#line 1 "No_61_\u30ea\u30d9\u30ea\u30aa\u30f3.cpp"
#define YRSD
#line 2 "YRS/all.hpp"

#line 2 "YRS/aa/head.hpp"

#include <iostream>
#include <algorithm>

#include <array>
#include <bitset>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <string>
#include <tuple>

#include <bit>
#include <chrono>
#include <functional>
#include <iomanip>
#include <utility>
#include <type_traits>
#include <cassert>
#include <cctype>
#include <cmath>
#include <cstring>
#include <ctime>
#include <limits>
#include <ranges>
#include <concepts>

#define TE template <typename T>
#define TES template <typename T, typename ...S>
#define Z auto
#define ep emplace_back
#define eb emplace
#define fi first
#define se second
#define all(x) (x).begin(), (x).end()

#define OV4(a, b, c, d, e, ...) e
#define FOR1(a) for (int _ = 0; _ < (a); ++_)
#define FOR2(i, a) for (int i = 0; i < (a); ++i)
#define FOR3(i, a, b) for (int i = (a); i < (b); ++i)
#define FOR4(i, a, b, c) for (int i = (a); i < (b); i += (c))
#define FOR(...) OV4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR1_R(a) for (int _ = (a) - 1; _ >= 0; --_)
#define FOR2_R(i, a) for (int i = (a) - 1; i >= 0; --i)
#define FOR3_R(i, a, b) for (int i = (b) - 1; i >= (a); --i)
#define FOR4_R(i, a, b, c) for (int i = (b) - 1; i >= (a); i -= (c))
#define FOR_R(...) OV4(__VA_ARGS__, FOR4_R, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define FOR_subset(t, s) for (int t = (s); t > -1; t = (t == 0 ? -1 : (t - 1) & s))

#define sort ranges::sort

using namespace std;

TE using vc = vector<T>;
TE using vvc = vc<vc<T>>;
TE using T1 = tuple<T>;
TE using T2 = tuple<T, T>;
TE using T3 = tuple<T, T, T>;
TE using T4 = tuple<T, T, T, T>;
TE using max_heap = priority_queue<T>;
TE using min_heap = priority_queue<T, vc<T>, greater<T>>;
using u8 = unsigned char; using uint = unsigned int; using ll = long long;      using ull = unsigned long long;
using ld = long double;   using i128 = __int128;     using u128 = __uint128_t;  using f128 = __float128;
using u16 = uint16_t;
using PII = pair<int, int>;   using PLL = pair<ll, ll>;

#ifdef YRSD
constexpr bool dbg = 1;
#else
constexpr bool dbg = 0;
#endif
#line 2 "YRS/IO/IO.hpp"

istream &operator>>(istream &I, i128 &x) {
  static string s;
  I >> s;
  int f = s[0] == '-';
  x = 0;
  const int N = (int)s.size();
  FOR(i, f, N) x = x * 10 + s[i] - '0';
  if (f) x = -x;
  return I;
}
ostream &operator<<(ostream &O, i128 x) {
  static string s;
  s.clear();
  bool f = x < 0;
  if (f) x = -x;
  while (x) s += '0' + x % 10, x /= 10;
  if (s.empty()) s += '0';
  if (f) s += '-';
  reverse(all(s));
  return O << s;
}
istream &operator>>(istream &I, f128 &x) {
  static string s;
  I >> s, x = stold(s);
  return I;
}
ostream &operator<<(ostream &O, const f128 x) { return O << ld(x); }
template <typename... S>
istream &operator>>(istream &I, tuple<S...> &t) {
  return apply([&I](Z &...s) { ((I >> s), ...); }, t), I;
}
template <typename T, typename U>
istream &operator>>(istream &I, pair<T, U> &x) {
  return I >> x.fi >> x.se;
}
template <typename T, typename U>
ostream &operator<<(ostream &O, const pair<T, U> &x) {
  return O << x.fi << ' ' << x.se;
}
TE requires requires(T &c) { begin(c); end(c); } and 
                          (not is_same_v<decay_t<T>, string>)
istream &operator>>(istream &I, T &c) {
  for (Z &e : c) I >> e;
  return I;
}
TE requires requires(const T &c) { begin(c); end(c); } and 
  (not is_same_v<decay_t<T>, const char*>) and 
  (not is_same_v<decay_t<T>, string>) and 
  (not is_array_v<remove_reference_t<T>> or 
   not is_same_v<remove_extent_t<remove_reference_t<T>>, char>)
ostream &operator<<(ostream &O, const T &a) {
  if (a.empty()) return O;
  Z i = a.begin();
  O << *i++;
  for (; i != a.end(); ++i) O << ' ' << *i;
  return O;
}
void IN() {}
TE void IN(T &x, Z &...s) { cin >> x, IN(s...); }
void print() { cout << '\n'; }
TES void print(T &&x, S &&...y) {
  cout << x;
  if constexpr (sizeof...(S)) cout << ' ';
  print(forward<S>(y)...);
}
void put() { cout << ' '; }
TES void put(T &&x, S &&...y) {
  cout << x;
  if constexpr (sizeof...(S)) cout << ' ';
  put(forward<S>(y)...);
}

#define INT(...)  int    __VA_ARGS__; IN(__VA_ARGS__)
#define UINT(...) uint   __VA_ARGS__; IN(__VA_ARGS__)
#define LL(...)   ll     __VA_ARGS__; IN(__VA_ARGS__)
#define ULL(...)  ull    __VA_ARGS__; IN(__VA_ARGS__)
#define I128(...) i128   __VA_ARGS__; IN(__VA_ARGS__)
#define STR(...)  string __VA_ARGS__; IN(__VA_ARGS__)
#define CH(...)   char   __VA_ARGS__; IN(__VA_ARGS__)
#define REAL(...) re     __VA_ARGS__; IN(__VA_ARGS__)
#define VEC(T, a, n) vc<T> a(n); IN(a)

void YES(bool o = 1) { print(o ? "YES" : "NO"); }
void Yes(bool o = 1) { print(o ? "Yes" : "No"); }
void yes(bool o = 1) { print(o ? "yes" : "no"); }
void NO(bool o = 1) { YES(not o); }
void No(bool o = 1) { Yes(not o); }
void no(bool o = 1) { yes(not o); }
void ALICE(bool o = 1) { print(o ? "ALICE" : "BOB"); }
void Alice(bool o = 1) { print(o ? "Alice" : "Bob"); }
void alice(bool o = 1) { print(o ? "alice" : "bob"); }
void BOB(bool o = 1) { ALICE(not o); }
void Bob(bool o = 1) { Alice(not o); }
void bob(bool o = 1) { alice(not o); }
void POSSIBLE(bool o = 1) { print(o ? "POSSIBLE" : "IMPOSSIBLE"); }
void Possible(bool o = 1) { print(o ? "Possible" : "Impossible"); }
void possible(bool o = 1) { print(o ? "possible" : "impossible"); }
void IMPOSSIBLE(bool o = 1) { POSSIBLE(not o); }
void Impossible(bool o = 1) { Possible(not o); }
void impossible(bool o = 1) { possible(not o); }
void TAK(bool o = 1) { print(o ? "TAK" : "NIE"); }
void NIE(bool o = 1) { TAK(not o); }
#line 5 "YRS/all.hpp"

#if (__cplusplus >= 202002L)
#include <numbers>
constexpr ld pi = numbers::pi;
#endif
TE constexpr T inf = numeric_limits<T>::max();
template <> constexpr i128 inf<i128> = i128(inf<ll>) * 2'000'000'000'000'000'000;
template <typename T, typename U>
constexpr pair<T, U> inf<pair<T, U>> = {inf<T>, inf<U>};

TE constexpr static inline int pc(T x) { return popcount(make_unsigned_t<T>(x)); }
constexpr static inline ll len(const Z &a) { return a.size(); }

void reverse(Z &a) { reverse(all(a)); }

void unique(Z &a) {
  sort(a);
  a.erase(unique(all(a)), a.end());
}
TE vc<int> inverse(const vc<T> &a) {
  int N = len(a);
  vc<int> b(N, -1);
  FOR(i, N) if (a[i] != -1) b[a[i]] = i;
  return b;
}

Z QMAX(const Z &a) { return *max_element(all(a)); }
Z QMIN(const Z &a) { return *min_element(all(a)); }
constexpr bool chmax(Z &a, const Z &b) { return (a < b ? a = b, 1 : 0); }
constexpr bool chmin(Z &a, const Z &b) { return (a > b ? a = b, 1 : 0); }

vc<int> argsort(const Z &a) {
  vc<int> I(len(a));
  iota(all(I), 0);
  sort(I, [&](int i, int k) { return a[i] < a[k] or (a[i] == a[k] and i < k); });
  return I;
}
TE vc<T> rearrange(const vc<T> &a, const vc<int> &I) {
  int N = len(I);
  vc<T> b(N);
  FOR(i, N) b[i] = a[I[i]];
  return b;
}
template <int of = 1, typename T> 
vc<T> pre_sum(const vc<T> &a) {
  int N = len(a);
  vc<T> c(N + 1);
  FOR(i, N) c[i + 1] = c[i] + a[i];
  if (of == 0) c.erase(c.begin());
  return c;
}

TE constexpr static int topbit(T x) {
  if (x == 0) return - 1;
  if constexpr (sizeof(T) <= 4) return 31 - __builtin_clz(x);
  else return 63 - __builtin_clzll(x);
}
TE constexpr static int lowbit(T x) {
  if (x == 0) return -1;
  if constexpr (sizeof(T) <= 4) return __builtin_ctz(x);
  else return __builtin_ctzll(x);
}

TE constexpr T floor(T x, T y) { return x / y - (x % y and (x ^ y) < 0); }
TE constexpr T ceil(T x, T y) { return floor(x + y - 1, y); }
TE constexpr T bmod(T x, T y) { return x - floor(x, y) * x; }
TE constexpr pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return pair{q, x - q * y};
}
template <typename T = ll>
T SUM(const Z &v) {
  return accumulate(all(v), T(0));
}
int lb(const Z &a, Z x) { return lower_bound(all(a), x) - a.begin(); }
int lb(Z l, Z r, Z x) requires(is_same_v<decltype(l), decltype(r)>) { return lower_bound(l, r, x) - l; }
int ub(const Z &a, Z x) { return upper_bound(all(a), x) - a.begin(); }
int ub(Z l, Z r, Z x) requires(is_same_v<decltype(l), decltype(r)>) { return upper_bound(l, r, x) - l; }

template <bool ck = 1>
ll bina(Z f, ll l, ll r) {
  if constexpr (ck) assert(f(l));
  while (abs(l - r) > 1) {
    ll x = (r + l) >> 1;
    (f(x) ? l : r) = x;
  }
  return l;
}
TE T bina_real(Z f, T l, T r, int c = 100) {
  while (c--) {
    T x = (l + r) / 2;
    (f(x) ? l : r) = x;
  }
  return (l + r) / 2;
}

Z pop(Z &s) {
  if constexpr (requires { s.pop_back(); }) {
    Z x = s.back();
    return s.pop_back(), x;
  } else if constexpr (requires { s.top(); }) {
    Z x = s.top();
    return s.pop(), x;
  } else {
    Z x = s.front();
    return s.pop(), x;
  }
}
void setp(int x) { cout << fixed << setprecision(x); }

TE inline void sh(vc<T> &a, int N) {
  a.resize(N, T(0));
}
#line 1 "YRS/debug.hpp"
#ifdef YRSD
void DBG() { cerr << "]" << endl; }
TES void DBG(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  DBG(forward<S>(y)...);
}
#define debug(...) cerr << "[" << __LINE__ << "]: [" #__VA_ARGS__ "] = [", DBG(__VA_ARGS__)
void ERR() { cerr << endl; }
TES void ERR(T &&x, S &&...y) {
  cerr << x;
  if constexpr (sizeof...(S)) cerr << ", ";
  ERR(forward<S>(y)...);
}
#define err(...) cerr << "[" << __LINE__ << "]: ", ERR(__VA_ARGS__)
#define asser assert
#else
#define debug(...) void(0721)
#define err(...)   void(0721)
#define asser(...) void(0721)
#endif
#line 4 "No_61_\u30ea\u30d9\u30ea\u30aa\u30f3.cpp"
// #include "YRS/IO/fast_io.hpp"
// #include "YRS/random/rng.hpp"
#line 2 "YRS/pr/crt.hpp"

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

struct Barrett {
  uint m;
  ull im;
  explicit Barrett(uint m = 1) : m(m), im(ull(-1) / m + 1) {}
  uint modulo(ull z) const {
    if (m == 1) return 0;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return (z - y + (z < y ? m : 0));
  }
  inline uint mul(uint a, uint b) const { return modulo(ull(a) * b); }
  uint pow(uint a, uint b) const {
    uint s = 1;
    for (; b; b >>= 1, a = mul(a, a))
      if (b & 1) s = mul(s, a);
    return s - (s >= m ? m : 0);
  }
  ull floor(ull z) const {
    if (m == 1) return z;
    ull x = (ull)(((u128)z * im) >> 64);
    ull y = x * m;
    return (z < y ? x - 1 : x);
  }
  pair<ull, uint> divmod(ull z) const {
    if (m == 1) return {z, 0};
    ull x = ull((u128(z) * im) >> 64), y = x * m;
    if (z < y) return {x - 1, z - y + m};
    return {x, z - y};
  }
  uint umod() const { return m; }
};

struct Barrett_64 {
  u128 m, mh, ml;
  explicit Barrett_64(ull mod = 1) : m(mod) {
    u128 m = u128(-1) / mod;
    if (m * mod + mod == u128(0)) ++m;
    mh = m >> 64;
    ml = m & ull(-1);
  }
  ull modulo(u128 x) const {
    u128 z = (x & ull(-1)) * ml;
    z = (x & ull(-1)) * mh + (x >> 64) * ml + (z >> 64);
    z = (x >> 64) * mh + (z >> 64);
    x -= z * m;
    return x < m ? x : x - m;
  }
  ull mul(ull a, ull b) const { return modulo(u128(a) * b); }
  ull pow(ull a, ull b) const {
    ull s = 1;
    for (; b; b >>= 1, a = mul(a, a)) 
      if (b & 1) s = mul(s, b);
    return s - (s >= m ? m : 0);
  }
  ull umod() const { return m; }
};
#line 2 "YRS/mod/mod_inv.hpp"

constexpr ll mod_inv(ll val, ll mod) {
  if (mod == 0) return 0;
  mod = abs(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 2 "YRS/pr/copr_bas.hpp"

// 分解出互质基底 return {bas, dec(a)} dec:= {{id, c}}
TE pair<vc<T>, vc<vc<PII>>> copr_bas(const vc<T> &a) {
  vc<T> s;
  for (T e : a) {
    vc<T> f;
    for (T x : s) {
      if (e == 1) {
        f.ep(x);
        continue;
      }
      vc<T> dat{e, x};
      FOR(p, 1, len(dat)) {
        FOR(i, p) {
          while (1) {
            if (dat[p] > 1 and dat[i] % dat[p] == 0) dat[i] /= dat[p];
            else if (dat[i] > 1 and dat[p] % dat[i] == 0) dat[p] /= dat[i];
            else break;
          }
          T g = gcd(dat[i], dat[p]);
          if (g == 1 or g == dat[i] or g == dat[p]) continue;
          dat[i] /= g, dat[p] /= g, dat.ep(g);
        }
      }
      e = dat[0];
      int sz = len(dat);
      FOR(i, 1, sz) if (dat[i] != 1) f.ep(dat[i]);
    }
    if (e > 1) f.ep(e);
    s.swap(f);
  }
  sort(s);
  int N = len(a), sz = len(s);
  vc<vc<PII>> res(N);
  FOR(i, N) {
    T x = a[i];
    FOR(k, sz) {
      int t = 0;
      while (x % s[k] == 0) x /= s[k], ++t;
      if (t) res[i].ep(k, t);
    }
  }
  return {s, res};
}
#line 2 "YRS/pr/factors.hpp"

#line 2 "YRS/pr/prims_test.hpp"

struct MM {
  using uu = unsigned __int128;
  inline static ull m, r, nn;
  static void set_mod(ull m) {
    MM::m = m;
    nn = -uu(m) % m;
    r = m;
    FOR(5) r *= 2 - m * r;
    r = -r;
  }
  static ull reduce(uu x) { return (x + uu(ull(x) * r) * m) >> 64; }

  ull x;
  MM() : x(0) {}
  MM(ull x) : x(reduce(uu(x) * nn)) {}
  ull val() const {
    ull y = reduce(x);
    return y >= m ? y - m : y;
  }
  MM &operator+=(MM y) {
    x += y.x - (m << 1);
    x = (ll(x) < 0 ? x + (m << 1) : x);
    return *this;
  }
  MM &operator-=(MM y) {
    x -= y.x;
    x = (ll(x) < 0 ? x + (m << 1) : x);
    return *this;
  }
  MM &operator*=(MM y) {
    x = reduce(uu(x) * y.x);
    return *this;
  }
  MM operator+(MM y) const { return MM(*this) += y; }
  MM operator-(MM y) const { return MM(*this) -= y; }
  MM operator*(MM y) const { return MM(*this) *= y; }
  bool operator==(MM y) const {
    return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
  }
  bool operator!=(MM y) const { return not operator==(y); }
  MM pow(ull k) const {
    MM r = 1, a = *this;
    for (; k; k >>= 1, a *= a) if (k & 1) r *= a;
    return r;
  }
};

bool primetest(const ull x) {
  if (x == 2 or x == 3 or x == 5 or x == 7) return 1;
  if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return 0;
  if (x < 121) return x > 1;
  const ull d = (x - 1) >> __builtin_ctzll(x - 1);
  MM::set_mod(x);
  const MM o(1), mo(x - 1);
  Z f = [&](ull a) -> bool {
    MM y = MM(a).pow(d);
    ull t = d;
    while (y != o and y != mo and t != x - 1) y *= y, t <<= 1;
    if (y != mo and t % 2 == 0) return 1;
    return 0;
  };
  if (x < (1ull << 32)) {
    for (ull a : {2, 7, 61}) if (f(a)) return 0;
  } else {
    for (ull a : {2, 325, 9'375, 281'78, 450'775, 978'050'4, 179'526'502'2}) {
      if (x <= a) return 1;
      if (f(a)) return 0;
    }
  }
  return 1;
}
ll rho(ll n, ll c) {
  MM::set_mod(n);
  const MM cc(c);
  Z f = [&](MM x) { return x * x + cc; };
  MM x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1ll << (__lg(n) / 5);
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(r) y = f(y);
    for (ll k = 0; k < r and g == 1; k += m) {
      z = y;
      FOR(i, min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n) do {
    z = f(z);
    g = gcd((x - z).val(), n);
  } while (g == 1);
  return g;
}
#line 2 "YRS/random/rng.hpp"

#include <random>

#ifdef MeIoN
std::mt19937 rg(0);
std::mt19937_64 rd_64(0);
#else
std::mt19937 rg(std::chrono::steady_clock::now().time_since_epoch().count());
std::mt19937_64 rd_64(std::chrono::steady_clock::now().time_since_epoch().count());
#endif

uint rng() { return rg(); }
uint rng(uint lim) { return rg() % lim; }
int rng(int l, int r) { return l + rg() % (r - l); }
ull rng_64() { return rd_64(); }
ull rng_64(ull lim) { return rd_64() % lim; }
ll rng_64(ll l, ll r) { return l + rd_64() % (r - l); }

template <typename T>
void shuffle(vector<T> &v) {
  const int N = len(v);
  FOR(i, 1, N) {
    int k = rng(0, i + 1);
    if (i != k) swap(v[i], v[k]);
  }
}
#line 5 "YRS/pr/factors.hpp"

// https://yukicoder.me/problems/no/36 => factor();

ll find_pr_e(ll x) {
  assert(x > 1);
  if (primetest(x)) return x;
  FOR(100) {
    ll e = rho(x, rng_64(x));
    if (primetest(e)) return e;
    x = e;
  }
  err("failed");
  assert(0);
  return -1;
}

vc<pair<ll, int>> factor(ll x) {
  assert(x >= 1);
  vc<pair<ll, int>> r;
  for (int e = 2; e < 100; ++e) {
    if (e * e > x) break;
    if (x % e == 0) {
      int c = 0;
      do {
        x /= e, c += 1;
      } while (x % e == 0);
      r.ep(e, c);
    }
  }
  while (x > 1) {
    ll e = find_pr_e(x);
    int c = 0;
    do {
      x /= e, c += 1;
    } while (x % e == 0);
    r.ep(e, c);
  }
  return sort(r), r;
}
vc<pair<ll, int>> factor_by_lpf(ll n, vc<int> &lpf) {
  vc<pair<ll, int>> s;
  while (n > 1) {
    int p = lpf[n], e = 0;
    while (n % p == 0) n /= p, ++e;
    s.ep(p, e);
  }
  return s;
}
#line 7 "YRS/pr/crt.hpp"

// x = a_i (mod b_i) return min x >= 0

TE bool reduce_by_fac(vc<T> &a, vc<T> &mods) {
  int N = len(a);
  unordered_map<T, pair<T, T>> mp;
  FOR(i, N) {
    for (Z [e, c] : factor(mods[i])) {
      T mod = 1;
      FOR(c) mod *= e;
      T x = a[i] % mod;
      if (not mp.contains(e)) {
        mp[e] = {mod, x};
        continue;
      }
      Z &[m, xx] = mp[e];
      if (mod > m) swap(m, mod), swap(x, xx);
      if (xx % mod != x) return 0;
    }
  }
  mods.clear(), a.clear();
  for (Z [p, x] : mp) mods.ep(x.fi), a.ep(x.se);
  return 1;
}

TE bool reduce_by_copr(vc<T> &a, vc<T> &mods) {
  Z [bs, pfs] = copr_bas(mods);
  int k = len(bs), N = len(a);
  vc<pair<T, T>> dat(k, {1, 0});
  FOR(i, N) {
    for (Z [id, c] : pfs[i]) {
      T mod = 1;
      FOR(c) mod *= bs[id];
      T val = a[i] % mod;
      Z &[mm, xx] = dat[id];
      if (mod > mm) swap(mod, mm), swap(val, xx);
      if (xx % mod != val) return 0;
    }
  }
  mods.clear(), a.clear();
  for (Z x : dat) mods.ep(x.fi), a.ep(x.se);
  return 1;
}

i128 crt_reduce(vc<int> a, vc<int> mods, ll mod = -1) {
  int N = len(a);
  FOR(i, N) a[i] = ((a[i] %= mods[i]) >= 0 ? a[i] : a[i] + mods[i]);
  bool ok = (N <= 10 ? reduce_by_copr(a, mods) : reduce_by_fac(a, mods));
  if (not ok) return -1;
  if (N == 0) return 0;
  vc<ll> s(N);
  FOR(i, N) {
    Barrett X(mods[i]);
    ll x = a[i], pr = 1;
    FOR(k, i) {
      x = X.modulo(x + s[k] * (mods[i] - pr));
      pr = X.mul(pr, mods[k]);
    }
    s[i] = X.mul(mod_inv(pr, mods[i]), x);
  }
  i128 res = 0, pr = 1;
  FOR(i, N) {
    res += pr * s[i], pr *= mods[i];
    if (mod != -1) res %= mod, pr %= mod;
  }
  return res;
}

i128 crt_reduce(vc<ll> a, vc<ll> mods, ll mod = -1) {
  int N = len(a);
  FOR(i, N) a[i] = ((a[i] %= mods[i]) >= 0 ? a[i] : a[i] + mods[i]);
  bool ok = (N <= 10 ? reduce_by_copr(a, mods) : reduce_by_fac(a, mods));
  if (not ok) return -1;
  if (N == 0) return 0;
  vc<ll> s(N);
  FOR(i, N) {
    Barrett_64 X(mods[i]);
    ll x = a[i], pr = 1;
    FOR(k, i) {
      x = X.modulo(x + s[k] * (mods[i] - pr));
      pr = X.mul(pr, mods[k]);
    }
    s[i] = X.mul(mod_inv(pr, mods[i]), x);
  }
  i128 res = 0, pr = 1;
  FOR(i, N) {
    res += pr * s[i], pr *= mods[i];
    if (mod != -1) res %= mod, pr %= mod;
  }
  return res;
}
i128 crt_copr(vc<int> a, vc<int> mods, int mod = -1) {
  int N = len(a);
  FOR(i, N) a[i] = ((a[i] %= mods[i]) >= 0 ? a[i] : a[i] + mods[i]);
  vc<ll> s(N);
  FOR(i, N) {
    Barrett X(mods[i]);
    ll x = a[i], pr = 1;
    FOR(k, i) {
      x = X.modulo(x + s[k] * (mods[i] - pr));
      pr = X.mul(pr, mods[k]);
    }
    s[i] = X.mul(mod_inv(pr, mods[i]), x);
  }
  i128 res = 0, pr = 1;
  FOR(i, N) {
    res += pr * s[i], pr *= mods[i];
    if (mod != -1) res %= mod, pr %= pr;
  }
  return res;
}
i128 crt_copr(vc<ll> a, vc<ll> mods, ll mod = -1) {
  int N = len(a);
  FOR(i, N) a[i] = ((a[i] %= mods[i]) >= 0 ? a[i] : a[i] + mods[i]);
  vc<ll> s(N);
  FOR(i, N) {
    Barrett_64 X(mods[i]);
    ll x = a[i], pr = 1;
    FOR(k, i) {
      x = X.modulo(x + s[k] * (mods[i] - pr));
      pr = X.mul(pr, mods[k]);
    }
    s[i] = X.mul(mod_inv(pr, mods[i]), x);
  }
  i128 res = 0, pr = 1;
  FOR(i, N) {
    res += pr * s[i], pr *= mods[i];
    if (mod != -1) res %= mod, pr %= pr;
  }
  return res;
}
#line 2 "YRS/pr/f/line_con_1.hpp"

#line 4 "YRS/pr/f/line_con_1.hpp"
// #include "../exgcd.hpp"

// ax = b (mod) return t(=== x), mod (reduced)
PLL line_con_1(int a, int b, int mod) {
  assert(mod != 0);
  // a = (a %= mod) < 0 ? a + mod : a;
  b = (b %= mod) < 0 ? b + mod : b;
  int g = gcd(abs(a), mod);
  if (b % g != 0) return {-1, -1};
  a /= g, b /= g, mod /= g;
  ll ia = mod_inv(a, mod);
  return {1ll * b * ia % mod, mod};
}
#line 8 "No_61_\u30ea\u30d9\u30ea\u30aa\u30f3.cpp"

#define tests 1
#define fl 0
#define DB 10
void Yorisou() {
  LL(N, M, D, tx, ty, sx, sy, vx, vy);
  ll d = gcd(vx, vy);
  vx /= d, vy /= d, D *= d;
  ll f[]{tx, tx, 2 * N - tx, 2 * N - tx}, g[]{ty, 2 * M - ty, ty, 2 * M - ty};
  FOR(i, 4) {
    int tx = f[i], ty = g[i];
    vc<ll> a, mds;
    Z [x, mod] = line_con_1(vx, tx - sx, 2 * N);
    if (x == -1) continue;
    a.ep(x);
    mds.ep(mod);
    tie(x, mod) = line_con_1(vy, ty - sy, 2 * M);
    if (x == -1) continue;
    a.ep(x);
    mds.ep(mod);
    ll st = crt_reduce(a, mds);
    if (st != -1 and st <= D) return print("Hit");
  }
  print("Miss");
}
#line 1 "YRS/aa/main.hpp"
int main() {
  cin.tie(nullptr)->sync_with_stdio(false);
  int T = 1;
  if (fl) cerr.tie(0);
  if (tests and not fl) IN(T);
  for (int i = 0; i < T or fl; ++i) {
    Yorisou();
    if (fl and i % DB == 0) cerr << "Case: " << i << '\n';
  }
  return 0;
}
#line 34 "No_61_\u30ea\u30d9\u30ea\u30aa\u30f3.cpp"