ts...> operator+=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> &a, const Tp &b) { return a += b; }

template <class T, class Add>
void offset(vc<T> &v, const Add &add) { for (auto &vi : v) vi += add; }
template <class T, class Add>
void offset(vvc<T> &v, const Add &add) { for (auto &vi : v) for (auto &vij : vi) vij += add; }
// ----------

// ----- 転置 -----
template <class T, const size_t m>
array<vc<T>, m> top(const vc<array<T, m>> &vt)
{
  const size_t n = vt.size();
  array<vc<T>, m> tv;
  tv.fill(vc<T>(n));
  for (size_t i = 0; i < n; i++)
    for (size_t j = 0; j < m; j++)
      tv[j][i] = vt[i][j];
  return tv;
}
template <class T, const size_t m>
vc<array<T, m>> top(const array<vc<T>, m> &tv)
{
  if (tv.empty()) return {};
  const size_t n = tv[0].size();
  vc<array<T, m>> vt(n);
  for (size_t j = 0; j < m; j++)
  {
    assert(tv[j].size() == n);
    for (size_t i = 0; i < n; i++)
      vt[i][j] = tv[j][i];
  }
  return vt;
}

template <class T, class U>
pair<vc<T>, vc<U>> top(const vc<pair<T, U>> &vt)
{
  const size_t n = vt.size();
  pair<vc<T>, vc<U>> tv;
  tv.first.resize(n), tv.second.resize(n);
  for (size_t i = 0; i < n; i++)
    tie(tv.first[i], tv.second[i]) = vt[i];
  return tv;
}
template <class T, class U>
vc<pair<T, U>> top(const pair<vc<T>, vc<U>> &tv)
{
  const size_t n = tv.first.size();
  assert(n == tv.second.size());
  vc<pair<T, U>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = make_pair(tv.first[i], tv.second[i]);
  return vt;
}

namespace internal
{

template <size_t... I, class V, class Tp>
auto vt_to_tv_impl(V &tv, const Tp &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }

template <size_t... I, class Tp>
auto tv_to_vt_impl(const Tp &tv, index_sequence<I...>, size_t index)
{ return make_tuple(get<I>(tv)[index]...); }

};

template <class... Ts>
auto top(const vc<tuple<Ts...>> &vt)
{
  const size_t n = vt.size();
  tuple<vc<Ts>...> tv;
  apply([&](auto &...v)
        { ((v.resize(n)), ...); }, tv);
  for (size_t i = 0; i < n; i++)
    internal::vt_to_tv_impl(tv, vt[i], make_index_sequence<tuple_size_v<decltype(tv)>>{}, i);
  return tv;
}

template <class... Ts>
auto top(const tuple<vc<Ts>...> &tv)
{
  size_t n = get<0>(tv).size();
  apply([&](auto &...v)
        { ((assert(v.size() == n)), ...); }, tv);
  vc<tuple<Ts...>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = internal::tv_to_vt_impl(tv, index_sequence_for<Ts...>{}, i);
  return vt;
}
// ----------





namespace internal
{

constexpr ll powmod32_constexpr(ll x, ll n, int m)
{
  if (m == 1)
    return 0;
  uint _m = (uint)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}

constexpr bool isprime32_constexpr(int n)
{
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr ll bases[3] = {2, 7, 61};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod32_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
template <int n>
constexpr bool isprime32 = isprime32_constexpr(n);

struct barrett32
{
  uint m;
  ull im;

  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = (ull)((u128(z)*im) >> 64);
    ull y = x * m;
    return (uint)(z - y + (z < y ? m : 0));
  }
};

}

namespace internal
{

constexpr ll powmod64_constexpr(ll x, ll n, ll m)
{
  if (m == 1)
    return 0;
  ull _m = (ull)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    u128 y128(y);
    if (n & 1)
      r = (y128 * r) % _m;
    y = (y128 * y) % _m;
    n >>= 1;
  }
  return r;
}

constexpr bool isprime64_constexpr(ll n)
{
  if (n <= INT_MAX)
    return isprime32_constexpr(n);
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod64_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = (u128(y) * y) % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
template <ll n>
constexpr bool isprime64 = isprime64_constexpr(n);

inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}

struct montgomery64odd
{
  ull m, im, sq;
  // sq = (2^64)^2 % m = (2^128 - m) % m = (-m % 2^128) % m
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m)
      t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const
  { return reduce(u128(v % m + m) * sq); }
};

// https://www.mathenachia.blog/even-mod-montgomery-impl/
struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;

  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b;  // m == 2^b * mx, mx is odd
    imx = inv64(mx);
    d = powmod64_constexpr((mx + 1) / 2, b, mx);  // 2^{-b} mod mx
    u128 sq = -u128(mx) % mx;  // 2^128 mod mx
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    ull p = x & MASK(b);  // x mod 2^b
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m)
    {
      t -= m;
      if (t >= m)
        t -= m;
    }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const
  { return reduce(u128(v % m + m) * q); }
};

}


namespace internal
{

#define REF static_cast<mint &>(*this)
#define CREF static_cast<const mint &>(*this)
#define VAL *static_cast<const mint *>(this)

template <class mint>
struct modint_base
{
  mint &operator+=(const mint &rhs)
  {
    mint &self = REF;
    self._v += rhs._v;
    if (self._v >= self.umod())
      self._v -= self.umod();
    return self;
  }
  mint &operator-=(const mint &rhs)
  {
    mint &self = REF;
    self._v -= rhs._v;
    if (self._v >= self.umod())
      self._v += self.umod();
    return self;
  }
  mint &operator/=(const mint &rhs)
  {
    mint &self = REF;
    return self = self * rhs.inv();
  }

  mint &operator++()
  {
    mint &self = REF;
    self._v++;
    if (self._v == self.umod())
      self._v = 0;
    return self;
  }
  mint &operator--()
  {
    mint &self = REF;
    if (self._v == 0)
      self._v = self.umod();
    self._v--;
    return self;
  }
  mint operator++(int)
  {
    mint res = VAL;
    ++REF;
    return res;
  }
  mint operator--(int)
  {
    mint res = VAL;
    --REF;
    return res;
  }

  mint operator+() const { return VAL; }
  mint operator-() const { return mint() - VAL; }

  mint pow(ll n) const
  {
    assert(n >= 0);
    mint x = VAL, r = 1;
    while (n)
    {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }

  friend mint operator+(const mint &lhs, const mint &rhs)
  { return mint(lhs) += rhs; }
  friend mint operator-(const mint &lhs, const mint &rhs)
  { return mint(lhs) -= rhs; }
  friend mint operator*(const mint &lhs, const mint &rhs)
  { return mint(lhs) *= rhs; }
  friend mint operator/(const mint &lhs, const mint &rhs)
  { return mint(lhs) /= rhs; }
  friend bool operator==(const mint &lhs, const mint &rhs)
  { return mint(lhs).eq(rhs); }
  friend bool operator!=(const mint &lhs, const mint &rhs)
  { return mint(lhs).neq(rhs); }
private:
  bool eq(const mint &rhs) { return REF._v == rhs._v; }
  bool neq(const mint &rhs) { return REF._v != rhs._v; }
};

}

#if defined LOCAL or not defined FAST_IO
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
istream &operator>>(istream &is, T &x)
{
  ll a;
  is >> a;
  x = a;
  return is;
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
ostream &operator<<(ostream &os, const T &x)
{
  os << x.val();
  return os;
}
#else
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void rd1(T &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void wt1(const T &x) { fastio::wt1(x.val()); }
#endif


/**
 * @brief 拡張ユークリッド互除法 (extgcd)
 * @docs docs/math/extgcd.md
 */

// g == gcd(x, y) >= 0, ax + by == g を満たす (g, x, y)
// max(|x|, |y|) <= max(|a|, |b|)
template <class T = ll>
constexpr tuple<T, T, T> extgcd(const T &a, const T &b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  
  // a*x1 + b*y1 == z1  ...(1)
  // a*x2 + b*y2 == z2  ...(2)
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    // (1)' = (2)
    // (2)' = (1) - q*(2)
    T q = z1 / z2;
    tie(x1, x2) = make_pair(x2, x1 - q * x2);
    tie(y1, y2) = make_pair(y2, y1 - q * y2);
    tie(z1, z2) = make_pair(z2, z1 - q * z2);
  }
  if (z1 < 0)
    x1 = -x1, y1 = -y1, z1 = -z1;
  return {z1, x1, y1};
}

/**
 * @brief modint (64 bit)
 * @docs docs/math/modint/modint64.md
 */

template <ll m>
struct static_modint64 : internal::modint_base<static_modint64<m>>
{
  using mint = static_modint64;
private:
  friend struct internal::modint_base<static_modint64<m>>;
  ull _v;
  static constexpr ull umod() { return m; }
  static constexpr bool prime = internal::isprime64<m>;

public:
  static constexpr ll mod() { return m; }
  static mint raw(ll v)
  {
    mint x;
    x._v = v;
    return x;
  }

  static_modint64() : _v(0) {}
  template <class T>
  static_modint64(T v)
  {
    if constexpr (is_unsigned_v<T>)
    {
      _v = (ull)(v % umod());
    }
    else
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (ull)x;
    }
  }

  ll val() const { return (ll)_v; }

  mint& operator*=(const mint &rhs)
  {
    u128 z = _v;
    z *= rhs._v;
    _v = (ull)(z % umod());
    return *this;
  }

  mint inv() const
  {
    if (prime)
    {
      assert(_v != 0);
      return CREF.pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<ll>(_v, m);
      assert(g == 1);
      return x;
    }
  }
};

template <int id>
struct dynamic_modint64_odd : internal::modint_base<dynamic_modint64_odd<id>>
{
  using mint = dynamic_modint64_odd;
private:
  friend struct internal::modint_base<dynamic_modint64_odd<id>>;
  ull _v; // montgomery expression
  static internal::montgomery64odd mg;
  static ull umod() { return mg.umod(); }

public:
  static ll mod() { return (ll)(mg.umod()); }
  static void set_mod(ll m)
  {
    assert(m >= 1 && m % 2 == 1);
    mg = internal::montgomery64odd(m);
  }

  dynamic_modint64_odd() : _v(0) {}
  dynamic_modint64_odd(i128 v)
  { _v = mg.inv_reduce(v); }

  ll val() const { return (ll)mg.reduce(_v); }

  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }

  mint inv() const
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::montgomery64odd dynamic_modint64_odd<id>::mg((1LL << 61) - 1);

template <int id>
struct dynamic_modint64 : internal::modint_base<dynamic_modint64<id>>
{
  using mint = dynamic_modint64;
private:
  friend struct internal::modint_base<dynamic_modint64<id>>;
  ull _v; // montgomery expression
  static internal::montgomery64 mg;
  static ull umod() { return mg.umod(); }

public:
  static ll mod() { return (ll)(mg.umod()); }
  static void set_mod(ll m)
  {
    assert(m >= 1);
    mg = internal::montgomery64(m);
  }

  dynamic_modint64() : _v(0) {}
  dynamic_modint64(i128 v)
  { _v = mg.inv_reduce(v); }

  ll val() const { return (ll)mg.reduce(_v); }

  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }

  mint inv() const
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::montgomery64 dynamic_modint64<id>::mg((1LL << 61) - 1);

using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64_odd = dynamic_modint64_odd<-1>;
using modint64 = dynamic_modint64<-1>;
using mint = modint61;

#ifdef LOCAL
const mint B = 10;
#else
const mint B = 4530108;
#endif

mint solve(const vc<pair<string, ll>> &SX)
{
  dump(SX);
  // i128 sum = 0;
  mint powsum = 1; // B^sum
  mint hash = 0;
  fec([ s, x ] : reversed(SX))
  {
    mint ss = 0;
    fec(c : s)
    {
      ss *= B;
      #ifdef LOCAL
      ss += c - '0';
      #else
      ss += c;
      #endif
    }

    ll len = s.size();
    // B^sum * s * (B^{len*x}-1)/(B^len-1)
    mint tmp = powsum * ss * (B.pow(len).pow(x) - 1) / (B.pow(len) - 1);
    hash += tmp;
    dump(tmp, hash);

    // sum += i128(len) * x;
    powsum *= B.pow(len).pow(x);
    dump(s, x, len);
  }
  return hash;
}

void init() {}

void main2()
{
  LL(N);
  using P = pair<string, ll>;
  VEC(P, N, SL);
  mint hash1 = solve(SL);
  reverse(ALL(SL));
  fem([s, l] : SL) reverse(ALL(s));
  mint hash2 = solve(SL);
  PRINT(hash1 == hash2 ? "Yes" : "No");
}

void test()
{
  /*
  local(
    rep(testcase, 100000)
    {
      cout << endl;
      dump(testcase);


      // ----- generate cases -----

      string S = to_string(rand() % 10000);
      string T = to_string(rand() % 10000);
      auto runsS = calc_rle(S), runsT = calc_rle(T);
      vpll CX, DY;
      fec(run : runsS) CX.eb(run.val - '0', run.len);
      fec(run : runsT) DY.eb(run.val - '0', run.len);

      // --------------------------

      // ------ check output ------
      #define INPUT CX, DY
      auto god = naive(INPUT);
      auto ans = solve(INPUT);
      if (god != ans)
      {
        dump(S, T);
        dump(INPUT);
        dump(god, ans);
        exit(0);
      }
      // --------------------------
    }
    dump("ok");
  );
  //*/
}

int main()
{
  cauto CERR = [](string val, string color)
  {
    string s = "\033[" + color + "m" + val + "\033[m";
    #ifdef LOCAL
    cerr << s;
    #endif
    /* コードテストで確認する際にコメントアウトを外す
    cerr << val;
    //*/
  };

  #if defined FAST_IO and not defined LOCAL
  CERR("\n[FAST_IO]\n\n", "32");
  #endif
  #if defined FAST_CIO and not defined LOCAL
  CERR("\n[FAST_CIO]\n\n", "32");
  cin.tie(0);
  ios::sync_with_stdio(false);
  #endif
  cout << fixed << setprecision(20);

  test();
  init();

  #if defined AOJ_TESTCASE or (defined LOCAL and defined SINGLE_TESTCASE)
  CERR("\n[AOJ_TESTCASE]\n\n", "35");
  while (true)
  {
    dump("new testcase");
    main2();
  }
  #elif defined SINGLE_TESTCASE
  CERR("\n[SINGLE_TESTCASE]\n\n", "36");
  main2();
  #elif defined MULTI_TESTCASE
  CERR("\n[MULTI_TESTCASE]\n\n", "33");
  dump("T");
  IN(uint, T);
  while (T--)
  {
    dump("new testcase");
    main2();
  }
  #endif
}