#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

// https://github.com/miscalculation53/library/tree/wip/template/template_all.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_all_but_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_types.hpp

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

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

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

using vstr = vc<string>;

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (int i = neg; i < (int)s.size(); i++)
    res = 10 * res + s[i] - '0';
  if (neg)
    return -i128(res - 1) - 1;
  return i128(res);
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  u128 ux;
  if (x < 0)
    ux = u128(-(x + 1)) + 1, sign = "-";
  else
    ux = x;
  while (ux > 0)
  {
    res += '0' + ux % 10;
    ux /= 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;
}

#define cauto const auto
// https://github.com/miscalculation53/library/tree/wip/template/template_rep.hpp

#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__)
// https://github.com/miscalculation53/library/tree/wip/template/template_math.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/is_integral_ext.hpp

template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;

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

template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;
// https://github.com/miscalculation53/library/tree/wip/utils/default_infty.hpp

namespace default_infty_detail
{

  template <class T>
  inline constexpr bool unsupported = false;
}

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

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<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, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }

// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

#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

#define LMD(x,fx) ([&](const auto &x) { return fx; })
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_infty.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_value.hpp

template <class V>
auto SUM(const V &v)
{
  typename V::value_type s{};
  fec(vi : v) s += vi;
  return s;
}
template <class V>
auto MAX(const V &v) { return *max_element(ALL(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;
}

template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;

namespace internal
{
}

constexpr array<pll, 4> DRULgrid = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
constexpr array<pll, 4> DRULplane = {{{0, -1}, {1, 0}, {0, 1}, {-1, 0}}};
// https://github.com/miscalculation53/library/tree/wip/template/template_binsearch.hpp

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;

#define DEFAULT_COMP ranges::less

namespace internal
{
};

// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp

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

inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }

inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }

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); }

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; }
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; }
// https://github.com/miscalculation53/library/tree/wip/template/template_inout.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump_map.hpp

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) 
#define local(...) 
#define oj(...) __VA_ARGS__
#define local_oj(a,b) (b)

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;
}

namespace internal
{

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

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

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

}; 

#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__)

#define ENDL endl

template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p);

template <class... Ts>
ostream &operator<<(ostream &os, const tuple<Ts...> &t);
template <class T, size_t n>
ostream &operator<<(ostream &os, const array<T, n> &a);
template <class T>
ostream &operator<<(ostream &os, const vc<T> &v);

namespace internal
{

template <class... Ts>
void COUTP(const Ts &...a)
{
  if constexpr (sizeof...(Ts))
  {
    int i = 0;
    ((cout << (i++ ? " " : "") << a), ...);
  }
  cout << ENDL;
}

}; 

#define WRITE internal::COUTW
#define PRINT internal::COUTP

#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)

namespace internal
{

}; 

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = zip(tuple{__VA_ARGS__})

// https://github.com/miscalculation53/library/tree/wip/template/template_random.hpp

mt19937_64 mt;

bool randbool(double p)
{
  assert(0 <= p && p <= 1);
  return bernoulli_distribution(p)(mt);
}

namespace internal
{
}; 

// https://github.com/miscalculation53/library/tree/wip/math/modint/template_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_static.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/larger_int.hpp

namespace larger_int_detail
{
}

template <class T>
struct larger_int
{
private:
  static constexpr bool check();
  static_assert(check());

public:
  using type = T;
};

#define LARGER_INT(T,U) template <> struct larger_int<T> { using type = U; };

LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)

LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)

#undef LARGER_INT

template <class T>
struct Rational;

template <class T>
struct larger_int<Rational<T>>
{
  using type = Rational<typename larger_int<T>::type>;
};

template <class T>
using larger_int_t = typename larger_int<T>::type;
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_isprime.hpp

namespace internal
{

template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}

template <auto n>
constexpr bool isprime = isprime_constexpr(n);

};

namespace internal
{

template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = larger_int_t<value_type>;

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_barrett32.hpp

namespace internal
{

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));
  }
};

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_montgomery64.hpp

namespace internal
{

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); }
};

struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(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); }
};

template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{(1LL << 61) - 1};
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v);
};

};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp

namespace internal
{

  template <class Policy>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;

  private:
    V _v;

  public:

    modint_impl();

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v);

    M val() const;

    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 M safe_hash_key(const mint &x) { return x.val(); }
  
    friend std::istream &operator>>(std::istream &is, mint &x)
    {
      long long a;
      is >> a;
      x = a;
      return is;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint &x)
    {
      os << x.val();
      return os;
    }
  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;

template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;

// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp

template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
  
public:
  inline static vc<T> fac_, finv_, inv_;

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp

// https://github.com/miscalculation53/library/tree/wip/math/svp2d.hpp

namespace cpp_dump
{
  template <class T>
  struct rat_value
  {
    T p, q;

    friend ostream &operator<<(ostream &os, const rat_value &x)
    {
      os << x.p;
      if (x.q != 1)
        os << '/' << x.q;
      return os;
    }
  };

  template <class T>
  inline constexpr bool rat_is_map = false;
  template <class... Args>
  inline constexpr bool rat_is_map<std::map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::multimap<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_multimap<Args...>> = true;

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {

  };

  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}

using mint = modint998244353;
// using mint = modint1000000007;
// using mint = static_modint<1000000000>;
// using mint = modint;

void init()
{
  oj(mt.seed(random_device()()));
}

// https://github.com/miscalculation53/library/tree/wip/algo/subset_sum.hpp

// https://github.com/miscalculation53/library/tree/wip/ds/dynamic_bitset.hpp

struct DynamicBitset
{
  using Word = ull;
  static constexpr int word_bits = numeric_limits<Word>::digits;

private:
  int n;
  vc<Word> dat;

  static int word_size(int n) { return (n + word_bits - 1) / word_bits; }
  static Word low_mask(int k)
  {
    return k == word_bits ? ~Word(0) : (Word(1) << k) - 1;
  }
  Word last_mask() const
  {
    return n % word_bits == 0 ? ~Word(0) : low_mask(n % word_bits);
  }
  void trim()
  {
    if (!dat.empty())
      dat.back() &= last_mask();
  }

  Word get_word(int i) const
  {
    int w = i / word_bits, b = i % word_bits;
    Word x = dat[w] >> b;
    if (b && w + 1 < SZ(dat))
      x |= dat[w + 1] << (word_bits - b);
    return x;
  }

  template <int op, class F>
  DynamicBitset &apply_slice(int l, int r, const DynamicBitset &b, int bl, const F &f)
  {
    assert(0 <= l && l <= r && r <= n);
    assert(0 <= bl && bl + r - l <= b.n);
    if (l == r)
      return *this;

    if constexpr (op == 2)
    {
      if (this == &b && bl == 0 && l > 0)
      {
        int dw = l / word_bits, db = l % word_bits;
        int lw = l / word_bits, rw = (r - 1) / word_bits;
        repi(w, rw, lw - 1, -1)
        {
          Word x = dat[w - dw] << db;
          if (db && w > dw)
            x |= dat[w - dw - 1] >> (word_bits - db);
          int lo = max(l, w * word_bits), hi = min(r, (w + 1) * word_bits);
          Word mask = low_mask(hi - lo) << (lo % word_bits);
          Word added = x & ~dat[w] & mask;
          dat[w] |= x & mask;
          if constexpr (!is_same_v<F, nullptr_t>)
          {
            while (added)
            {
              f(w * word_bits + __builtin_ctzll(added));
              added &= added - 1;
            }
          }
        }
        return *this;
      }
    }

    auto apply_word = [&](int w)
    {
      int lo = max(l, w * word_bits), hi = min(r, (w + 1) * word_bits);
      int len = hi - lo, shift = lo % word_bits;
      Word mask = low_mask(len) << shift;
      Word x = (b.get_word(bl + lo - l) & low_mask(len)) << shift;
      Word old = dat[w], y;
      if constexpr (op == 0)
        y = x;
      else if constexpr (op == 1)
        y = old & x;
      else if constexpr (op == 2)
        y = old | x;
      else
        y = old ^ x;
      dat[w] = (old & ~mask) | (y & mask);
      if constexpr (!is_same_v<F, nullptr_t>)
      {
        Word added = dat[w] & ~old & mask;
        while (added)
        {
          f(w * word_bits + __builtin_ctzll(added));
          added &= added - 1;
        }
      }
    };

    int lw = l / word_bits, rw = (r - 1) / word_bits;
    if (this == &b && bl < l && l < bl + r - l)
      repi(w, rw, lw - 1, -1) apply_word(w);
    else
      repi(w, lw, rw + 1) apply_word(w);
    return *this;
  }

public:
  struct Reference
  {
    DynamicBitset *bs;
    int pos;

    operator bool() const { return bs->test(pos); }
    Reference &operator=(bool value)
    {
      bs->set(pos, value);
      return *this;
    }
    Reference &operator=(const Reference &ref) { return *this = bool(ref); }
    Reference &flip()
    {
      bs->flip(pos);
      return *this;
    }
  };

  DynamicBitset() : n(0) {}
  
  DynamicBitset(int n, bool value = false) : n(n), dat(n < 0 ? 0 : word_size(n), value ? ~Word(0) : 0)
  {
    assert(n >= 0);
    trim();
  }
  
  explicit DynamicBitset(const string &s, char zero = '0', char one = '1') : n(0)
  {
    assign(s, zero, one);
  }

  int size() const { return n; }
  
  bool empty() const { return n == 0; }
  
  int capacity() const { return int(dat.capacity()) * word_bits; }

  void reserve(int n)
  {
    assert(n >= 0);
    dat.reserve(word_size(n));
  }

  void resize(int n, bool value = false)
  {
    assert(n >= 0);
    int old = this->n;
    dat.resize(word_size(n));
    this->n = n;
    if (value && old < n)
      set_range(old, n);
    trim();
  }

  DynamicBitset &assign(const string &s, char zero = '0', char one = '1')
  {
    assert(zero != one);
    n = int(s.size());
    dat.assign(word_size(n), 0);
    repi(i, n)
    {
      char c = s[n - 1 - i];
      assert(c == zero || c == one);
      if (c == one)
        dat[i / word_bits] |= Word(1) << (i % word_bits);
    }
    return *this;
  }

  bool test(int i) const
  {
    assert(0 <= i && i < n);
    return dat[i / word_bits] >> (i % word_bits) & 1;
  }
  bool operator[](int i) const { return test(i); }
  Reference operator[](int i)
  {
    assert(0 <= i && i < n);
    return {this, i};
  }

  DynamicBitset &set(int i, bool value = true)
  {
    assert(0 <= i && i < n);
    Word mask = Word(1) << (i % word_bits);
    if (value)
      dat[i / word_bits] |= mask;
    else
      dat[i / word_bits] &= ~mask;
    return *this;
  }
  
  DynamicBitset &reset(int i) { return set(i, false); }
  
  DynamicBitset &flip(int i)
  {
    assert(0 <= i && i < n);
    dat[i / word_bits] ^= Word(1) << (i % word_bits);
    return *this;
  }
  
  DynamicBitset &set()
  {
    fill(ALL(dat), ~Word(0));
    trim();
    return *this;
  }
  
  DynamicBitset &reset()
  {
    fill(ALL(dat), 0);
    return *this;
  }
  
  DynamicBitset &flip()
  {
    fem(x : dat) x = ~x;
    trim();
    return *this;
  }

  DynamicBitset &set_range(int l, int r, bool value = true)
  {
    assert(0 <= l && l <= r && r <= n);
    if (!value)
      return reset_range(l, r);
    if (l == r)
      return *this;
    int lw = l / word_bits, rw = (r - 1) / word_bits;
    if (lw == rw)
      dat[lw] |= low_mask((r - 1) % word_bits + 1) & ~low_mask(l % word_bits);
    else
    {
      dat[lw] |= ~low_mask(l % word_bits);
      fill(dat.begin() + lw + 1, dat.begin() + rw, ~Word(0));
      dat[rw] |= low_mask((r - 1) % word_bits + 1);
    }
    trim();
    return *this;
  }

  DynamicBitset &reset_range(int l, int r)
  {
    assert(0 <= l && l <= r && r <= n);
    if (l == r)
      return *this;
    int lw = l / word_bits, rw = (r - 1) / word_bits;
    if (lw == rw)
      dat[lw] &= ~(low_mask((r - 1) % word_bits + 1) & ~low_mask(l % word_bits));
    else
    {
      dat[lw] &= low_mask(l % word_bits);
      fill(dat.begin() + lw + 1, dat.begin() + rw, 0);
      dat[rw] &= ~low_mask((r - 1) % word_bits + 1);
    }
    return *this;
  }

  DynamicBitset &flip_range(int l, int r)
  {
    assert(0 <= l && l <= r && r <= n);
    if (l == r)
      return *this;
    int lw = l / word_bits, rw = (r - 1) / word_bits;
    if (lw == rw)
      dat[lw] ^= low_mask((r - 1) % word_bits + 1) & ~low_mask(l % word_bits);
    else
    {
      dat[lw] ^= ~low_mask(l % word_bits);
      repi(w, lw + 1, rw) dat[w] = ~dat[w];
      dat[rw] ^= low_mask((r - 1) % word_bits + 1);
    }
    trim();
    return *this;
  }

  int count() const
  {
    int res = 0;
    fec(x : dat) res += __builtin_popcountll(x);
    return res;
  }

  int count(int l, int r) const
  {
    assert(0 <= l && l <= r && r <= n);
    if (l == r)
      return 0;
    int lw = l / word_bits, rw = (r - 1) / word_bits;
    if (lw == rw)
      return __builtin_popcountll(dat[lw] & low_mask((r - 1) % word_bits + 1) & ~low_mask(l % word_bits));
    int res = __builtin_popcountll(dat[lw] & ~low_mask(l % word_bits));
    repi(w, lw + 1, rw) res += __builtin_popcountll(dat[w]);
    return res + __builtin_popcountll(dat[rw] & low_mask((r - 1) % word_bits + 1));
  }

  bool any() const
  {
    fec(x : dat) if (x) return true;
    return false;
  }
  
  bool any(int l, int r) const { return count(l, r) != 0; }
  
  bool none() const { return !any(); }
  
  bool none(int l, int r) const { return !any(l, r); }
  
  bool all() const { return count() == n; }
  
  bool all(int l, int r) const { return count(l, r) == r - l; }

  int find_first(bool value = true) const { return find_next(-1, value); }
  
  int find_next(int i, bool value = true) const
  {
    assert(-1 <= i && i < n);
    i++;
    if (i == n)
      return n;
    int w = i / word_bits, b = i % word_bits;
    Word x = (value ? dat[w] : ~dat[w]) & (~Word(0) << b);
    while (true)
    {
      if (x)
      {
        int res = w * word_bits + __builtin_ctzll(x);
        return min(res, n);
      }
      if (++w == SZ(dat))
        return n;
      x = value ? dat[w] : ~dat[w];
    }
  }
  
  int find_last(bool value = true) const { return find_prev(n, value); }
  
  int find_prev(int i, bool value = true) const
  {
    assert(0 <= i && i <= n);
    if (i == 0)
      return -1;
    i--;
    int w = i / word_bits, b = i % word_bits;
    Word x = (value ? dat[w] : ~dat[w]) & low_mask(b + 1);
    while (true)
    {
      if (x)
        return w * word_bits + 63 - __builtin_clzll(x);
      if (w-- == 0)
        return -1;
      x = value ? dat[w] : ~dat[w];
    }
  }

  DynamicBitset &operator&=(const DynamicBitset &b)
  {
    assert(n == b.n);
    repi(i, SZ(dat)) dat[i] &= b.dat[i];
    return *this;
  }
  DynamicBitset &operator|=(const DynamicBitset &b)
  {
    assert(n == b.n);
    repi(i, SZ(dat)) dat[i] |= b.dat[i];
    return *this;
  }
  DynamicBitset &operator^=(const DynamicBitset &b)
  {
    assert(n == b.n);
    repi(i, SZ(dat)) dat[i] ^= b.dat[i];
    return *this;
  }
  friend DynamicBitset operator&(DynamicBitset a, const DynamicBitset &b) { return a &= b; }
  friend DynamicBitset operator|(DynamicBitset a, const DynamicBitset &b) { return a |= b; }
  friend DynamicBitset operator^(DynamicBitset a, const DynamicBitset &b) { return a ^= b; }
  DynamicBitset operator~() const { return DynamicBitset(*this).flip(); }

  DynamicBitset &operator<<=(int k)
  {
    assert(k >= 0);
    if (k >= n)
      return reset();
    if (k == 0)
      return *this;
    int dw = k / word_bits, db = k % word_bits;
    repi(i, SZ(dat) - 1, dw - 1, -1)
    {
      Word x = dat[i - dw] << db;
      if (db && i > dw)
        x |= dat[i - dw - 1] >> (word_bits - db);
      dat[i] = x;
    }
    fill(dat.begin(), dat.begin() + dw, 0);
    trim();
    return *this;
  }

  DynamicBitset &operator>>=(int k)
  {
    assert(k >= 0);
    if (k >= n)
      return reset();
    if (k == 0)
      return *this;
    int dw = k / word_bits, db = k % word_bits, m = SZ(dat);
    repi(i, m - dw)
    {
      Word x = dat[i + dw] >> db;
      if (db && i + dw + 1 < m)
        x |= dat[i + dw + 1] << (word_bits - db);
      dat[i] = x;
    }
    fill(dat.end() - dw, dat.end(), 0);
    trim();
    return *this;
  }
  DynamicBitset operator<<(int k) const { return DynamicBitset(*this) <<= k; }
  DynamicBitset operator>>(int k) const { return DynamicBitset(*this) >>= k; }

  DynamicBitset &assign_slice(int l, int r, const DynamicBitset &b, int bl)
  {
    return apply_slice<0>(l, r, b, bl, nullptr);
  }
  DynamicBitset &and_slice(int l, int r, const DynamicBitset &b, int bl)
  {
    return apply_slice<1>(l, r, b, bl, nullptr);
  }
  DynamicBitset &or_slice(int l, int r, const DynamicBitset &b, int bl)
  {
    return apply_slice<2>(l, r, b, bl, nullptr);
  }
  template <class F>
  DynamicBitset &or_slice(int l, int r, const DynamicBitset &b, int bl, const F &f)
  {
    return apply_slice<2>(l, r, b, bl, f);
  }
  [[gnu::always_inline]] DynamicBitset &xor_slice(int l, int r, const DynamicBitset &b, int bl)
  {
    if (l == bl && l % word_bits == 0 && r == n && n == b.n)
    {
      assert(0 <= l && l <= r);
      repi(w, l / word_bits, SZ(dat)) dat[w] ^= b.dat[w];
      return *this;
    }
    return apply_slice<3>(l, r, b, bl, nullptr);
  }

  void push_back(bool value)
  {
    int i = n;
    resize(n + 1);
    if (value)
      set(i);
  }
  
  void pop_back()
  {
    assert(n > 0);
    resize(n - 1);
  }

  string to_string(char zero = '0', char one = '1') const
  {
    assert(zero != one);
    string res(n, zero);
    repi(i, n) if (test(i)) res[n - 1 - i] = one;
    return res;
  }

  friend bool operator==(const DynamicBitset &a, const DynamicBitset &b)
  {
    return a.n == b.n && a.dat == b.dat;
  }
  friend bool operator!=(const DynamicBitset &a, const DynamicBitset &b) { return !(a == b); }
  friend bool operator<(const DynamicBitset &a, const DynamicBitset &b)
  {
    assert(a.n == b.n);
    repi(i, SZ(a.dat) - 1, -1, -1)
      if (a.dat[i] != b.dat[i])
        return a.dat[i] < b.dat[i];
    return false;
  }
  friend bool operator>(const DynamicBitset &a, const DynamicBitset &b) { return b < a; }
  friend bool operator<=(const DynamicBitset &a, const DynamicBitset &b) { return !(b < a); }
  friend bool operator>=(const DynamicBitset &a, const DynamicBitset &b) { return !(a < b); }
};

istream &operator>>(istream &is, DynamicBitset &a)
{
  string s;
  is >> s;
  if (is)
    a.assign(s);
  return is;
}
ostream &operator<<(ostream &os, const DynamicBitset &a)
{
  return os << a.to_string();
}

struct SubsetSum;

struct SubsetSumFromFrequency
{
private:
  friend struct SubsetSum;

  int s;
  vc<pair<int, int>> val_cnt;
  DynamicBitset dp;
  vc<int> time;
  vc<DynamicBitset> history;
  bool keep_history;

  void build(const vc<pair<int, int>> &freq, int smax)
  {
    s = smax;
    assert(s >= 0);
    val_cnt.clear();
    history.clear();

    ll sm = 0;
    for (auto [v, c] : freq)
    {
      assert(v > 0 && c >= 0);
      if (v > s)
        continue;
      chmin(c, s / v);
      sm = min<ll>(s, sm + ll(v) * c);
    }
    s = int(sm);

    dp.resize(s + 1);
    dp.reset();
    dp.set(0);
    for (auto [v, c] : freq)
    {
      if (v > s)
        continue;
      chmin(c, s / v);
      for (ll k = 1; c > 0; k *= 2)
      {
        int x = int(min<ll>(c, k));
        c -= x;
        val_cnt.eb(v, x);
      }
    }

    ll bitset_bytes = ll((s + DynamicBitset::word_bits) / DynamicBitset::word_bits) * ll(sizeof(DynamicBitset::Word));
    keep_history = ll(val_cnt.size()) * bitset_bytes < ll(s + 1) * ll(sizeof(int));
    if (keep_history)
      history.reserve(val_cnt.size());
    else
    {
      time.assign(s + 1, -1);
      time[0] = 0;
    }

    int hi = 0;
    repi(t, SZ(val_cnt))
    {
      auto [v, c] = val_cnt[t];
      int w = v * c;
      int r = min(s, hi + w) + 1;
      if (keep_history)
      {
        dp.or_slice(w, r, dp, 0);
        history.eb(dp);
      }
      else
        dp.or_slice(w, r, dp, 0, [&](int i) { time[i] = t + 1; });
      hi = r - 1;
      if (hi == s && (t & 63) == 63 && dp.all())
      {
        val_cnt.resize(t + 1);
        if (keep_history)
          history.resize(t + 1);
        break;
      }
    }
  }

public:
  SubsetSumFromFrequency() : s(0), dp(1), time(1, 0), keep_history(false) { dp.set(0); }

  bool exists(int x) const
  {
    return 0 <= x && x <= s && dp.test(x);
  }

  const DynamicBitset &reachable() const { return dp; }

  pair<bool, vc<pair<int, int>>> answer(int x) const
  {
    if (!exists(x))
      return {false, {}};

    vc<pair<int, int>> res;
    int y = x;
    repi(t, SZ(val_cnt) - 1, -1, -1)
    {
      auto [v, c] = val_cnt[t];
      int w = v * c;
      bool can = false;
      if (y >= w)
      {
        if (keep_history)
          can = t == 0 ? y == w : history[t - 1].test(y - w);
        else
          can = time[y - w] != -1 && time[y - w] <= t;
      }
      if (can)
      {
        y -= w;
        if (res.empty() || res.back().first != v)
          res.eb(v, 0);
        res.back().second += c;
      }
    }
    return {true, res};
  }
};

struct SubsetSum
{
private:
  int n, s, xmax;
  ll negsum;
  vc<bool> isneg;
  vc<pair<int, int>> elems;
  SubsetSumFromFrequency ss;

public:
  SubsetSum() : n(0), s(-1), xmax(-1), negsum(0) {}

  template <class T>
  SubsetSum(const vc<T> &a, int smax) : n(SZ<int>(a)), xmax(smax), negsum(0)
  {
    static_assert(is_integral_ext<T>);
    isneg.assign(n, false);
    repi(i, n) if (a[i] < 0)
    {
      isneg[i] = true;
      negsum += ll(a[i]);
    }
    ll shifted_s = ll(smax) - negsum;
    if (shifted_s < 0)
    {
      s = -1;
      return;
    }

    ll sm = 0;
    fec(a_i : a)
    {
      ll x = ll(a_i);
      if (x < 0) x = -x;
      if (x <= shifted_s)
        sm = min(shifted_s, sm + x);
    }
    s = int(sm);

    repi(i, n)
    {
      ll x = ll(a[i]);
      if (x < 0)
      {
        if (x < -ll(s))
          continue;
        x = -x;
      }
      if (x <= s)
      {
        int v = int(x);
        if (v)
          elems.eb(v, i);
      }
    }
    sort(elems.begin(), elems.end());
    vc<pair<int, int>> freq;
    for (auto [v, i] : elems)
    {
      if (freq.empty() || freq.back().first != v)
        freq.eb(v, 0);
      ++freq.back().second;
    }
    ss.build(freq, s);
    s = ss.reachable().size() - 1;
  }

  bool exists(int x) const
  {
    ll y = ll(x) - negsum;
    return x <= xmax && 0 <= y && y <= s && ss.exists(int(y));
  }

  const DynamicBitset &reachable() const
  {
    static const DynamicBitset empty;
    return s < 0 ? empty : ss.reachable();
  }

  ll min_sum() const { return negsum; }

  pair<bool, vc<bool>> answer(int x) const
  {
    ll y = ll(x) - negsum;
    if (x > xmax || !(0 <= y && y <= s))
      return {false, {}};
    auto [ok, cnt] = ss.answer(int(y));
    if (!ok)
      return {false, {}};

    vc<bool> res(n, false);
    int p = elems.size();
    for (auto [v, c] : cnt)
    {
      while (p && elems[p - 1].first > v) p--;
      int q = p;
      while (q && elems[q - 1].first == v) q--;
      assert(p - q >= c);
      repi(k, c) res[elems[q + k].second] = true;
      p = q;
    }
    repi(i, n) if (isneg[i]) res[i] = !res[i];
    return {true, res};
  }
};

void main2()
{
  LL(N, M);
  VEC(ll, N, U);
  VEC(ll, M, T);

  SubsetSum ss(T, SUM(T) + 1);
  auto res = ss.reachable();
  local(rep(i, SZ(res)) if (res.test(i)) dump(i););
  DynamicBitset res2(SUM(T) + MAX(U) + 1);
  fe(u : U) res2.or_slice(u, min(SZ(res2), u + SZ(res)), res, 0);
  PRINT(res2.count());
  local(rep(i, SZ(res2)) if (res2.test(i)) dump(i););
}

void test()
{

}

// https://github.com/miscalculation53/library/tree/wip/template/template_main.hpp

template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      
    };
  
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}