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

#define FAST_IO
// #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
 */

/**
 * 参考：
 * 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
 */

inline bool chmin(auto &a, cauto &b) { return a > b ? a = b, true : false; }
inline bool chmax(auto &a, cauto &b) { return a < b ? a = b, true : false; }

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

template <class T = ll>
constexpr T ipow(auto a, auto b)
{
  assert(b >= 0);
  if (b == 0) return 1;
  if (a == 0 || a == 1) return a;
  if (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>
T mul_limited(cauto &a, cauto &b, cauto &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>
T pow_limited(cauto &a, cauto &b, cauto &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;
  repi(_, b)
  {
    if (res > T(m) / T(a))
      return T(m);
    res *= T(a);
  }
  return res;
}

template <class T = ll>
constexpr T iroot(cauto &a, cauto &k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;

  auto isok = [&](const T &x) -> bool
  {
    if (x == 0)
      return true;
    T tmp = 1;
    repi(_, k)
    {
      if (tmp > T(a) / x)
        return false;
      tmp *= x;
    }
    return tmp <= T(a);
  };

  T ok = 0, ng = 1;
  while (isok(ng))
    ok = ng, ng <<= 1;
  while (ng - ok > 1)
  {
    T mid = ((ng - ok) >> 1) + ok;
    if (isok(mid))
      ok = mid;
    else
      ng = mid;
  }
  return ok;
}

// https://misawa.github.io/others/avoid_errors/techniques_to_avoid_errors.html
template <class D = decltype(EPS)>
int sgn(cauto &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）
template <class T = ll>
vc<T> base_repr(auto val, int base)
{
  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>
vc<T> base_repr(auto val, int base, int n)
{
  vc<T> a(n);
  repi(i, n)
  {
    a[i] = val % base;
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
string base_repr_str(auto val, int base, bool use_upper = true)
{
  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;
}
string base_repr_str(auto val, int base, int n, bool use_upper = true)
{
  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>
inline T SZ(cauto &x) { return x.size(); }

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>
auto dvec(cauto (&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>
T vecget(const vc<T> &v, cauto &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];
}


/**
 * @brief テンプレート（関数オブジェクト）
 * @docs docs/template/template_func.md
 */

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

template <class T = ll>
struct max_op
{
  T operator()(const T &a, const T &b) const { return max(a, b); }
};
template <class T = ll>
struct min_op
{
  T operator()(const T &a, const T &b) const { return min(a, b); }
};

template <class T, const T val>
struct const_fn
{
  T operator()() const { return val; }
};
using max_e = const_fn<ll, -INF>;
using min_e = const_fn<ll, INF>;
using zero_fn = const_fn<ll, 0LL>;

#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 T, class U = ll>
U SUM(const vc<T> &v) { return accumulate(ALL(v), U(0)); }
template <class T>
T MAX(const vc<T> &v) { return *max_element(ALL(v)); }
template <class T>
T MIN(const vc<T> &v) { return *min_element(ALL(v)); }
template <class T, class U = ll>
U ARGMAX(const vc<T> &v) { return max_element(ALL(v)) - v.begin(); }
template <class T, class U = ll>
U ARGMIN(const vc<T> &v) { return min_element(ALL(v)) - v.begin(); }

template<class T, class U = ll>
U mex(const vector<T> &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.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(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;
  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, class F = decltype(plus<>())>
vc<T> cuml(const vc<T> &v, const F &op = plus<>(), const T &e = 0)
{
  const int n = v.size();
  vc<T> res(n + 1, e);
  repi(i, n) res[i + 1] = op(res[i], v[i]);
  return res;
}
template <class T, class F = decltype(plus<>())>
vc<T> cumr(const vc<T> &v, const F &op = plus<>(), const T &e = 0)
{ return reversed(cuml<T, F>(reversed(v), op, e)); }
template <class T>
vc<T> cumlmax(const vc<T> &v) { return cuml(v, max_op<T>(), max_e()()); }
template <class T>
vc<T> cumrmax(const vc<T> &v) { return cumr(v, max_op<T>(), max_e()()); }
template <class T>
vc<T> cumlmin(const vc<T> &v) { return cuml(v, min_op<T>(), min_e()()); }
template <class T>
vc<T> cumrmin(const vc<T> &v) { return cumr(v, min_op<T>(), min_e()()); }

template <class T>
vc<T> adjd(const vc<T> &v)
{
  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];
  return res;
}

template <class T = ll>
struct direct_product
{
private:
  vc<T> a;
public:
  direct_product(const vc<T> &a) : a(a)
  {
    assert(!a.empty());
    fec(ai : a) assert(ai >= 1);
  }
  struct Iterator
  {
  private:
    vc<T> b;
    const direct_product &prod;

  public:
    Iterator(const vc<T> &b, const direct_product &prod) : b(b), prod(prod) {}
    vc<T> operator*() const { return b; }
    Iterator& operator++()
    {
      b.back()++;
      repi(i, SZ<int>(prod.a) - 1, 0, -1)
      {
        if (b[i] == prod.a[i])
        {
          b[i] = 0;
          b[i - 1]++;
        }
        else
          break;
      }
      return *this;
    }
    bool operator!=(const Iterator &other) const { return b != other.b; }
  };
  Iterator begin() const { return Iterator(vc<T>(a.size(), 0), *this); }
  Iterator end() const
  {
    vc<T> c(a.size(), 0);
    c[0] = a[0];
    return Iterator(c, *this);
  }
};

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
 */

#if __cplusplus < 202002L
// val <= v[i] となる最小の i (val 未満の値の個数)
// 引数: comp
template <class T = ll, class V, class... Args>
inline T LB(const V &v, Args&&... args)
{ return lower_bound(ALL(v), forward<Args>(args)...) - v.begin(); }
// val < v[i] となる最小の i (val 以下の値の個数)
// 引数: comp
template <class T = ll, class V, class... Args>
inline T UB(const V &v, Args&&... args)
{ return upper_bound(ALL(v), forward<Args>(args)...) - v.begin(); }
#else
// val <= v[i] となる最小の i (val 未満の値の個数)
// 引数: comp, proj
template <class T = ll, class V, class... Args>
inline T LB(const V &v, Args&&... args)
{ return ranges::lower_bound(v, forward<Args>(args)...) - v.begin(); }
// val < v[i] となる最小の i (val 以下の値の個数)
// 引数: comp, proj
template <class T = ll, class V, class... Args>
inline T UB(const V &v, Args&&... args)
{ return ranges::upper_bound(v, forward<Args>(args)...) - v.begin(); }
#endif

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;

template <class T = ll, class V, class... Args>
inline auto lt_max(const V &v, Args&&... args)
{
  if constexpr (is_random_access_iterator_v<typename V::iterator>)
    return LB<T>(v, forward<Args>(args)...) - 1;
  else
  {
    auto it = v.lower_bound(forward<Args>(args)...);
    if (it == v.begin())
      return v.end();
    else
      return prev(it);
  }
}
template <class T = ll, class V, class... Args>
inline auto leq_max(const V &v, Args&&... args)
{
  if constexpr (is_random_access_iterator_v<typename V::iterator>)
    return UB<T>(v, forward<Args>(args)...) - 1;
  else
  {
    auto it = v.upper_bound(forward<Args>(args)...);
    if (it == v.begin())
      return v.end();
    else
      return prev(it);
  }
}
template <class T = ll, class V, class... Args>
inline auto gt_min(const V &v, Args&&... args)
{
  if constexpr (is_random_access_iterator_v<typename V::iterator>)
    return UB<T>(v, forward<Args>(args)...);
  else
    return v.upper_bound(forward<Args>(args)...);
}
template <class T = ll, class V, class... Args>
inline auto geq_min(const V &v, Args&&... args)
{
  if constexpr (is_random_access_iterator_v<typename V::iterator>)
    return LB<T>(v, forward<Args>(args)...);
  else
    return v.lower_bound(forward<Args>(args)...);
}

template <class T = ll, class V, class... Args>
inline T lt_cnt(const V &v, Args&&... args)
{ return LB<T>(v, forward<Args>(args)...); }
template <class T = ll, class V, class... Args>
inline T leq_cnt(const V &v, Args&&... args)
{ return UB<T>(v, forward<Args>(args)...); }
template <class T = ll, class V, class... Args>
inline T gt_cnt(const V &v, Args&&... args)
{ return SZ<T>(v) - UB<T>(v, forward<Args>(args)...); }
template <class T = ll, class V, class... Args>
inline T geq_cnt(const V &v, Args&&... args)
{ return SZ<T>(v) - LB<T>(v, forward<Args>(args)...); }

template <class T = ll, class V, class... Args>
inline T in_cnt(const V &v, auto l, auto r, Args&&... args)
{
  if (l > r)
    return 0;
  return lt_cnt<T>(v, r, forward<Args>(args)...) - lt_cnt<T>(v, l, forward<Args>(args)...);
}

template <class T = ll>
pair<T, T> binsearch(cauto &judge, cauto &init_ok, cauto &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 make_pair(ok, ng);
}
template <class T = ld>
T binsearch_real(cauto &judge, cauto &init_ok, cauto &init_ng, const 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;
}
template <class T = ll>
pair<T, T> expsearch(cauto &judge, cauto &init_val, const 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 make_pair(ok, ng);
}


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

inline constexpr ull pow2(auto k) { return 1ULL << k; }
inline constexpr ull MASK(auto 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 bit_width((ull)x); }
// 0, 1, 2, 2, 4, 4, 4, 4, 8, 8, ...
inline constexpr ll bit_floor(ll x) { return bit_floor((ull)x); }
// 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, ...
inline constexpr ll bit_ceil(ll x) { return bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return 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; }

template <class T>
struct bsubsets
{
private:
  T x;
public:
  bsubsets(T x) : x(x) {}
  struct Iterator
  {
  private:
    T y;
    bool is_end;
    const bsubsets &bs;
  public:
    Iterator(T y, bool is_end, const bsubsets &bs) : y(y), is_end(is_end), bs(bs) {}
    T operator*() const { return y; }
    Iterator& operator++()
    {
      if (y == 0)
        is_end = true;
      y = (y - 1) & bs.x;
      return *this;
    }
    bool operator!=(const Iterator &other) const { return y != other.y || is_end != other.is_end; }
  };
  Iterator begin() const { return Iterator(x, false, *this); }
  Iterator end() const { return Iterator(x, true, *this); }
};
template <class T>
struct bsupsets
{
private:
  int n;
  T x;
public:
  bsupsets(int n, T x) : n(n), x(x) {}
  struct Iterator
  {
  private:
    T y;
    const bsupsets &bs;
  public:
    Iterator(T y, const bsupsets &bs) : y(y), bs(bs) {}
    T operator*() const { return y; }
    Iterator& operator++()
    {
      y = (y + 1) | bs.x;
      return *this;
    }
    bool operator!=(const Iterator &other) const { return y != other.y; }
  };
  Iterator begin() const { return Iterator(x, *this); }
  Iterator end() const { return Iterator((T(1) << n) | x, *this); }
};


/**
 * @brief テンプレート（入出力）
 * @docs docs/template/template_inout.md
 */
/**
 * 参考：
 * https://trap.jp/post/1224/
 */

// ---- 入力 ----
template <class T, class U>
istream &operator>>(istream &is, pair<T, U> &p)
{
  cin >> p.first >> p.second;
  return is;
}
template <class T, size_t n>
istream &operator>>(istream &is, array<T, n> &a)
{
  for (size_t i = 0; i < n; i++)
    cin >> a[i];
  return is;
}
template <class... Ts>
istream &operator>>(istream &is, tuple<Ts...> &t)
{
  apply([&](auto &...a)
        { (is >> ... >> a); }, t);
  return is;
}

template <class... Ts>
void CIN(Ts &...a) { (cin >> ... >> a); }

template <class T>
void CINVEC(int n, vc<T> &v)
{
  v.resize(n);
  repi(i, n) cin >> v[i];
}
template <class T, class... Ts>
void CINVEC(int n, vc<T> &v, vc<Ts> &...vs)
{ CINVEC(n, v), CINVEC(n, vs...); }

template <class T>
void CINVEC2(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  repi(i, n) repi(j, m) cin >> v[i][j];
}
template <class T, class... Ts>
void CINVEC2(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ CINVEC2(n, m, v), CINVEC2(n, m, vs...); }

#define IN(T, ...) T __VA_ARGS__; CIN(__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__; CIN(__VA_ARGS__)

#define VEC(T, n, ...) vc<T> __VA_ARGS__; CINVEC(n, __VA_ARGS__)
#define VEC2(T, n, m, ...) vvc<T> __VA_ARGS__; CINVEC2(n, m, __VA_ARGS__)
// ----------

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

template <class T>
void PRINT(const T &a) { cout << a << ENDL; }
template <class T, class... Ts>
void PRINT(const T &a, const Ts &...b)
{
  cout << a;
  (cout << ... << (cout << ' ', b));
  cout << ENDL;
}
#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

template <class T>
void PRINTVEC(const vc<T> &v)
{
  const int n = v.size();
  repi(i, n) cout << v[i] << (i == n - 1 ? "" : " ");
  cout << ENDL;
}
template <class T>
void PRINTVECT(const vc<T> &v) { for (auto &vi : v) cout << vi << ENDL; }
template <class T>
void PRINTVEC2(const vvc<T> &v) { for (auto &vi : v) PRINTVEC(vi); }
// ----------

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

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

template <size_t... I>
auto tuple_add_impl(auto &a, cauto &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}
template <class... Ts>
tuple<Ts...> operator+=(tuple<Ts...> &a, cauto &b)
{ return tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }
template <class... Ts>
tuple<Ts...> operator+(tuple<Ts...> &a, cauto &b) { return a += b; }

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

template <size_t... I>
auto vt_to_tv_impl(auto &tv, cauto &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }
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++)
    vt_to_tv_impl(tv, vt[i], make_index_sequence<tuple_size_v<decltype(tv)>>{}, i);
  return tv;
}

template <size_t... I>
auto tv_to_vt_impl(cauto &tv, index_sequence<I...>, size_t index)
{ return make_tuple(get<I>(tv)[index]...); }
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] = tv_to_vt_impl(tv, index_sequence_for<Ts...>{}, i);
  return vt;
}
// ----------


/**
 * @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




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 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 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;
}
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 <int n>
constexpr bool isprime32 = isprime32_constexpr(n);
template <ll n>
constexpr bool isprime64 = isprime64_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));
  }
};

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;
  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, sq;
  uint b;

  explicit montgomery64(ull m) : m(m), sq(-u128(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
  }
  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) * sq); }
};

}


/**
 * @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 (32 bit)
 * @docs docs/math/modint/modint.md
 */

template <int m>
struct static_modint
{
  using mint = static_modint;
private:
  uint _v;
  static constexpr uint umod() { return m; }
  static constexpr bool prime = internal::isprime32<m>;

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

  static_modint() : _v(0) {}
  template <class T>
  static_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }

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

  mint& operator++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int)
  {
    mint res = *this;
    ++*this;
    return res;
  }
  mint operator--(int)
  {
    mint res = *this;
    --*this;
    return res;
  }

  mint& operator+=(const mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    ull z = _v;
    z *= rhs._v;
    _v = (uint)(z % umod());
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(ll n) const
  {
    assert(n >= 0);
    mint x = *this, r = 1;
    while (n)
    {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const
  {
    if (prime)
    {
      assert(_v != 0);
      return pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<int>(_v, m);
      assert(g == 1);
      return x;
    }
  }

  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 lhs._v == rhs._v; }
  friend bool operator!=(const mint &lhs, const mint &rhs)
  { return lhs._v != rhs._v; }

  friend istream &operator>>(istream &is, mint &x)
  {
    ll a;
    is >> a;
    x = a;
    return is;
  }
  friend ostream &operator<<(ostream &os, const mint &x)
  {
    os << x.val();
    return os;
  }
};

template <int id>
struct dynamic_modint
{
  using mint = dynamic_modint;
private:
  uint _v;
  static internal::barrett32 bt;
  static uint umod() { return bt.umod(); }

public:
  static int mod() { return (int)(bt.umod()); }
  static void set_mod(int m)
  {
    assert(m >= 1);
    bt = internal::barrett32(m);
  }
  static mint raw(int v)
  {
    mint x;
    x._v = v;
    return x;
  }

  dynamic_modint() : _v(0) {}
  template <class T>
  dynamic_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }

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

  mint& operator++()
  {
    _v++;
    if (_v == umod())
      _v = 0;
    return *this;
  }
  mint& operator--()
  {
    if (_v == 0)
      _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int)
  {
    mint res = *this;
    ++*this;
    return res;
  }
  mint operator--(int)
  {
    mint res = *this;
    --*this;
    return res;
  }

  mint& operator+=(const mint& rhs)
  {
    _v += rhs._v;
    if (_v >= umod())
      _v -= umod();
    return *this;
  }
  mint& operator-=(const mint &rhs)
  {
    _v -= rhs._v;
    if (_v >= umod())
      _v += umod();
    return *this;
  }
  mint& operator*=(const mint &rhs)
  {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint& operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }

  mint pow(ll n) const
  {
    assert(n >= 0);
    mint x = *this, r = 1;
    while (n)
    {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const
  {
    auto [g, x, y] = extgcd<int>(_v, mod());
    assert(g == 1);
    return x;
  }

  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 lhs._v == rhs._v; }
  friend bool operator!=(const mint &lhs, const mint &rhs)
  { return lhs._v != rhs._v; }

  friend istream &operator>>(istream &is, mint &x)
  {
    ll a;
    is >> a;
    x = a;
    return is;
  }
  friend ostream &operator<<(ostream &os, const mint &x)
  {
    os << x.val();
    return os;
  }
};
template <int id>
internal::barrett32 dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
using mint = modint998244353;
// using mint = modint1000000007;
// using mint = static_modint<1000000000>;
// using mint = modint;

void init() {}

void main2()
{
  LL(N, M);
  PRINT(base_repr_str(M, N));
}

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


      // ----- generate cases -----
      ll N = 1 + rand() % 5;
      vl A(N);
      rep(i, N) A.at(i) = 1 + rand() % 10;
      // --------------------------

      // ------ check output ------
      #define INPUT A
      auto god = naive(INPUT);
      auto ans = solve(INPUT);
      if (god != ans)
      {
        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
  };

  #if defined FAST_IO and not defined LOCAL
  CERR("\n[FAST_IO]\n\n", "34");
  cin.tie(0);
  ios::sync_with_stdio(false);
  #endif
  cout << fixed << setprecision(20);

  test();
  init();

  #if defined AOJ_TESTCASE or (not defined NOT_AOJ and 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");
  uint T;
  cin >> T;
  while (T--)
  {
    dump("new testcase");
    main2();
  }
  #endif
}