#line 1 "library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) \
  for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sum = 0;
  for (auto &&a: A) sum += a;
  return sum;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) \
  sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  assert(!que.empty());
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  assert(!que.empty());
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x));
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}
#endif
#line 1 "library/other/io.hpp"
// based on yosupo's fastio
#include <unistd.h>

namespace fastio {
#define FASTIO
// クラスが read(), print() を持っているかを判定するメタ関数
struct has_write_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.write(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_write : public decltype(has_write_impl::check<T>(std::declval<T>())) {
};

struct has_read_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.read(), std::true_type{});

  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_read : public decltype(has_read_impl::check<T>(std::declval<T>())) {};

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <typename T,
            typename enable_if<has_read<T>::value>::type * = nullptr>
  inline bool read_single(T &x) {
    x.read();
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <size_t N = 0, typename T>
  void read_single_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
      auto &x = std::get<N>(t);
      read_single(x);
      read_single_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool read_single(tuple<T...> &tpl) {
    read_single_tuple(tpl);
    return true;
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <typename T,
            typename enable_if<has_write<T>::value>::type * = nullptr>
  inline void write(T x) {
    x.write();
  }
  template <class T>
  void write(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <size_t N = 0, typename T>
  void write_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
      if constexpr (N > 0) { write(' '); }
      const auto x = std::get<N>(t);
      write(x);
      write_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool write(tuple<T...> tpl) {
    write_tuple(tpl);
    return true;
  }
  template <class T, size_t S>
  void write(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if (val < 0) {
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if (negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);
void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}
} // namespace fastio
using fastio::print;
using fastio::flush;
using fastio::read;

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

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

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

#line 2 "library/mod/barrett.hpp"

// https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp
struct Barrett {
  u32 m;
  u64 im;
  explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {}
  u32 umod() const { return m; }
  u32 modulo(u64 z) {
    if (m == 1) return 0;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z - y + (z < y ? m : 0));
  }
  u64 floor(u64 z) {
    if (m == 1) return z;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z < y ? x - 1 : x);
  }
  pair<u64, u32> divmod(u64 z) {
    if (m == 1) return {z, 0};
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    if (z < y) return {x - 1, z - y + m};
    return {x, z - y};
  }
  u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); }
};
#line 3 "library/mod/mod_pow.hpp"

// int
ll mod_pow(ll a, ll n, int mod) {
  assert(n >= 0);
  a %= mod;
  if (a < 0) a += mod;
  Barrett bt(mod);
  ll p = a, v = bt.modulo(1);
  while (n) {
    if (n & 1) v = bt.mul(v, p);
    p = bt.mul(p, p);
    n >>= 1;
  }
  return v;
}

ll mod_pow_long(ll a, ll n, ll mod) {
  assert(n >= 0);
  a %= mod;
  if (a < 0) a += mod;
  ll p = a, v = 1 % mod;
  while (n) {
    if (n & 1) v = i128(v) * p % mod;
    p = i128(p) * p % mod;
    n >>= 1;
  }
  return v;
}
#line 2 "library/mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n);
  if (n >= mod) return 0;
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if (dense) return C_dense<mint>(n, k);
  if (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d] (1-x) ^ {-n} の計算
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "library/mod/modint.hpp"

template <int mod>
struct modint {
  static_assert(mod < (1 << 30));
  int val;
  constexpr modint(const ll val = 0) noexcept
      : val(val >= 0 ? val % mod : (mod - (-val) % mod) % mod) {}
  bool operator<(const modint &other) const {
    return val < other.val;
  } // To use std::map
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= mod) val -= mod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += mod - p.val) >= mod) val -= mod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = (int)(1LL * val * p.val % mod);
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint(-val); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint(u);
  }
  modint pow(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
#ifdef FASTIO
  void write() { fastio::printer.write(val); }
  void read() { fastio::scanner.read(val); }
#endif
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 998244353) return {23, 31};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 2 "library/mod/mod_inv.hpp"
// long でも大丈夫
ll mod_inv(ll val, ll mod) {
  val %= mod;
  if (val < 0) val += mod;
  ll a = val, b = mod, u = 1, v = 0, t;
  while (b > 0) {
    t = a / b;
    swap(a -= t * b, b), swap(u -= t * v, v);
  }
  if (u < 0) u += mod;
  return u;
}
#line 1 "library/poly/convolution_naive.hpp"
template <class T>
vector<T> convolution_naive(const vector<T>& a, const vector<T>& b) {
  int n = int(a.size()), m = int(b.size());
  vector<T> ans(n + m - 1);
  if (n < m) {
    FOR(j, m) FOR(i, n) ans[i + j] += a[i] * b[j];
  } else {
    FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
  }
  return ans;
}
#line 2 "library/poly/ntt.hpp"

template <class mint>
void ntt(vector<mint>& a, bool inverse) {
  assert(mint::can_ntt());
  const int rank2 = mint::ntt_info().fi;
  const int mod = mint::get_mod();
  static array<mint, 30> root, iroot;
  static array<mint, 30> rate2, irate2;
  static array<mint, 30> rate3, irate3;

  static bool prepared = 0;
  if (!prepared) {
    prepared = 1;
    root[rank2] = mint::ntt_info().se;
    iroot[rank2] = mint(1) / root[rank2];
    FOR_R(i, rank2) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    mint prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 2; i++) {
      rate2[i] = root[i + 2] * prod;
      irate2[i] = iroot[i + 2] * iprod;
      prod *= iroot[i + 2];
      iprod *= root[i + 2];
    }
    prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 3; i++) {
      rate3[i] = root[i + 3] * prod;
      irate3[i] = iroot[i + 3] * iprod;
      prod *= iroot[i + 3];
      iprod *= root[i + 3];
    }
  }

  int n = int(a.size());
  int h = topbit(n);
  assert(n == 1 << h);
  if (!inverse) {
    int len = 0;
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        FOR(s, 1 << len) {
          int offset = s << (h - len);
          FOR(i, p) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          rot *= rate2[topbit(~s & -~s)];
        }
        len++;
      } else {
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            u64 mod2 = u64(mod) * mod;
            u64 a0 = a[i + offset].val;
            u64 a1 = u64(a[i + offset + p].val) * rot.val;
            u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val;
            u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val;
            u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val;
            u64 na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          rot *= rate3[topbit(~s & -~s)];
        }
        len += 2;
      }
    }
  } else {
    mint coef = mint(1) / mint(len(a));
    FOR(i, len(a)) a[i] *= coef;
    int len = h;
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        FOR(s, 1 << (len - 1)) {
          int offset = s << (h - len + 1);
          FOR(i, p) {
            u64 l = a[i + offset].val;
            u64 r = a[i + offset + p].val;
            a[i + offset] = l + r;
            a[i + offset + p] = (mod + l - r) * irot.val;
          }
          irot *= irate2[topbit(~s & -~s)];
        }
        len--;
      } else {
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        FOR(s, (1 << (len - 2))) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            u64 a0 = a[i + offset + 0 * p].val;
            u64 a1 = a[i + offset + 1 * p].val;
            u64 a2 = a[i + offset + 2 * p].val;
            u64 a3 = a[i + offset + 3 * p].val;
            u64 x = (mod + a2 - a3) * iimag.val % mod;
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val;
            a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val;
            a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val;
          }
          irot *= irate3[topbit(~s & -~s)];
        }
        len -= 2;
      }
    }
  }
}
#line 1 "library/poly/fft.hpp"
namespace CFFT {
using real = double;

struct C {
  real x, y;

  C() : x(0), y(0) {}

  C(real x, real y) : x(x), y(y) {}
  inline C operator+(const C& c) const { return C(x + c.x, y + c.y); }
  inline C operator-(const C& c) const { return C(x - c.x, y - c.y); }
  inline C operator*(const C& c) const {
    return C(x * c.x - y * c.y, x * c.y + y * c.x);
  }

  inline C conj() const { return C(x, -y); }
};

const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};

void ensure_base(int nbase) {
  if (nbase <= base) return;
  rev.resize(1 << nbase);
  rts.resize(1 << nbase);
  for (int i = 0; i < (1 << nbase); i++) {
    rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
  }
  while (base < nbase) {
    real angle = PI * 2.0 / (1 << (base + 1));
    for (int i = 1 << (base - 1); i < (1 << base); i++) {
      rts[i << 1] = rts[i];
      real angle_i = angle * (2 * i + 1 - (1 << base));
      rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
    }
    ++base;
  }
}

void fft(vector<C>& a, int n) {
  assert((n & (n - 1)) == 0);
  int zeros = __builtin_ctz(n);
  ensure_base(zeros);
  int shift = base - zeros;
  for (int i = 0; i < n; i++) {
    if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); }
  }
  for (int k = 1; k < n; k <<= 1) {
    for (int i = 0; i < n; i += 2 * k) {
      for (int j = 0; j < k; j++) {
        C z = a[i + j + k] * rts[j + k];
        a[i + j + k] = a[i + j] - z;
        a[i + j] = a[i + j] + z;
      }
    }
  }
}
} // namespace CFFT
#line 7 "library/poly/convolution.hpp"

template <class mint>
vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
  if (a.empty() || b.empty()) return {};
  int n = int(a.size()), m = int(b.size());
  int sz = 1;
  while (sz < n + m - 1) sz *= 2;

  // sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。
  if ((n + m - 3) <= sz / 2) {
    auto a_last = a.back(), b_last = b.back();
    a.pop_back(), b.pop_back();
    auto c = convolution(a, b);
    c.resize(n + m - 1);
    c[n + m - 2] = a_last * b_last;
    FOR(i, len(a)) c[i + len(b)] += a[i] * b_last;
    FOR(i, len(b)) c[i + len(a)] += b[i] * a_last;
    return c;
  }

  a.resize(sz), b.resize(sz);
  bool same = a == b;
  ntt(a, 0);
  if (same) {
    b = a;
  } else {
    ntt(b, 0);
  }
  FOR(i, sz) a[i] *= b[i];
  ntt(a, 1);
  a.resize(n + m - 1);
  return a;
}

template <typename mint>
vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  static const long long nttprimes[] = {754974721, 167772161, 469762049};
  using mint0 = modint<754974721>;
  using mint1 = modint<167772161>;
  using mint2 = modint<469762049>;
  vc<mint0> a0(n), b0(m);
  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
  FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
  auto c0 = convolution_ntt<mint0>(a0, b0);
  auto c1 = convolution_ntt<mint1>(a1, b1);
  auto c2 = convolution_ntt<mint2>(a2, b2);
  static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
  static const long long m0_inv_m1 = mint1(nttprimes[0]).inverse().val;
  static const long long m01_inv_m2 = mint2(m01).inverse().val;
  const int mod = mint::get_mod();
  auto garner = [&](mint0 x0, mint1 x1, mint2 x2) -> mint {
    int r0 = x0.val, r1 = x1.val, r2 = x2.val;
    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * mint2(m01_inv_m2);
    return mint(r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val);
  };
  vc<mint> c(len(c0));
  FOR(i, len(c)) c[i] = garner(c0[i], c1[i], c2[i]);
  return c;
}

template <typename R>
vc<double> convolution_fft(const vc<R>& a, const vc<R>& b) {
  using C = CFFT::C;
  int need = (int)a.size() + (int)b.size() - 1;
  int nbase = 1;
  while ((1 << nbase) < need) nbase++;
  CFFT::ensure_base(nbase);
  int sz = 1 << nbase;
  vector<C> fa(sz);
  for (int i = 0; i < sz; i++) {
    int x = (i < (int)a.size() ? a[i] : 0);
    int y = (i < (int)b.size() ? b[i] : 0);
    fa[i] = C(x, y);
  }
  CFFT::fft(fa, sz);
  C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
  for (int i = 0; i <= (sz >> 1); i++) {
    int j = (sz - i) & (sz - 1);
    C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
    fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
    fa[i] = z;
  }
  for (int i = 0; i < (sz >> 1); i++) {
    C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
    C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * CFFT::rts[(sz >> 1) + i];
    fa[i] = A0 + A1 * s;
  }
  CFFT::fft(fa, sz >> 1);
  vector<double> ret(need);
  for (int i = 0; i < need; i++) {
    ret[i] = (i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
  }
  return ret;
}

vector<ll> convolution(const vector<ll>& a, const vector<ll>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  if (min(n, m) <= 60) return convolution_naive(a, b);
  ll abs_sum_a = 0, abs_sum_b = 0;
  ll LIM = 1e15;
  FOR(i, n) abs_sum_a = min(LIM, abs_sum_a + abs(a[i]));
  FOR(i, m) abs_sum_b = min(LIM, abs_sum_b + abs(b[i]));
  if (i128(abs_sum_a) * abs_sum_b < 1e15) {
    vc<double> c = convolution_fft<ll>(a, b);
    vc<ll> res(len(c));
    FOR(i, len(c)) res[i] = ll(floor(c[i] + .5));
    return res;
  }

  static constexpr unsigned long long MOD1 = 754974721; // 2^24
  static constexpr unsigned long long MOD2 = 167772161; // 2^25
  static constexpr unsigned long long MOD3 = 469762049; // 2^26
  static constexpr unsigned long long M2M3 = MOD2 * MOD3;
  static constexpr unsigned long long M1M3 = MOD1 * MOD3;
  static constexpr unsigned long long M1M2 = MOD1 * MOD2;
  static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

  static const unsigned long long i1 = mod_inv(MOD2 * MOD3, MOD1);
  static const unsigned long long i2 = mod_inv(MOD1 * MOD3, MOD2);
  static const unsigned long long i3 = mod_inv(MOD1 * MOD2, MOD3);

  using mint1 = modint<MOD1>;
  using mint2 = modint<MOD2>;
  using mint3 = modint<MOD3>;

  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  vc<mint3> a3(n), b3(m);
  FOR(i, n) a1[i] = a[i], a2[i] = a[i], a3[i] = a[i];
  FOR(i, m) b1[i] = b[i], b2[i] = b[i], b3[i] = b[i];

  auto c1 = convolution_ntt<mint1>(a1, b1);
  auto c2 = convolution_ntt<mint2>(a2, b2);
  auto c3 = convolution_ntt<mint3>(a3, b3);

  vc<ll> c(n + m - 1);
  FOR(i, n + m - 1) {
    u64 x = 0;
    x += (c1[i].val * i1) % MOD1 * M2M3;
    x += (c2[i].val * i2) % MOD2 * M1M3;
    x += (c3[i].val * i3) % MOD3 * M1M2;
    ll diff = c1[i].val - ((long long)(x) % (long long)(MOD1));
    if (diff < 0) diff += MOD1;
    static constexpr unsigned long long offset[5]
        = {0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
    x -= offset[diff % 5];
    c[i] = x;
  }
  return c;
}

template <typename mint>
vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  if (mint::can_ntt()) {
    if (min(n, m) <= 50) return convolution_naive(a, b);
    return convolution_ntt(a, b);
  }
  if (min(n, m) <= 200) return convolution_naive(a, b);
  return convolution_garner(a, b);
}
#line 2 "library/nt/primetable.hpp"

template <typename T = long long>
vc<T> primetable(int LIM) {
  ++LIM;
  const int S = 32768;
  static int done = 2;
  static vc<T> primes = {2}, sieve(S + 1);

  if (done < LIM) {
    done = LIM;

    primes = {2}, sieve.assign(S + 1, 0);
    const int R = LIM / 2;
    primes.reserve(int(LIM / log(LIM) * 1.1));
    vc<pair<int, int>> cp;
    for (int i = 3; i <= S; i += 2) {
      if (!sieve[i]) {
        cp.eb(i, i * i / 2);
        for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1;
      }
    }
    for (int L = 1; L <= R; L += S) {
      array<bool, S> block{};
      for (auto& [p, idx]: cp)
        for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1;
      FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1);
    }
  }
  int k = LB(primes, LIM + 1);
  return {primes.begin(), primes.begin() + k};
}
#line 8 "main.cpp"

#line 2 "library/random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 10 "main.cpp"

ll calc_F(ll p) {
  ll X = ((p - 1) % 4 == 0 ? p - 1 : p + 1);
  // x^2 = 0, x^2 = 1
  ll x0 = 0, x1 = 1;
  x0 = (mod_pow_long(2, (p - 1) / 2, p) == 1 ? 2 : 0);
  x1 = 2;
  // x=0,1,-1
  X -= x0 + x1 + x1;
  // y=0,1,-1
  X -= x0 + x1 + x1;
  // 両方
  X += 4;

  // 4 で割る
  assert(X % 4 == 0);
  return X / 4;
}

vc<int> leq_convolution(vc<int> F, vc<int> G) {
  ll N = len(F);

  vc<int> Fc = cumsum<int>(F);

  using mint = modint998;
  vc<mint> f, g;
  for (auto&& x: F) f.eb(mint(x));
  for (auto&& x: G) g.eb(mint(x));

  // i<=j で畳む
  vc<mint> h(N + N);

  // i=j
  FOR(i, N) h[i + i] += f[i] * g[i];

  auto dfs = [&](auto& dfs, int L, int R) -> void {
    if (R == L + 1) return;
    int M = (L + R) / 2;
    dfs(dfs, L, M);
    dfs(dfs, M, R);
    if (Fc[M] - Fc[L] == 0) return;
    if (M + R <= N) return;
    // [L,M) x [M,R)
    vc<mint> A = {f.begin() + L, f.begin() + M};
    vc<mint> B = {g.begin() + M, g.begin() + R};
    while (len(A) && A.back() == mint(0)) POP(A);
    while (len(B) && B.back() == mint(0)) POP(B);
    A = convolution(A, B);
    FOR(i, len(A)) h[L + M + i] += A[i];
  };
  dfs(dfs, 0, N);

  vc<int> H(N + N);
  FOR(i, len(H)) H[i] = h[i].val;
  return H;
}

// [<0, >0, >0]
ll calc_NG(ll p, ll t) {
  assert(2 * t < p);

  // legendre
  vc<int> dp(2 * t + 1, 1);
  dp[0] = 0;
  for (auto&& x: primetable(2 * t)) {
    if (x > 2 * t) break;
    ll pow = mod_pow_long(x, (p - 1) / 2, p);
    assert(pow == 1 || pow == p - 1);
    pow = (pow == 1 ? 1 : -1);

    ll pp = x;
    while (pp <= 2 * t) {
      int idx = pp;
      while (idx <= 2 * t) dp[idx] *= pow, idx += pp;
      pp *= x;
    }
  }

  // [0, 2t - 1]
  vc<bool> right(2 * t);
  FOR(i, 1, 2 * t) {
    right[i] = (dp[i] == 1);
    // assert(right[i] == (mod_pow_long(i, (p - 1) / 2, p) == 1));
  }

  // [-0, -(t - 1)]
  vc<bool> left(t);
  FOR(i, 1, t) {
    if (p % 4 == 1) {
      left[i] = right[i];
    } else {
      left[i] = !right[i];
    }
    // assert(left[i] == (mod_pow_long(p - i, (p - 1) / 2, p) == 1));
  }

  // 添字偶奇, true/false -> 数える累積和
  int N = 2 * t;
  vvv(int, dat, 2, 2, N);
  FOR(parity, 2) {
    FOR(ok, 2) {
      FOR(i, N) {
        if ((i - parity) % 2 == 0 && 1 * right[i] == ok) {
          dat[parity][ok][i] = 1;
        }
      }
    }
  }
  FOR(parity, 2) FOR(ok, 2) dat[parity][ok] = cumsum<int>(dat[parity][ok]);

  ll aa_ = 0, ab_ = 0;
  /*
  FOR(i, 1, t) {
    FOR(j, 1, 2 * t) {
      // -i, j, 2j+i
      if (j + i <= t) {
        if (left[i] == right[j]) aa_++;
        if (left[i] != right[j]) ab_++;
      }
    }
  }
  */
  FOR(i, 1, t) {
    ll lo = 1, hi = 2 * t - 1;
    // j + i <= t
    chmin(hi, t - i);
    if (lo > hi) continue;
    ++hi;
    FOR(pa, 2) {
      if (!left[i]) aa_ += dat[pa][0][hi] - dat[pa][0][lo];
      if (!left[i]) ab_ += dat[pa][1][hi] - dat[pa][1][lo];
      if (left[i]) aa_ += dat[pa][1][hi] - dat[pa][1][lo];
      if (left[i]) ab_ += dat[pa][0][hi] - dat[pa][0][lo];
    }
  }

  ll a_a = 0, a_b = 0;
  /*
  FOR(i, 1, t) {
    FOR(k, i + 1, 2 * t) {
      if (k + i > 2 * t) continue;
      if ((k - i) % 2 == 0) {
        if (left[i] == right[k]) a_a++;
        if (left[i] != right[k]) a_b++;
      }
    }
  }
  */
  FOR(i, 1, t) {
    int pa = i % 2;
    ll lo = i + 1, hi = 2 * t - 1;
    chmin(hi, 2 * t - i);
    if (lo > hi) continue;
    ++hi;
    if (!left[i]) a_a += dat[pa][0][hi] - dat[pa][0][lo];
    if (!left[i]) a_b += dat[pa][1][hi] - dat[pa][1][lo];
    if (left[i]) a_a += dat[pa][1][hi] - dat[pa][1][lo];
    if (left[i]) a_b += dat[pa][0][hi] - dat[pa][0][lo];
  }

  ll _aa = 0, _ab = 0;
  /*
  FOR(k, 1, 2 * t) {
    FOR(j, 1, k) {
      if (k - j > t) continue;
      if (2 * j - k < 0) {
        if (right[j] == right[k]) _aa++;
        if (right[j] != right[k]) _ab++;
      }
    }
  }
  */
  FOR(k, 1, 2 * t) {
    ll lo = 1, hi = k - 1;
    chmax(lo, k - t);
    chmin(hi, (k - 1) / 2);
    if (lo > hi) continue;
    ++hi;
    FOR(pa, 2) {
      if (!right[k]) _aa += dat[pa][0][hi] - dat[pa][0][lo];
      if (!right[k]) _ab += dat[pa][1][hi] - dat[pa][1][lo];
      if (right[k]) _aa += dat[pa][1][hi] - dat[pa][1][lo];
      if (right[k]) _ab += dat[pa][0][hi] - dat[pa][0][lo];
    }
  }

  ll ANS = aa_ * 2 + a_a * 2 + _aa * 2;
  ANS -= ab_ + a_b + _ab;
  assert(ANS % 6 == 0);
  return ANS / 6;
}

void solve() {
  LL(p, t_max);
  ll F = calc_F(p);
  chmin(t_max, p / 2);
  ll ANS = F * t_max;

  ANS -= calc_NG(p, t_max) * 2;
  print(ANS);
}

void test() {
  ll N = 1e6;
  vc<int> F(N + N), G(N + N);
  FOR(i, N, N + N) F[i] = RNG(0, 2);
  FOR(i, N + N) G[i] = RNG(0, 2);
  leq_convolution(G, F);
}

signed main() {
  // test();
  INT(T);
  FOR(T) solve();
  return 0;
}