#line 1 "main.cpp" #define PROBLEM "https://yukicoder.me/problems/no/2362" #line 1 "library/my_template.hpp" #if defined(LOCAL) #include #else // https://codeforces.com/blog/entry/96344 #pragma GCC optimize("Ofast,unroll-loops") // いまの CF だとこれ入れると動かない? // #pragma GCC target("avx2,popcnt") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template constexpr T infty = 0; template <> constexpr int infty = 1'010'000'000; template <> constexpr ll infty = 2'020'000'000'000'000'000; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * 2'000'000'000'000'000'000; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) vector>> name(h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name(a, vector>>(b, vector>(c, vector(__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); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(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 T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template T ceil(T x, T y) { return floor(x + y - 1, y); } template T bmod(T x, T y) { return x - y * floor(x, y); } template pair divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #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 T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { T a = que.top(); que.pop(); return a; } template T POP(vc &que) { T a = que.back(); que.pop_back(); return a; } template 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; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector 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 vector argsort(const vector &A) { vector 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 vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } template void concat(vc &first, const Vectors &... others) { vc &res = first; (res.insert(res.end(), others.begin(), others.end()), ...); } #endif #line 1 "library/other/io.hpp" #define FASTIO #include // https://judge.yosupo.jp/submission/21623 namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; 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, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(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 void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(u32 &x) { rd_integer(x); } void rd(u64 &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } void rd(f128 &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d: x) rd(d); } template void rd(vc &x) { for (auto &d: x) rd(d); } void read() {} template void read(H &h, T &... t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c: s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(u32 x) { wt_integer(x); } void wt(u64 x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } void wt(f128 x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #if defined(LOCAL) #define SHOW(...) SHOW_IMPL(__VA_ARGS__, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, SHOW1)(__VA_ARGS__) #define SHOW_IMPL(_1, _2, _3, _4, _5, _6, NAME, ...) NAME #define SHOW1(x) print(#x, "=", (x)), flush() #define SHOW2(x, y) print(#x, "=", (x), #y, "=", (y)), flush() #define SHOW3(x, y, z) print(#x, "=", (x), #y, "=", (y), #z, "=", (z)), flush() #define SHOW4(x, y, z, w) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush() #define SHOW5(x, y, z, w, v) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v)), flush() #define SHOW6(x, y, z, w, v, u) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v), #u, "=", (u)), flush() #else #define SHOW(...) #endif #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ read(__VA_ARGS__) #define U64(...) \ u64 __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(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 4 "main.cpp" #line 1 "library/mod/floor_sum_of_linear_polynomial.hpp" #line 2 "library/alg/monoid_pow.hpp" // chat gpt template struct has_power_method { private: // ヘルパー関数の実装 template static auto check(int) -> decltype(std::declval().power(std::declval(), std::declval()), std::true_type{}); template static auto check(...) -> std::false_type; public: // メソッドの有無を表す型 static constexpr bool value = decltype(check(0))::value; }; template typename Monoid::X monoid_pow(typename Monoid::X x, ll exp) { using X = typename Monoid::X; if constexpr (has_power_method::value) { return Monoid::power(x, exp); } else { assert(exp >= 0); X res = Monoid::unit(); while (exp) { if (exp & 1) res = Monoid::op(res, x); x = Monoid::op(x, x); exp >>= 1; } return res; } } #line 2 "library/mod/floor_monoid_product.hpp" // https://yukicoder.me/submissions/883884 // https://qoj.ac/contest/1411/problem/7620 // U は範囲内で ax+b がオーバーフローしない程度 // yyy x yyyy x ... yyy x yyy (x を N 個) // k 個目の x までに floor(ak+b,m) 個の y がある // my<=ax+b における lattice path における辺の列と見なせる template X floor_monoid_product(X x, X y, U N, U a, U b, U m) { U c = (a * N + b) / m; X pre = Monoid::unit(), suf = Monoid::unit(); while (1) { const U p = a / m, q = b / m; a %= m, b %= m; x = Monoid::op(x, monoid_pow(y, p)); pre = Monoid::op(pre, monoid_pow(y, q)); c -= (p * N + q); if (c == 0) break; const U d = (m * c - b - 1) / a + 1; suf = Monoid::op(y, Monoid::op(monoid_pow(x, N - d), suf)); b = m - b - 1 + a, N = c - 1, c = d; swap(m, a), swap(x, y); } x = monoid_pow(x, N); return Monoid::op(Monoid::op(pre, x), suf); } #line 1 "library/alg/monoid/monoid_for_floor_sum.hpp" // sum i^k1floor^k2: floor path で (x,y) から x 方向に進むときに x^k1y^k2 を足す template struct Monoid_for_floor_sum { using ARR = array, K1 + 1>; struct Data { ARR dp; T dx, dy; }; using value_type = Data; using X = value_type; static X op(X a, X b) { static constexpr int n = max(K1, K2); static T comb[n + 1][n + 1]; if (comb[0][0] != T(1)) { comb[0][0] = T(1); FOR(i, n) FOR(j, i + 1) { comb[i + 1][j] += comb[i][j], comb[i + 1][j + 1] += comb[i][j]; } } array pow_x; array pow_y; pow_x[0] = 1, pow_y[0] = 1; FOR(i, K1) pow_x[i + 1] = pow_x[i] * a.dx; FOR(i, K2) pow_y[i + 1] = pow_y[i] * a.dy; // +dy FOR(i, K1 + 1) { FOR_R(j, K2 + 1) { T x = b.dp[i][j]; FOR(k, j + 1, K2 + 1) b.dp[i][k] += comb[k][j] * pow_y[k - j] * x; } } // +dx FOR(j, K2 + 1) { FOR_R(i, K1 + 1) { FOR(k, i, K1 + 1) a.dp[k][j] += comb[k][i] * pow_x[k - i] * b.dp[i][j]; } } a.dx += b.dx, a.dy += b.dy; return a; } static X to_x() { X x = unit(); x.dp[0][0] = 1, x.dx = 1; return x; } static X to_y() { X x = unit(); x.dy = 1; return x; } static constexpr X unit() { return {ARR{}, T(0), T(0)}; } static constexpr bool commute = 0; }; #line 4 "library/mod/floor_sum_of_linear_polynomial.hpp" // 全部非負, T は答, U は ax+b がオーバーフローしない template array, K1 + 1> floor_sum_of_linear_polynomial_nonnegative(U N, U a, U b, U mod) { static_assert(is_same_v || is_same_v); assert(a == 0 || N < (U(-1) - b) / a); using Mono = Monoid_for_floor_sum; auto x = floor_monoid_product(Mono::to_x(), Mono::to_y(), N, a, b, mod); return x.dp; }; // sum_{L<=x array, K1 + 1> floor_sum_of_linear_polynomial(I L, I R, I a, I b, I mod) { static_assert(is_same_v || is_same_v); assert(L <= R && mod > 0); if (a < 0) { auto ANS = floor_sum_of_linear_polynomial(-R + 1, -L + 1, -a, b, mod); FOR(i, K1 + 1) { if (i % 2 == 1) { FOR(j, K2 + 1) ANS[i][j] = -ANS[i][j]; } } return ANS; } assert(a >= 0); I ADD_X = L; I N = R - L; b += a * L; I ADD_Y = floor(b, mod); b -= ADD_Y * mod; assert(a >= 0 && b >= 0); using Mono = Monoid_for_floor_sum; using Data = typename Mono::Data; using U = std::conditional_t, i128, u128>; Data A = floor_monoid_product(Mono::to_x(), Mono::to_y(), N, a, b, mod); Data offset = Mono::unit(); offset.dx = T(ADD_X), offset.dy = T(ADD_Y); A = Mono::op(offset, A); return A.dp; }; #line 6 "main.cpp" void solve() { u64 N, M, X, Y; read(N, M, X, Y); u64 ANS = 0; { auto a = floor_sum_of_linear_polynomial_nonnegative(N, X, 0, M); ANS -= a[0][1] * N - a[1][1]; } { auto a = floor_sum_of_linear_polynomial_nonnegative(N, X, Y, M); ANS -= a[0][1] * (N - 1) - 2 * a[1][1]; } print(ANS); } signed main() { INT(T); FOR(T) solve(); return 0; }