// BEGIN: main.cpp
#line 1 "main.cpp"
// BEGIN: my_template.hpp
#line 1 "my_template.hpp"
#if defined(USE_PCH)
#include <my_template_compiled.hpp>
#else
#if defined(__GNUC__)
#include <bits/allocator.h>
#pragma GCC optimize("Ofast,unroll-loops")
// 環境によってはコンパイル成功かつ実行時エラー
#pragma GCC target("avx2,popcnt")
#endif
#include <bits/stdc++.h>
#include <cassert>

using namespace std;

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

template <class>
constexpr bool dependent_false = false;

template <class T>
constexpr T infty = [] {
  static_assert(dependent_false<T>, "infty<T> is not defined");
  return T{};
}();
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
template <>
constexpr double infty<double> = infty<i128>;
template <>
constexpr long double infty<long double> = infty<i128>;

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 pq_max = priority_queue<T>;
template <class T>
using pq_min = 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 = ll(a) - 1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = ll(a) - 1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = ll(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 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

// require y > 0
template <typename T>
T floor(T x, T y) {
  return x / y - (x % y < 0);
}

// require y > 0
template <typename T>
T ceil(T x, T y) {
  return (x / y) + (x % y > 0);
}

// require y > 0
template <typename T>
T bmod(T x, T y) {
  T r = x % y;
  return (r < 0 ? r + y : r);
}

// require y > 0
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = x / y, r = x % y;
  if (r < 0) --q, r += y;
  return {q, r};
}

constexpr auto TEN = [] {
  array<u64, 20> A{};
  A[0] = 1;
  for (int i = 1; i < 20; ++i) A[i] = 10 * A[i - 1];
  return A;
}();

template <typename T, typename U>
T SUM(const U &A) {
  return std::accumulate(A.begin(), A.end(), T{});
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
template <class C, class T>
inline long long LB(const C &c, const T &x) {
  return lower_bound(c.begin(), c.end(), x) - c.begin();
}
template <class C, class T>
inline long long UB(const C &c, const T &x) {
  return upper_bound(c.begin(), c.end(), x) - c.begin();
}
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end())

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <class T, class Container, class Compare>
T POP(priority_queue<T, Container, Compare> &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>
i128 binary_search(F check, i128 ok, i128 ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (1) {
    i128 x = (ok + ng) / 2;
    if (x == ok || x == ng) break;
    (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) {
  T c = max<T>(a, b);
  bool changed = (c != a);
  a = c;
  return changed;
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  T c = min<T>(a, b);
  bool changed = (c != a);
  a = c;
  return changed;
}

// ? 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>
vc<T> cumsum(const vc<U> &A, int off = 1) {
  int N = A.size();
  vc<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>
vc<int> argsort(const vc<T> &A) {
  vc<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
      [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

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

template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &...others) {
  first.reserve(first.size() + (others.size() + ... + 0));
  (first.insert(first.end(), others.begin(), others.end()), ...);
}

// i128
template <class T, enable_if_t<is_same_v<T, i128>, int> = 0>
constexpr i128 abs(T x) {
  return x < 0 ? -x : x;
}

constexpr i128 gcd(i128 a, i128 b) {
  while (b != 0) {
    i128 c = a % b;
    a = b, b = c;
  }
  return abs(a);
}
#endif
// END: my_template.hpp
#line 2 "main.cpp"
// BEGIN: other/io.hpp
#line 1 "other/io.hpp"
#define FASTIO

// 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;
bool input_eof = false;

template <class T>
constexpr bool is_signed_integer_v = is_signed_v<T> || is_same_v<T, i128>;

template <class T>
struct unsigned_integer {
  using type = make_unsigned_t<T>;
};
template <>
struct unsigned_integer<i128> {
  using type = u128;
};
template <>
struct unsigned_integer<u128> {
  using type = u128;
};
template <class T>
using unsigned_integer_t = typename unsigned_integer<T>::type;

[[noreturn]] inline void input_error(const char *message) {
  fputs(message, stderr);
  fputc('\n', stderr);
  exit(EXIT_FAILURE);
}

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() {
  uint32_t n = pir - pil;
  memmove(ibuf, ibuf + pil, n);
  pil = 0;
  pir = n;
  if (input_eof) return;

  pir += fread(ibuf + pir, 1, SZ - pir, stdin);
  if (ferror(stdin)) input_error("fastio: input error");
  if (feof(stdin)) {
    input_eof = true;
    // Allows the last token to end exactly at EOF without a trailing
    // whitespace.
    if (pir < SZ) ibuf[pir++] = '\n';
  }
}

inline char get_char() {
  if (pil == pir) {
    load();
    if (pil == pir) input_error("fastio: unexpected EOF");
  }
  return ibuf[pil++];
}

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

void rd(char &c) {
  do c = get_char();
  while (isspace(static_cast<unsigned char>(c)));
}

void rd(string &x) {
  x.clear();
  char c;
  do c = get_char();
  while (isspace(static_cast<unsigned char>(c)));
  do {
    x += c;
    c = get_char();
  } while (!isspace(static_cast<unsigned char>(c)));
}

template <typename T>
void rd_real(T &x) {
  string s;
  rd(s);
  x = stod(s);
}

template <typename T>
void rd_integer_slow(T &x) {
  char c;
  do c = get_char();
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed_integer_v<T>) {
    if (c == '-') {
      minus = 1, c = get_char();
    }
  }
  x = 0;
  assert('0' <= c && c <= '9');
  while ('0' <= c && c <= '9') {
    x = x * 10 + (c & 15), c = get_char();
  }
  assert(isspace(static_cast<unsigned char>(c)));
  if constexpr (is_signed_integer_v<T>) {
    if (minus) x = -x;
  }
}

template <typename T>
void rd_integer(T &x) {
  if (pil + 100 > pir) {
    load();
    if (pil + 100 > pir) {
      rd_integer_slow(x);
      return;
    }
  }
  char c;
  do c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed_integer_v<T>) {
    if (c == '-') {
      minus = 1, c = ibuf[pil++];
    }
  }
  x = 0;
  assert('0' <= c && c <= '9');
  while ('0' <= c && c <= '9') {
    x = x * 10 + (c & 15), c = ibuf[pil++];
  }
  assert(isspace(static_cast<unsigned char>(c)));
  if constexpr (is_signed_integer_v<T>) {
    if (minus) x = -x;
  }
}

template <class T>
enable_if_t<is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>> rd(
    T &x) {
  rd_integer(x);
}

template <class T>
enable_if_t<is_floating_point_v<T> || is_same_v<T, f128>> rd(T &x) {
  rd_real(x);
}

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

template <class T, size_t N>
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);
}

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

inline void wt_range(const char *s, size_t n) {
  size_t i = 0;
  while (i < n) {
    if (por == SZ) flush();
    size_t chunk = min(n - i, (size_t)(SZ - por));
    memcpy(obuf + por, s + i, chunk);
    por += chunk;
    i += chunk;
  }
}

void wt(const char c) {
  if (por == SZ) flush();
  obuf[por++] = c;
}
void wt(const char *s) { wt_range(s, strlen(s)); }
void wt(const string &s) { wt_range(s.data(), s.size()); }

template <typename T>
void wt_integer(T x) {
  if (por > SZ - 100) flush();
  using U = unsigned_integer_t<T>;
  U y = static_cast<U>(x);
  if constexpr (is_signed_integer_v<T>) {
    if (x < 0) {
      obuf[por++] = '-';
      y = U(0) - y;
    }
  }
  int outi;
  for (outi = 96; y >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[y % 10000], 4);
    y /= 10000;
  }
  if (y >= 1000) {
    memcpy(obuf + por, pre.num[y], 4);
    por += 4;
  } else if (y >= 100) {
    memcpy(obuf + por, pre.num[y] + 1, 3);
    por += 3;
  } else if (y >= 10) {
    int q = (y * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (y - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = y | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
inline void wt_real(T x) {
  static char buf[1000];
  int n = std::snprintf(buf, sizeof(buf), "%.15f", (double)x);
  wt_range(buf, (size_t)n);
}

template <class T>
enable_if_t<is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>> wt(
    T x) {
  wt_integer(x);
}

template <class T>
enable_if_t<is_floating_point_v<T> || is_same_v<T, f128>> wt(T x) {
  wt_real(x);
}

inline void wt(bool b) { wt(static_cast<char>('0' + (b ? 1 : 0))); }

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 < tuple_size<T>::value) {
    if constexpr (N > 0) wt(' ');
    wt(get<N>(t));
    wt_tuple<N + 1>(t);
  }
}
template <class... T>
void wt(const 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(forward<Head>(head));
  ((wt(' '), wt(forward<Tail>(tail))), ...);
  wt('\n');
}

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

#if defined(LOCAL)
#define HDR "[DEBUG:", __func__, __LINE__, "]"
#define SHOW(...)                                                         \
  SHOW_IMPL(__VA_ARGS__, SHOW8, SHOW7, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, \
            SHOW1)                                                        \
  (__VA_ARGS__)
#define SHOW_IMPL(_1, _2, _3, _4, _5, _6, _7, _8, NAME, ...) NAME
#define SHOW1(x) print(HDR, #x, "=", (x)), flush()
#define SHOW2(x, y) print(HDR, #x, "=", (x), #y, "=", (y)), flush()
#define SHOW3(x, y, z) \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z)), flush()
#define SHOW4(x, y, z, w) \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush()
#define SHOW5(x, y, z, w, v)                                                  \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", \
        (v)),                                                                 \
      flush()
#define SHOW6(x, y, z, w, v, u)                                               \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", \
        (v), #u, "=", (u)),                                                   \
      flush()
#define SHOW7(x, y, z, w, v, u, t)                                            \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", \
        (v), #u, "=", (u), #t, "=", (t)),                                     \
      flush()
#define SHOW8(x, y, z, w, v, u, t, s)                                         \
  print(HDR, #x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", \
        (v), #u, "=", (u), #t, "=", (t), #s, "=", (s)),                       \
      flush()
#else
#define SHOW(...)
#endif

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

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

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
void YA(bool t = 1) { print(t ? "YA" : "TIDAK"); }
void TIDAK(bool t = 1) { YA(!t); }
void Alice(bool t = 1) { print(t ? "Alice" : "Bob"); }
void Bob(bool t = 1) { Alice(!t); }// END: other/io.hpp
#line 3 "main.cpp"

// BEGIN: ds/bit_array.hpp
#line 1 "ds/bit_array.hpp"
#line 2 "ds/bit_array.hpp"
#line 3 "ds/bit_array.hpp"

// BEGIN: other/bit.hpp
#line 1 "other/bit.hpp"

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_sgn(int x) { return (__builtin_parity(unsigned(x)) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
// (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 kth_bit(int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return T(1) << k;
}
template <typename T>
bool has_kth_bit(T x, int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return x >> k & 1;
}

template <typename UINT>
struct all_bit {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_bit(UINT s) : s(s) {}
  struct iter {
    UINT s;
    int operator*() const { return lowbit(s); }
    void operator++() { s &= s - 1; }
    bool operator!=(nullptr_t) const { return s; }
  };
  iter begin() const { return {s}; }
  nullptr_t end() const { return nullptr; }
};

template <typename UINT>
struct all_subset {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_subset(UINT s) : s(s) {}
  struct iter {
    UINT s, t;
    bool done = false;
    UINT operator*() const { return t; }
    void operator++() {
      done = (t == 0);
      t = (t - 1) & s;
    }
    bool operator!=(nullptr_t) const { return !done; }
  };
  iter begin() const { return {s, s}; }
  nullptr_t end() const { return nullptr; }
};

constexpr u64 full_mask(int n) {
  assert(0 <= n && n <= 64);
  return n == 64 ? -1ULL : (1ULL << n) - 1;
}

u64 bit_reverse(u64 x) {
  x = ((x & 0x5555555555555555ULL) << 1) | ((x >> 1) & 0x5555555555555555ULL);
  x = ((x & 0x3333333333333333ULL) << 2) | ((x >> 2) & 0x3333333333333333ULL);
  x = ((x & 0x0f0f0f0f0f0f0f0fULL) << 4) | ((x >> 4) & 0x0f0f0f0f0f0f0f0fULL);
  x = ((x & 0x00ff00ff00ff00ffULL) << 8) | ((x >> 8) & 0x00ff00ff00ff00ffULL);
  x = ((x & 0x0000ffff0000ffffULL) << 16) | ((x >> 16) & 0x0000ffff0000ffffULL);
  x = (x << 32) | (x >> 32);
  return x;
}// END: other/bit.hpp
#line 5 "ds/bit_array.hpp"

/*
01 列を管理する．内部では 64 bit ごとにまとめて保持する．主な用途として，
- 可変長でスライス操作が可能な bitset
- F_2 上の多項式
などを想定している．
*/
struct Bit_Array {
  using T = Bit_Array;
  int N;
  vc<u64> dat;

  // x で埋める
  Bit_Array(int N = 0, int x = 0) : N(N) {
    assert(N >= 0);
    assert(x == 0 || x == 1);
    u64 v = (x == 0 ? 0 : -1);
    dat.assign((N + 63) >> 6, v);
    resize(N);
  }

  int size() const { return N; }

  void resize(int n) {
    assert(n >= 0);
    dat.resize((n + 63) >> 6);
    int r = n & 63;
    if (r) dat.back() &= full_mask(r);
    N = n;
  }

  void fill0() { fill(all(dat), u64(0)); }
  void fill1() {
    fill(all(dat), u64(-1));
    resize(N);
  }

  void push_back(bool b) {
    resize(N + 1);
    (*this)[N - 1] = b;
  }

  static T from_string(const string &S) {
    int N = len(S);
    T ANS(N);
    FOR(i, N) ANS[i] = (S[i] == '1');
    return ANS;
  }

  class Proxy {
   public:
    Proxy(vc<u64> &d, int i) : dat(d), index(i) {}
    operator bool() const { return (dat[index >> 6] >> (index & 63)) & 1; }
    Proxy &operator=(bool value) {
      u64 mask = u64(1) << (index & 63);
      if (value)
        dat[index >> 6] |= mask;
      else
        dat[index >> 6] &= ~mask;
      return *this;
    }
    Proxy &operator=(const Proxy &p) { return *this = bool(p); }

    // bit operations
    Proxy &operator^=(bool x) {
      if (x) dat[index >> 6] ^= u64(1) << (index & 63);
      return *this;
    }
    Proxy &operator|=(bool x) {
      if (x) dat[index >> 6] |= u64(1) << (index & 63);
      return *this;
    }
    Proxy &operator&=(bool x) {
      if (!x) dat[index >> 6] &= ~(u64(1) << (index & 63));
      return *this;
    }
    // finite field F_2
    Proxy &operator+=(bool x) { return *this ^= x; }
    Proxy &operator-=(bool x) { return *this ^= x; }
    Proxy &operator*=(bool x) { return *this &= x; }
    Proxy &operator/=(bool x) {
      assert(x);
      return *this;
    }
    bool inverse() const {
      assert(bool(*this));
      return true;
    }
    void flip() { *this ^= true; }

   private:
    vc<u64> &dat;
    int index;
  };

  Proxy operator[](int i) {
    assert(0 <= i && i < N);
    return Proxy(dat, i);
  }
  bool operator[](int i) const {
    assert(0 <= i && i < N);
    return (dat[i >> 6] >> (i & 63)) & 1;
  }

  bool operator==(const T &p) const {
    if (N != p.N) return false;
    FOR(i, len(dat)) if (dat[i] != p.dat[i]) return false;
    return true;
  }

  T &operator&=(const T &p) {
    assert(N == p.N);
    FOR(i, len(dat)) dat[i] &= p.dat[i];
    return *this;
  }
  T &operator|=(const T &p) {
    assert(N == p.N);
    FOR(i, len(dat)) dat[i] |= p.dat[i];
    return *this;
  }
  T &operator^=(const T &p) {
    assert(N == p.N);
    FOR(i, len(dat)) dat[i] ^= p.dat[i];
    return *this;
  }
  T operator&(const T &p) const { return T(*this) &= p; }
  T operator|(const T &p) const { return T(*this) |= p; }
  T operator^(const T &p) const { return T(*this) ^= p; }
  T operator~() const {
    T p = (*this);
    p.flip_range(0, N);
    return p;
  }

  void set_minus_inplace(const T &other) {
    assert(N == other.N);
    FOR(i, len(dat)) dat[i] = dat[i] & (~other.dat[i]);
  }

  T set_minus(T other) const {
    assert(N == other.N);
    FOR(i, len(dat)) other.dat[i] = dat[i] & ~other.dat[i];
    return other;
  }

  int count() const {
    int ans = 0;
    for (u64 val : dat) ans += popcnt(val);
    return ans;
  }

  // size of set intersection, not modulo 2
  int dot(const T &p) const {
    assert(N == p.N);
    int ans = 0;
    FOR(i, len(dat)) ans += popcnt(dat[i] & p.dat[i]);
    return ans;
  }

  // minimum j >= i with (*this)[j] = 1, or N if none
  int next(int i) const {
    chmax(i, 0);
    if (i >= N) return N;
    int k = i >> 6;
    {
      u64 x = dat[k];
      int s = i & 63;
      x = (x >> s) << s;
      if (x) return (k << 6) | lowbit(x);
    }
    FOR(idx, k + 1, len(dat)) {
      if (dat[idx] == 0) continue;
      return (idx << 6) | lowbit(dat[idx]);
    }
    return N;
  }

  // maximum j <= i with (*this)[j] = 1, or -1 if none
  int prev(int i) const {
    chmin(i, N - 1);
    if (i <= -1) return -1;
    int k = i >> 6;
    if ((i & 63) < 63) {
      u64 x = dat[k];
      x &= (u64(1) << ((i & 63) + 1)) - 1;
      if (x) return (k << 6) | topbit(x);
      --k;
    }
    FOR_R(idx, k + 1) {
      if (dat[idx] == 0) continue;
      return (idx << 6) | topbit(dat[idx]);
    }
    return -1;
  }

  Bit_Array slice(int L, int R) const {
    assert(0 <= L && L <= R && R <= N);
    Bit_Array p(R - L);
    int rm = (R - L) & 63;
    FOR(rm) {
      p[R - L - 1] = bool((*this)[R - 1]);
      --R;
    }
    int n = (R - L) >> 6;
    int hi = L & 63;
    int lo = 64 - hi;
    int s = L >> 6;
    if (hi == 0) {
      FOR(i, n) { p.dat[i] = dat[s + i]; }
    } else {
      FOR(i, n) { p.dat[i] = (dat[s + i] >> hi) | (dat[s + i + 1] << lo); }
    }
    return p;
  }

  int count_range(int L, int R) const {
    assert(0 <= L && L <= R && R <= N);
    int cnt = 0;
    while ((L < R) && (L & 63)) cnt += (*this)[L++];
    while ((L < R) && (R & 63)) cnt += (*this)[--R];
    int l = L >> 6, r = R >> 6;
    FOR(i, l, r) cnt += popcnt(dat[i]);
    return cnt;
  }

  // [L,R) に p を代入
  void assign_to_range(int L, int R, const Bit_Array &p) {
    assert(0 <= L && L <= R && R <= N);
    assert(p.N == R - L);
    int a = 0, b = p.N;
    while (L < R && (L & 63)) {
      (*this)[L++] = bool(p[a++]);
    }
    while (L < R && (R & 63)) {
      (*this)[--R] = bool(p[--b]);
    }
    // p[a:b] を [L:R] に
    int l = L >> 6, r = R >> 6;
    int s = a >> 6;
    int n = r - l;
    if (!(a & 63)) {
      FOR(i, n) dat[l + i] = p.dat[s + i];
    } else {
      int hi = a & 63;
      int lo = 64 - hi;
      FOR(i, n) dat[l + i] = (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
    }
  }

  // [L,R) に p を xor
  void xor_to_range(int L, int R, const Bit_Array &p) {
    assert(0 <= L && L <= R && R <= N);
    assert(p.N == R - L);
    int a = 0, b = p.N;
    while (L < R && (L & 63)) {
      dat[L >> 6] ^= u64(p[a]) << (L & 63);
      ++a, ++L;
    }
    while (L < R && (R & 63)) {
      --b, --R;
      dat[R >> 6] ^= u64(p[b]) << (R & 63);
    }
    // p[a:b] を [L:R] に
    int l = L >> 6, r = R >> 6;
    int s = a >> 6;
    int n = r - l;
    if (!(a & 63)) {
      FOR(i, n) dat[l + i] ^= p.dat[s + i];
    } else {
      int hi = a & 63;
      int lo = 64 - hi;
      FOR(i, n) dat[l + i] ^= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
    }
  }

  // [L,R) に p を and
  void and_to_range(int L, int R, const Bit_Array &p) {
    assert(0 <= L && L <= R && R <= N);
    assert(p.N == R - L);
    int a = 0, b = p.N;
    while (L < R && (L & 63)) {
      if (!p[a]) (*this)[L] = 0;
      a++, L++;
    }
    while (L < R && (R & 63)) {
      --b, --R;
      if (!p[b]) (*this)[R] = 0;
    }
    // p[a:b] を [L:R] に
    int l = L >> 6, r = R >> 6;
    int s = a >> 6;
    int n = r - l;
    if (!(a & 63)) {
      FOR(i, n) dat[l + i] &= p.dat[s + i];
    } else {
      int hi = a & 63;
      int lo = 64 - hi;
      FOR(i, n) dat[l + i] &= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
    }
  }

  // [L,R) に p を or
  void or_to_range(int L, int R, const Bit_Array &p) {
    assert(0 <= L && L <= R && R <= N);
    assert(p.N == R - L);
    int a = 0, b = p.N;
    while (L < R && (L & 63)) {
      dat[L >> 6] |= u64(p[a]) << (L & 63);
      ++a, ++L;
    }
    while (L < R && (R & 63)) {
      --b, --R;
      dat[R >> 6] |= u64(p[b]) << (R & 63);
    }
    // p[a:b] を [L:R] に
    int l = L >> 6, r = R >> 6;
    int s = a >> 6;
    int n = r - l;
    if (!(a & 63)) {
      FOR(i, n) dat[l + i] |= p.dat[s + i];
    } else {
      int hi = a & 63;
      int lo = 64 - hi;
      FOR(i, n) dat[l + i] |= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo);
    }
  }

  // p は [i:N) にしかないとして p を xor する
  // 行列基本変形などで利用可能
  void xor_suffix(int i, const Bit_Array &p) {
    assert(N == p.N && 0 <= i && i < N);
    FOR(k, i / 64, len(dat)) { dat[k] ^= p.dat[k]; }
  }

  // [L,R) を 1 に変更
  void set_range(int L, int R) {
    assert(0 <= L && L <= R && R <= N);
    while (L < R && (L & 63)) set(L++);
    while (L < R && (R & 63)) set(--R);
    FOR(i, L >> 6, R >> 6) dat[i] = u64(-1);
  }

  // [L,R) を 0 に変更
  void reset_range(int L, int R) {
    assert(0 <= L && L <= R && R <= N);
    while (L < R && (L & 63)) reset(L++);
    while (L < R && (R & 63)) reset(--R);
    FOR(i, L >> 6, R >> 6) dat[i] = u64(0);
  }

  // [L,R) を flip
  void flip_range(int L, int R) {
    assert(0 <= L && L <= R && R <= N);
    while (L < R && (L & 63)) flip(L++);
    while (L < R && (R & 63)) flip(--R);
    FOR(i, L >> 6, R >> 6) dat[i] ^= u64(-1);
  }

  // bitset に仕様を合わせる
  void set(int i) { (*this)[i] = 1; }
  void reset(int i) { (*this)[i] = 0; }
  void flip(int i) { (*this)[i].flip(); }
  void set() { set_range(0, N); }
  void reset() { reset_range(0, N); }
  void flip() { flip_range(0, N); }
  bool any() const {
    FOR(i, len(dat)) {
      if (dat[i]) return true;
    }
    return false;
  }

  bool has_intersection(const T &other) const {
    assert(N == other.N);
    FOR(i, len(dat)) if (dat[i] & other.dat[i]) return true;
    return false;
  }

  template <typename F>
  void enumerate_intersection(const T &other, F f) const {
    assert(N == other.N);
    bool end = false;
    FOR(i, len(dat)) {
      u64 x = dat[i] & other.dat[i];
      while (x) {
        int k = lowbit(x);
        f(64 * i + k, end);
        if (end) return;
        x &= x - 1;
      }
    }
  }

  bool ALL() const {
    int r = N & 63;
    if (r != 0 && dat.back() != full_mask(r)) return 0;
    for (int i = 0; i < N / 64; ++i)
      if (dat[i] != u64(-1)) return false;
    return true;
  }

  Bit_Array reversed() const {
    int M = ceil(N, 64) * 64;
    Bit_Array a = *this;
    a.resize(M);
    reverse(all(a.dat));
    for (u64 &x : a.dat) x = bit_reverse(x);
    return a.slice(M - N, M);
  }

  // bs[i]==true であるような i 全体
  vc<int> collect_idx() const {
    vc<int> I;
    FOR(i, N) if ((*this)[i]) I.eb(i);
    return I;
  }

  bool is_subset(const T &other) const {
    assert(other.N == N);
    FOR(i, len(dat)) {
      u64 a = dat[i], b = other.dat[i];
      if ((a & b) != a) return false;
    }
    return true;
  }

  int _Find_first() const { return next(0); }
  int _Find_next(int p) const { return next(p + 1); }

  template <typename F>
  void enumerate(int L, int R, F f) const {
    assert(0 <= L && L <= R && R <= N);
    if (L == R) return;
    int p = ((*this)[L] ? L : _Find_next(L));
    while (p < R) {
      f(p);
      p = _Find_next(p);
    }
  }

  inline static string TO_STR[256];
  string to_string() const {
    if (TO_STR[0].empty()) precompute();
    string S;
    for (u64 x : dat) {
      FOR(i, 8) S += TO_STR[(x >> (8 * i) & 255)];
    }
    S.resize(N);
    return S;
  }

  static void precompute() {
    FOR(s, 256) {
      string x;
      FOR(i, 8) x += '0' + (s >> i & 1);
      TO_STR[s] = x;
    }
  }

  void prefix_xor_sum() {
    int carry = 0;
    for (u64 &a : dat) {
      a ^= carry;
      carry = __builtin_parityll(a);
      a ^= a << (1 << 0);
      a ^= a << (1 << 1);
      a ^= a << (1 << 2);
      a ^= a << (1 << 3);
      a ^= a << (1 << 4);
      a ^= a << (1 << 5);
    }
    resize(N);
  }
};// END: ds/bit_array.hpp
#line 5 "main.cpp"

using BS = Bit_Array;

void solve() {
  LL(N, S);
  BS dp(S + 1);
  dp[0] = 1;

  FOR(N) {
    INT(a);
    if (a > S) continue;
    BS x = dp.slice(0, S - a + 1);
    dp.or_to_range(a, S + 1, x);
  }
  int ANS = dp.prev(S);
  print(ANS);
}

signed main() {
  INT(T);
  FOR(T) solve();
}
// END: main.cpp