#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>
namespace {
using namespace std;

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>>;

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;
}

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

#define overload4(_1,_2,_3,_4,name,...) name
#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>;

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 divround(U a, V b) { return divfloor<T>(2 * T(a) + T(b), 2 * T(b)); }
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); }

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

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

template <class T = ll, class A, class K>
constexpr T iroot(A a, K k)
{
  assert(a >= 0 && k >= 1);
  if (a <= 1 || k == 1)
    return a;
  if (k == 2)
  {
    const T aa = T(a);
    T x = T(sqrtl((long double)a));
    while (x > aa / x)
      x--;
    while (x < numeric_limits<T>::max())
    {
      const T y = x + 1;
      if (y > aa / y)
        break;
      x = y;
    }
    return x;
  }

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

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

template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base)
{
  assert(val >= 0);
  assert(base >= 2);
  if (val == 0)
    return {0};
  vc<T> a;
  while (val > 0)
  {
    a.emplace_back(val % base);
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}

template <class T = ll, class U, class V>
vc<T> base_repr(U val, V base, int n)
{
  assert(val >= 0);
  assert(base >= 2);
  assert(n >= 0);
  vc<T> a(n);
  repi(i, n)
  {
    a[i] = val % base;
    val /= base;
  }
  reverse(a.begin(), a.end());
  return a;
}
template <const bool use_upper = true, class U>
string base_repr_str(U val, int base, int n)
{
  assert(val >= 0);
  assert(2 <= base && base <= 36);
  assert(n >= 0);
  auto a = base_repr(val, base, n);
  string s = "";
  for (cauto &ai : a)
    s += (ai < 10 ? '0' + ai : (use_upper ? 'A' : 'a') + (ai - 10));
  return s;
}
// 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

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

template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}

template <class T = ll>
T ctol(const char &c, const string &s)
{
  repi(i, SZ<int>(s)) if (s[i] == c) return i;
  return -1;
}
template <class T = ll>
vc<T> stov(const string &s, const string &t)
{
  return gen_vec(SZ<int>(s), [&](int i) -> T
                 { return ctol(s[i], t); });
}
template <class T>
string vtos(const vc<T> &v, const string &t)
{
  string res = "";
  fe(vi : v) res += t[vi];
  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;
}

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

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

template <class 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 T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs)
{
  return permuted(permuted(p, q), rs...);
}

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

template <class V, class Equal = equal_to<>>
void unique(V &v, Equal equal = {}) { v.erase(std::unique(ALL(v), equal), v.end()); }

template <class V, class Compare = less<>, class Equal = equal_to<>>
void sortunique(V &v, Compare comp = {}, Equal equal = {})
{
  sort(ALL(v), comp);
  unique(v, equal);
}

template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  if (n == 0)
    return;
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}

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;
}

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;

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
  template <class I, class = decltype(declval<S>() * declval<I>())>
  static constexpr S pow(const S &a, I k) { return a * k; }
};
template <class T, const T infty = INF>
struct MonoidMin
{
  using S = T;
  static constexpr S op(S a, S b) { return min(a, b); }
  static constexpr S e() { return infty; }
  template <class I>
  static constexpr S pow(const S &a, I k) { return k == 0 ? e() : a; }
};
template <class T, const T infty = INF>
struct MonoidMax
{
  using S = T;
  static constexpr S op(S a, S b) { return max(a, b); }
  static constexpr S e() { return -infty; }
  template <class I>
  static constexpr S pow(const S &a, I k) { return k == 0 ? e() : a; }
};

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

template <class M, class I>
typename M::S pow_monoid(typename M::S a, I k)
{
  if constexpr (is_signed_ext<I>)
    assert(k >= 0);
  if constexpr (internal::HasMonoidPow<M, I>::value)
    return M::pow(a, k);
  else
  {
    typename M::S c = M::e();
    for (; k; k >>= 1)
    {
      if (k & 1)
        c = M::op(c, a);
      a = M::op(a, a);
    }
    return c;
  }
}

template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0)
{
  const int n = v.size();
  vc<typename M::S> res(n + 1);
  res[0] = M::e();
  repi(i, n) res[i + 1] = M::op(res[i], v[i]);
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}

template <class M>
vc<typename M::S> cumr(const vc<typename M::S> &v, int right_index = 0)
{ return reversed(cuml<M>(reversed(v), right_index)); }

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;

template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T LB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::lower_bound(v, val, comp, proj) - v.begin(); }

template <class T = ll, class V, class Value, class Comp = ranges::less, class Proj = identity>
inline T UB(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
{ return ranges::upper_bound(v, val, comp, proj) - v.begin(); }
#define DEFAULT_COMP ranges::less

template <class T = ll, class V, class Value, class Comp = DEFAULT_COMP, class Proj = identity>
inline auto lt_cnt(const V &v, const Value &val, Comp comp = {}, Proj proj = {})
-> enable_if_t<is_random_access_iterator_v<typename V::iterator>, T>
{ return LB<T>(v, val, comp, proj); }

template <class V, class Value>
inline auto lt_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.lower_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}

template <class V, class Value>
inline auto leq_max(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{
  auto it = v.upper_bound(val);
  return it == v.begin() ? v.end() : prev(it);
}

template <class V, class Value>
inline auto gt_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.upper_bound(val); }

template <class V, class Value>
inline auto geq_min(const V &v, const Value &val)
-> enable_if_t<!is_random_access_iterator_v<typename V::iterator>, typename V::const_iterator>
{ return v.lower_bound(val); }

namespace internal
{
template <class T>
bool binsearch_adjacent(T a, T b)
{
  if (a < b)
    return a + 1 == b;
  if (b < a)
    return b + 1 == a;
  return false;
}
};

template <class T = ll, class Judge, class InitOk, class InitNg>
pair<T, T> binsearch(const Judge &judge, InitOk init_ok, InitNg init_ng, bool check_ok = true, bool check_ng = true)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  while (!internal::binsearch_adjacent(ok, ng))
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}

// 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); }

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

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

#define dump(...) 
#define local(...) 
#define oj(...) __VA_ARGS__

template <class T, class Sequence>
vc<T> content(queue<T, Sequence> que)
{
  vc<T> res;
  while (!que.empty())
  {
    res.eb(que.front());
    que.pop();
  }
  return res;
}
template <class T, class Sequence, class Compare>
vc<T> content(priority_queue<T, Sequence, Compare> pque)
{
  vc<T> res;
  while (!pque.empty())
  {
    res.eb(pque.top());
    pque.pop();
  }
  return res;
}
template <class T>
auto content(const T &obj) { return obj.content(); }

namespace fastio {
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;

static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];

uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}

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

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

template <typename T>
void rd1_real(T &x) {
  string s;
  rd1(s);
  if constexpr (is_same_v<T, long double>)
    x = stold(s);
  else
    x = stod(s);
}

template <typename T>
void rd1_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  using U = unsigned_integer_t<T>;
  U val = 0;
  while ('0' <= c) { val = val * 10 + (c & 15), c = ibuf[pil++]; }
  pil--;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus)
    {
      const U min_abs = U(numeric_limits<T>::max()) + 1;
      assert(val <= min_abs);
      x = val == min_abs ? numeric_limits<T>::lowest() : -T(val);
    }
    else
    {
      assert(val <= U(numeric_limits<T>::max()));
      x = T(val);
    }
  }
  else
    x = T(val);
}

void rd1(ll &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }

template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  apply([](auto &...x) { (rd1(x), ...); }, tpl);
}

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

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

void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(const string s) {
  for (char c: s) wt1(c);
}

template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  using U = unsigned_integer_t<T>;
  U ux;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (x < 0)
      obuf[por++] = '-', ux = U(0) - U(x);
    else
      ux = U(x);
  }
  else
    ux = x;
  int outi;
  for (outi = 96; ux >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[ux % 10000], 4);
    ux /= 10000;
  }
  if (ux >= 1000) {
    memcpy(obuf + por, pre.num[ux], 4);
    por += 4;
  } else if (ux >= 100) {
    memcpy(obuf + por, pre.num[ux] + 1, 3);
    por += 3;
  } else if (ux >= 10) {
    int q = (ux * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (ux - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = ux | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

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

template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }

template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  if constexpr (sizeof...(T))
  {
    int i = 0;
    apply([&](const auto &...x)
          { ((i++ ? wt1(' ') : void(), wt1(x)), ...); }, tpl);
  }
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}

template <class... T>
void write(T &&...x) {
  (wt1(std::forward<T>(x)), ...);
}

template <class... T>
void print(T &&...x) {
  if constexpr (sizeof...(T))
  {
    int i = 0;
    ((i++ ? wt1(' ') : void(), wt1(std::forward<T>(x))), ...);
  }
  wt1('\n');
}

} 

struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;

namespace internal
{

template <class... Ts>
void READnodump(Ts &...a) { fastio::read(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 LL(...) IN(ll, __VA_ARGS__)

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

#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)

#define PRINT fastio::print

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

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

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

namespace internal
{

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

}; 

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

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

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

namespace internal
{

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

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

};

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

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

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

mt19937_64 mt;

template <class T = ll, class U1, class U2>
T randint(U1 l, U2 r)
{
  assert(T(l) <= T(r));
  return uniform_int_distribution<T>(T(l), T(r))(mt);
}

namespace internal
{
template <bool does_sort, class V, class T>
void random_sample_range(V &res, T l, T r)
{
  int k = res.size();
  T n = r - l;
  if (k <= 256)
  {
    repi(i, k)
    {
      T j = n - T(k) + T(i), x = randint<T>(0, j);
      if (find(res.begin(), res.begin() + i, x) != res.begin() + i)
        x = j;
      res[i] = x;
    }
  }
  else
  {
    unordered_set<T> used;
    used.reserve(2 * size_t(k));
    repi(i, k)
    {
      T j = n - T(k) + T(i), x = randint<T>(0, j);
      if (!used.insert(x).second)
        x = j, used.insert(x);
      res[i] = x;
    }
  }
  for (T &x : res) x += l;
  if constexpr (does_sort)
    sort(res.begin(), res.end());
  else
    shuffle(res.begin(), res.end(), mt);
}
}; 

// 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

template <class T>
struct larger_int
{
  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>
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 <class T>
constexpr bool isprime_constexpr(T n)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_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;

  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };

  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}

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 mod_type mod() { return M; }
  static constexpr value_type umod() { return M; }

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v) { return v; }
  static constexpr value_type mul(value_type a, value_type b)
  {
    return (value_type)((calc_type(a) * b) % M);
  }
};

};
// 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 void set_mod(mod_type m) { reducer = barrett32(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v) { return v; }
  static value_type mul(value_type a, value_type b) { return reducer.mul(a, b); }
};

};
// 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 void set_mod(mod_type m) { reducer = montgomery64odd(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

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 void set_mod(mod_type m) { reducer = montgomery64(m); }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

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

template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    
    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};
}

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:
    static constexpr M mod() { return Policy::mod(); }

    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m)) { return T::set_mod(m); }

    static mint raw(V v)
    {
      mint x;
      x._v = v;
      return x;
    }

    modint_impl() : _v(0) {}

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(Policy::umod());
        if (x < 0)
          x += Policy::umod();
        rem = x;
      }
      else
        rem = V(v % Policy::umod());
      _v = Policy::init(rem);
    };

    M val() const { return Policy::val(_v); }

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

    mint &operator++()
    {
      _v++;
      if (_v == Policy::umod())
        _v = 0;
      return *this;
    }
    mint &operator--()
    {
      if (_v == 0)
        _v = Policy::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 { return *this; }
    mint operator-() const { return mint() - *this; }

    template <class T>
    mint pow(T 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 constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(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 void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

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;

template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};

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

template <class mint>
struct PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;

public:
  PowerTable() {}
  PowerTable(const mint &base) : mod(mint::mod()), base(base), pw(1, 1) {}

  void reserve(int n)
  {
    if (mod != mint::mod())
    {
      mod = mint::mod();
      pw = {1};
    }
    int i = pw.size();
    if (n < i)
      return;
    pw.resize(n + 1);
    for (; i <= n; i++)
      pw[i] = pw[i - 1] * base;
  }

  mint pow(int n)
  {
    reserve(n);
    return pw[n];
  }
};
// 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_;
  static void reserve(int n)
  {
    if constexpr (is_dynamic_modint_v<T>)
    {
      if (mod != T::mod())
      {
        mod = T::mod();
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
      }
    }
    else
    {
      if (fac_.empty())
        fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    if (n < SZ(fac_))
      return;
    chmin(n, T::mod() - 1);
    int si = fac_.size();
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    repi(i, si, n + 1)
    {
      fac_[i] = fac_[i - 1] * T::raw(i);
      inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  static T inv(int n)
  {
    n %= T::mod();
    if (n < 0)
      n += T::mod();
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }

  static T C(int n, int k)
  {
    if (n < k)
      return 0;
    if (n < 0 || k < 0)
      return 0;
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n] * finv_[k] * finv_[n - k];
  }
};
// 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

template <class T = ll, class U = larger_int_t<T>>
pair<T, T> svp2d(const pair<T, T> &a, const pair<T, T> &b)
{
  assert((a != pair<T, T>{0, 0} && b != pair<T, T>{0, 0}));
  auto [a1, a2] = a;
  auto [b1, b2] = b;
  if ((U)a1 * a1 + (U)a2 * a2 < (U)b1 * b1 + (U)b2 * b2)
    swap(a1, b1), swap(a2, b2);
  while ((U)a1 * a1 + (U)a2 * a2 > (U)b1 * b1 + (U)b2 * b2)
  {
    swap(a1, b1), swap(a2, b2);
    T k = divround<U>((U)a1 * b1 + (U)a2 * b2, (U)a1 * a1 + (U)a2 * a2);
    b1 -= k * a1, b2 -= k * a2;
    if (b1 == 0 && b2 == 0)
      return {a1, a2};
  }
  return {a1, a2};
}

template <class mint>
pair<decltype(mint(0).val()), decltype(mint(0).val())>
mint_to_rat(const mint &x)
{
  auto [p, q] = svp2d({x.val(), 1}, {mint::mod(), 0});
  if (q < 0)
    p = -p, q = -q;
  return {p, q};
}

namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;

        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };

  template <typename T>
    requires(!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)
  constexpr auto operator|(T &&t, const rat_closure &c)
  {
    return c(std::forward<T>(t));
  }

}

using mint = modint998244353;

using bi = Binomial<mint>;

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;

    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);
    assert(s.size() <= size_t(numeric_limits<int>::max()));
    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;
  }

  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(int l, int r) const { return !any(l, r); }
  
  bool all(int l, int r) const { return count(l, r) == r - l; }

  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_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 &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);
  }

  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); }
};

struct SubsetSumFromFrequency
{
private:
  int s;
  vc<pair<int, int>> val_cnt;
  vc<int> time;

public:
  SubsetSumFromFrequency() : s(0), time(1, 0) {}

  template <class T>
  SubsetSumFromFrequency(const vc<T> &freq, int smax) : s(smax)
  {
    static_assert(is_integral_ext<T>);
    assert(s >= 0);
    time.assign(s + 1, -1);
    time[0] = 0;
    DynamicBitset dp(s + 1);
    dp.set(0);
    int vmax = min(s, SZ<int>(freq) - 1);
    repi(v, 1, vmax + 1)
    {
      assert(freq[v] >= 0);
      int c = freq[v] < T(s / v) ? int(freq[v]) : s / v;
      for (ll k = 1; c > 0; k *= 2)
      {
        int x = int(min<ll>(c, k));
        c -= x;
        int t = val_cnt.size();
        val_cnt.eb(v, x);
        int w = v * x;
        dp.or_slice(w, s + 1, dp, 0, [&](int i) { time[i] = t + 1; });
      }
    }
  }

  bool exists(int x) const
  {
    return 0 <= x && x <= s && time[x] != -1;
  }

  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;
      if (y >= w && time[y - w] != -1 && time[y - w] <= t)
      {
        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;
  i128 negsum;
  vc<bool> isneg;
  vvc<int> ids;
  SubsetSumFromFrequency ss;

public:
  SubsetSum() : n(0), s(0), 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 += i128(a[i]);
    }
    i128 shifted_s = i128(smax) - negsum;
    assert(shifted_s <= numeric_limits<int>::max());
    s = shifted_s < 0 ? 0 : int(shifted_s);

    ids.resize(s + 1);
    vc<int> freq(s + 1);
    repi(i, n)
    {
      i128 x = i128(a[i]);
      if (x < 0)
      {
        if (x < -i128(s))
          continue;
        x = -x;
      }
      if (x <= s)
      {
        int v = int(x);
        ids[v].eb(i);
        freq[v]++;
      }
    }
    ss = SubsetSumFromFrequency(freq, s);
  }

  bool exists(int x) const
  {
    i128 y = i128(x) - negsum;
    return x <= xmax && 0 <= y && y <= s && ss.exists(int(y));
  }

  pair<bool, vc<bool>> answer(int x) const
  {
    i128 y = i128(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);
    for (auto [v, c] : cnt)
    {
      assert(c <= SZ(ids[v]));
      repi(k, c) res[ids[v][k]] = true;
    }
    repi(i, n) if (isneg[i]) res[i] = !res[i];
    return {true, res};
  }
};

void main2()
{
  LL(N, S);
  VEC(ll, N, A);
  SubsetSum ss(A, S);
  rep(s, S, -1, -1) if (ss.exists(s)) PRINTRETURN(s);
}

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";
      
    };
  
    CERR("\n[FAST_IO]\n\n", "32");
    
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[MULTI_TESTCASE]\n\n", "33");
    local(while (true))
    {
      dump("T");
      IN(uint, T);
      while (T--)
      {
        dump("new testcase");
        main2();
      }
    }
    
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}