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#include "poly/multivar_convolution_cyclic.hpp"
#include "poly/multipoint.hpp"
#include "mod/primitive_root.hpp"
/*
example : ns = (2, 3)
[a0, a1, a2, a3, a4, a5] = [a(0,0), a(1,0), a(0,1), a(1,1), a(0,2), a(1,2)]
[b0, b1, b2, b3, b4, b5] = [b(0,0), b(1,0), b(0,1), b(1,1), b(0,2), b(1,2)]
*/
template <typename mint>
vc<mint> multivar_convolution_cyclic(vc<int> ns, vc<mint> f, vc<mint>& g) {
int p = mint::get_mod();
for (auto&& n: ns) assert((p - 1) % n == 0);
mint r = primitive_root(p);
mint ir = r.inverse();
int K = len(ns);
int N = 1;
for (auto&& n: ns) N *= n;
assert(len(f) == N);
assert(len(g) == N);
vc<mint> root(K), iroot(K);
FOR(k, K) { root[k] = r.pow((p - 1) / ns[k]); }
FOR(k, K) { iroot[k] = ir.pow((p - 1) / ns[k]); }
int step = 1;
FOR(k, K) {
int n = ns[k];
FOR(i, N) if (i % (step * n) < step) {
vc<mint> a(n), b(n);
FOR(j, n) {
a[j] = f[i + step * j];
b[j] = g[i + step * j];
}
a = multipoint_eval_on_geom_seq(a, mint(1), root[k], n);
b = multipoint_eval_on_geom_seq(b, mint(1), root[k], n);
FOR(j, n) {
f[i + step * j] = a[j];
g[i + step * j] = b[j];
}
}
step *= n;
}
FOR(i, N) f[i] *= g[i];
step = 1;
FOR(k, K) {
int n = ns[k];
FOR(i, N) if (i % (step * n) < step) {
vc<mint> a(n);
FOR(j, n) { a[j] = f[i + step * j]; }
a = multipoint_eval_on_geom_seq(a, mint(1), iroot[k], n);
FOR(j, n) { f[i + step * j] = a[j]; }
}
step *= n;
}
mint cf = mint(N).inverse();
for (auto&& x: f) x *= cf;
return f;
}
template <typename mint>
vc<vc<mint>> multivar_convolution_cyclic_2d(vc<vc<mint>>& f, vc<vc<mint>>& g) {
int H = len(f);
int W = len(f[0]);
assert(len(g) == H);
assert(len(g[0]) == W);
vc<mint> F(H * W), G(H * W);
FOR(x, H) FOR(y, W) F[x + H * y] = f[x][y];
FOR(x, H) FOR(y, W) G[x + H * y] = g[x][y];
F = multivar_convolution_cyclic(vc<int>({H, W}), F, G);
vv(mint, h, H, W);
FOR(x, H) FOR(y, W) h[x][y] = F[x + H * y];
return h;
}
#line 2 "poly/multipoint.hpp"
#line 2 "poly/middle_product.hpp"
#line 2 "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;
assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));
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 2 "mod/crt3.hpp"
constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) {
a %= mod;
u64 res = 1;
FOR(32) {
if (n & 1) res = res * a % mod;
a = a * a % mod, n /= 2;
}
return res;
}
template <typename T, u32 p0, u32 p1>
T CRT2(u64 a0, u64 a1) {
static_assert(p0 < p1);
static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1);
u64 c = (a1 - a0 + p1) * x0_1 % p1;
return a0 + c * p0;
}
template <typename T, u32 p0, u32 p1, u32 p2>
T CRT3(u64 a0, u64 a1, u64 a2) {
static_assert(p0 < p1 && p1 < p2);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
return T(ans_1) + T(c) * T(p01);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3>
T CRT4(u64 a0, u64 a1, u64 a2, u64 a3) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = (a3 - ans_2 % p3 + p3) * x3 % p3;
return T(ans_2) + T(c) * T(p01) * T(p2);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3, u32 p4>
T CRT5(u64 a0, u64 a1, u64 a2, u64 a3, u64 a4) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3 && p3 < p4);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 x4 = mod_pow_constexpr(u64(p0) * p1 % p4 * p2 % p4 * p3 % p4, p4 - 2, p4);
static constexpr u64 p01 = u64(p0) * p1;
static constexpr u64 p23 = u64(p2) * p3;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = static_cast<u64>(a3 - ans_2 % p3 + p3) * x3 % p3;
u128 ans_3 = ans_2 + static_cast<u128>(c * p2) * p01;
c = static_cast<u64>(a4 - ans_3 % p4 + p4) * x4 % p4;
return T(ans_3) + T(c) * T(p01) * T(p23);
}
#line 2 "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::raw(q));
}
return dat[n];
}
template <typename mint>
mint fact(int n) {
static const int mod = mint::get_mod();
assert(0 <= n && n < mod);
static vector<mint> dat = {1, 1};
while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(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) {
assert(n >= 0);
if (k < 0 || n < k) return 0;
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 constexpr (dense) return C_dense<mint>(n, k);
if constexpr (!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}
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 "mod/modint.hpp"
template <int mod>
struct modint {
static constexpr u32 umod = u32(mod);
static_assert(umod < u32(1) << 31);
u32 val;
static modint raw(u32 v) {
modint x;
x.val = v;
return x;
}
constexpr modint() : val(0) {}
constexpr modint(u32 x) : val(x % umod) {}
constexpr modint(u64 x) : val(x % umod) {}
constexpr modint(u128 x) : val(x % umod) {}
constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
bool operator<(const modint &other) const { return val < other.val; }
modint &operator+=(const modint &p) {
if ((val += p.val) >= umod) val -= umod;
return *this;
}
modint &operator-=(const modint &p) {
if ((val += umod - p.val) >= umod) val -= umod;
return *this;
}
modint &operator*=(const modint &p) {
val = u64(val) * p.val % umod;
return *this;
}
modint &operator/=(const modint &p) {
*this *= p.inverse();
return *this;
}
modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
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;
}
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 == 120586241) return {20, 74066978};
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 == 943718401) return {22, 663003469};
if (mod == 998244353) return {23, 31};
if (mod == 1004535809) return {21, 582313106};
if (mod == 1012924417) return {21, 368093570};
return {-1, -1};
}
static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};
#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
fastio::rd(x.val);
x.val %= mod;
// assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
fastio::wt(x.val);
}
#endif
using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 6 "poly/middle_product.hpp"
// n, m 次多項式 (n>=m) a, b → n-m 次多項式 c
// c[i] = sum_j b[j]a[i+j]
template <typename mint>
vc<mint> middle_product(vc<mint>& a, vc<mint>& b) {
assert(len(a) >= len(b));
if (b.empty()) return vc<mint>(len(a) - len(b) + 1);
if (min(len(b), len(a) - len(b) + 1) <= 60) {
return middle_product_naive(a, b);
}
if (!(mint::can_ntt())) {
return middle_product_garner(a, b);
} else {
int n = 1 << __lg(2 * len(a) - 1);
vc<mint> fa(n), fb(n);
copy(a.begin(), a.end(), fa.begin());
copy(b.rbegin(), b.rend(), fb.begin());
ntt(fa, 0), ntt(fb, 0);
FOR(i, n) fa[i] *= fb[i];
ntt(fa, 1);
fa.resize(len(a));
fa.erase(fa.begin(), fa.begin() + len(b) - 1);
return fa;
}
}
template <typename mint>
vc<mint> middle_product_garner(vc<mint>& a, vc<mint> b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
static constexpr int p0 = 167772161;
static constexpr int p1 = 469762049;
static constexpr int p2 = 754974721;
using mint0 = modint<p0>;
using mint1 = modint<p1>;
using mint2 = modint<p2>;
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 = middle_product<mint0>(a0, b0);
auto c1 = middle_product<mint1>(a1, b1);
auto c2 = middle_product<mint2>(a2, b2);
vc<mint> c(len(c0));
FOR(i, n - m + 1) {
c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val);
}
return c;
}
template <typename mint>
vc<mint> middle_product_naive(vc<mint>& a, vc<mint>& b) {
vc<mint> res(len(a) - len(b) + 1);
FOR(i, len(res)) FOR(j, len(b)) res[i] += b[j] * a[i + j];
return res;
}
#line 2 "mod/all_inverse.hpp"
template <typename mint>
vc<mint> all_inverse(vc<mint>& X) {
for (auto&& x: X) assert(x != mint(0));
int N = len(X);
vc<mint> res(N + 1);
res[0] = mint(1);
FOR(i, N) res[i + 1] = res[i] * X[i];
mint t = res.back().inverse();
res.pop_back();
FOR_R(i, N) {
res[i] *= t;
t *= X[i];
}
return res;
}
#line 2 "poly/fps_div.hpp"
#line 2 "poly/count_terms.hpp"
template<typename mint>
int count_terms(const vc<mint>& f){
int t = 0;
FOR(i, len(f)) if(f[i] != mint(0)) ++t;
return t;
}
#line 2 "mod/mod_inv.hpp"
// long でも大丈夫
// (val * x - 1) が mod の倍数になるようにする
// 特に mod=0 なら x=0 が満たす
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 "poly/convolution_naive.hpp"
template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vector<T> ans(n + m - 1);
FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
return ans;
}
template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vc<T> ans(n + m - 1);
if (n <= 16 && (T::get_mod() < (1 << 30))) {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u64 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = sm;
}
} else {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u128 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = T::raw(sm % T::get_mod());
}
}
return ans;
}
#line 2 "poly/convolution_karatsuba.hpp"
// 任意の環でできる
template <typename T>
vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& g) {
const int thresh = 30;
if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g);
int n = max(len(f), len(g));
int m = ceil(n, 2);
vc<T> f1, f2, g1, g2;
if (len(f) < m) f1 = f;
if (len(f) >= m) f1 = {f.begin(), f.begin() + m};
if (len(f) >= m) f2 = {f.begin() + m, f.end()};
if (len(g) < m) g1 = g;
if (len(g) >= m) g1 = {g.begin(), g.begin() + m};
if (len(g) >= m) g2 = {g.begin() + m, g.end()};
vc<T> a = convolution_karatsuba(f1, g1);
vc<T> b = convolution_karatsuba(f2, g2);
FOR(i, len(f2)) f1[i] += f2[i];
FOR(i, len(g2)) g1[i] += g2[i];
vc<T> c = convolution_karatsuba(f1, g1);
vc<T> F(len(f) + len(g) - 1);
assert(2 * m + len(b) <= len(F));
FOR(i, len(a)) F[i] += a[i], c[i] -= a[i];
FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i];
if (c.back() == T(0)) c.pop_back();
FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i];
return F;
}
#line 8 "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 constexpr int p0 = 167772161;
static constexpr int p1 = 469762049;
static constexpr int p2 = 754974721;
using mint0 = modint<p0>;
using mint1 = modint<p1>;
using mint2 = modint<p2>;
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);
vc<mint> c(len(c0));
FOR(i, n + m - 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); }
return c;
}
vector<ll> convolution(vector<ll> a, vector<ll> b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
if (min(n, m) <= 2500) return convolution_naive(a, b);
ll mi_a = MIN(a), mi_b = MIN(b);
for (auto& x: a) x -= mi_a;
for (auto& x: b) x -= mi_b;
assert(MAX(a) * MAX(b) <= 1e18);
auto Ac = cumsum<ll>(a), Bc = cumsum<ll>(b);
vi res(n + m - 1);
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
res[k] += (t - s) * mi_a * mi_b;
res[k] += mi_a * (Bc[k - s + 1] - Bc[k - t + 1]);
res[k] += mi_b * (Ac[t] - Ac[s]);
}
static constexpr u32 MOD1 = 1004535809;
static constexpr u32 MOD2 = 1012924417;
using mint1 = modint<MOD1>;
using mint2 = modint<MOD2>;
vc<mint1> a1(n), b1(m);
vc<mint2> a2(n), b2(m);
FOR(i, n) a1[i] = a[i], a2[i] = a[i];
FOR(i, m) b1[i] = b[i], b2[i] = b[i];
auto c1 = convolution_ntt<mint1>(a1, b1);
auto c2 = convolution_ntt<mint2>(a2, b2);
FOR(i, n + m - 1) { res[i] += CRT2<u64, MOD1, MOD2>(c1[i].val, c2[i].val); }
return res;
}
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_karatsuba<mint>(a, b);
return convolution_ntt(a, b);
}
if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b);
return convolution_garner(a, b);
}
#line 4 "poly/fps_inv.hpp"
template <typename mint>
vc<mint> fps_inv_sparse(const vc<mint>& f) {
int N = len(f);
vc<pair<int, mint>> dat;
FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]);
vc<mint> g(N);
mint g0 = mint(1) / f[0];
g[0] = g0;
FOR(n, 1, N) {
mint rhs = 0;
for (auto&& [k, fk]: dat) {
if (k > n) break;
rhs -= fk * g[n - k];
}
g[n] = rhs * g0;
}
return g;
}
template <typename mint>
vc<mint> fps_inv_dense_ntt(const vc<mint>& F) {
vc<mint> G = {mint(1) / F[0]};
ll N = len(F), n = 1;
G.reserve(N);
while (n < N) {
vc<mint> f(2 * n), g(2 * n);
FOR(i, min(N, 2 * n)) f[i] = F[i];
FOR(i, n) g[i] = G[i];
ntt(f, false), ntt(g, false);
FOR(i, 2 * n) f[i] *= g[i];
ntt(f, true);
FOR(i, n) f[i] = 0;
ntt(f, false);
FOR(i, 2 * n) f[i] *= g[i];
ntt(f, true);
FOR(i, n, min(N, 2 * n)) G.eb(-f[i]);
n *= 2;
}
return G;
}
template <typename mint>
vc<mint> fps_inv_dense(const vc<mint>& F) {
if (mint::can_ntt()) return fps_inv_dense_ntt(F);
const int N = len(F);
vc<mint> R = {mint(1) / F[0]};
vc<mint> p;
int m = 1;
while (m < N) {
p = convolution(R, R);
p.resize(m + m);
vc<mint> f = {F.begin(), F.begin() + min(m + m, N)};
p = convolution(p, f);
R.resize(m + m);
FOR(i, m + m) R[i] = R[i] + R[i] - p[i];
m += m;
}
R.resize(N);
return R;
}
template <typename mint>
vc<mint> fps_inv(const vc<mint>& f) {
assert(f[0] != mint(0));
int n = count_terms(f);
int t = (mint::can_ntt() ? 160 : 820);
return (n <= t ? fps_inv_sparse<mint>(f) : fps_inv_dense<mint>(f));
}
#line 5 "poly/fps_div.hpp"
// f/g. f の長さで出力される.
template <typename mint, bool SPARSE = false>
vc<mint> fps_div(vc<mint> f, vc<mint> g) {
if (SPARSE || count_terms(g) < 200) return fps_div_sparse(f, g);
int n = len(f);
g.resize(n);
g = fps_inv<mint>(g);
f = convolution(f, g);
f.resize(n);
return f;
}
// f/g ただし g は sparse
template <typename mint>
vc<mint> fps_div_sparse(vc<mint> f, vc<mint>& g) {
if (g[0] != mint(1)) {
mint cf = g[0].inverse();
for (auto&& x: f) x *= cf;
for (auto&& x: g) x *= cf;
}
vc<pair<int, mint>> dat;
FOR(i, 1, len(g)) if (g[i] != mint(0)) dat.eb(i, -g[i]);
FOR(i, len(f)) {
for (auto&& [j, x]: dat) {
if (i >= j) f[i] += x * f[i - j];
}
}
return f;
}
#line 2 "poly/ntt_doubling.hpp"
#line 4 "poly/ntt_doubling.hpp"
// 2^k 次多項式の長さ 2^k が与えられるので 2^k+1 にする
template <typename mint, bool transposed = false>
void ntt_doubling(vector<mint>& a) {
static array<mint, 30> root;
static bool prepared = 0;
if (!prepared) {
prepared = 1;
const int rank2 = mint::ntt_info().fi;
root[rank2] = mint::ntt_info().se;
FOR_R(i, rank2) { root[i] = root[i + 1] * root[i + 1]; }
}
if constexpr (!transposed) {
const int M = (int)a.size();
auto b = a;
ntt(b, 1);
mint r = 1, zeta = root[topbit(2 * M)];
FOR(i, M) b[i] *= r, r *= zeta;
ntt(b, 0);
copy(begin(b), end(b), back_inserter(a));
} else {
const int M = len(a) / 2;
vc<mint> tmp = {a.begin(), a.begin() + M};
a = {a.begin() + M, a.end()};
transposed_ntt(a, 0);
mint r = 1, zeta = root[topbit(2 * M)];
FOR(i, M) a[i] *= r, r *= zeta;
transposed_ntt(a, 1);
FOR(i, M) a[i] += tmp[i];
}
}
#line 2 "poly/transposed_ntt.hpp"
template <class mint>
void transposed_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;
assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));
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 = h;
while (len > 0) {
if (len == 1) {
int p = 1 << (h - len);
mint rot = 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) * rot.val;
}
rot *= rate2[topbit(~s & -~s)];
}
len--;
} else {
int p = 1 << (h - len);
mint rot = 1, imag = root[2];
FOR(s, (1 << (len - 2))) {
int offset = s << (h - len + 2);
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
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) * imag.val % mod;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] = (a0 + mod - a1 + x) * rot.val;
a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * rot2.val;
a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * rot3.val;
}
rot *= rate3[topbit(~s & -~s)];
}
len -= 2;
}
}
} else {
mint coef = mint(1) / mint(len(a));
FOR(i, len(a)) a[i] *= coef;
int len = 0;
while (len < h) {
if (len == h - 1) {
int p = 1 << (h - len - 1);
mint irot = 1;
FOR(s, 1 << len) {
int offset = s << (h - len);
FOR(i, p) {
auto l = a[i + offset];
auto r = a[i + offset + p] * irot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
irot *= irate2[topbit(~s & -~s)];
}
len++;
} else {
int p = 1 << (h - len - 2);
mint irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << len); s++) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
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) * irot.val;
u64 a2 = u64(a[i + offset + 2 * p].val) * irot2.val;
u64 a3 = u64(a[i + offset + 3 * p].val) * irot3.val;
u64 a1na3imag = (a1 + mod2 - a3) % mod * iimag.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);
}
irot *= irate3[topbit(~s & -~s)];
}
len += 2;
}
}
}
}
#line 8 "poly/multipoint.hpp"
template <typename mint>
struct SubproductTree {
int m;
int sz;
vc<vc<mint>> T;
SubproductTree(const vc<mint>& x) {
m = len(x);
sz = 1;
while (sz < m) sz *= 2;
T.resize(2 * sz);
FOR(i, sz) T[sz + i] = {1, (i < m ? -x[i] : 0)};
FOR3_R(i, 1, sz) T[i] = convolution(T[2 * i], T[2 * i + 1]);
}
vc<mint> evaluation(vc<mint> f) {
int n = len(f);
if (n == 0) return vc<mint>(m, mint(0));
f.resize(2 * n - 1);
vc<vc<mint>> g(2 * sz);
g[1] = T[1];
g[1].resize(n);
g[1] = fps_inv(g[1]);
g[1] = middle_product(f, g[1]);
g[1].resize(sz);
FOR3(i, 1, sz) {
g[2 * i] = middle_product(g[i], T[2 * i + 1]);
g[2 * i + 1] = middle_product(g[i], T[2 * i]);
}
vc<mint> vals(m);
FOR(i, m) vals[i] = g[sz + i][0];
return vals;
}
vc<mint> interpolation(vc<mint>& y) {
assert(len(y) == m);
vc<mint> a(m);
FOR(i, m) a[i] = T[1][m - i - 1] * (i + 1);
a = evaluation(a);
vc<vc<mint>> t(2 * sz);
FOR(i, sz) t[sz + i] = {(i < m ? y[i] / a[i] : 0)};
FOR3_R(i, 1, sz) {
t[i] = convolution(t[2 * i], T[2 * i + 1]);
auto tt = convolution(t[2 * i + 1], T[2 * i]);
FOR(k, len(t[i])) t[i][k] += tt[k];
}
t[1].resize(m);
reverse(all(t[1]));
return t[1];
}
};
template <typename mint>
vc<mint> multipoint_evaluation_ntt(vc<mint> f, vc<mint> point) {
using poly = vc<mint>;
int n = 1, k = 0;
while (n < len(point)) n *= 2, ++k;
vv(mint, F, k + 1, 2 * n);
FOR(i, len(point)) F[0][2 * i] = -point[i];
FOR(d, k) {
int b = 1 << d;
for (int L = 0; L < 2 * n; L += 4 * b) {
poly f1 = {F[d].begin() + L, F[d].begin() + L + b};
poly f2 = {F[d].begin() + L + 2 * b, F[d].begin() + L + 3 * b};
ntt_doubling(f1), ntt_doubling(f2);
FOR(i, b) f1[i] += 1, f2[i] += 1;
FOR(i, b, 2 * b) f1[i] -= 1, f2[i] -= 1;
copy(all(f1), F[d].begin() + L);
copy(all(f2), F[d].begin() + L + 2 * b);
FOR(i, 2 * b) { F[d + 1][L + i] = f1[i] * f2[i] - 1; }
}
}
vc<mint> P = {F[k].begin(), F[k].begin() + n};
ntt(P, 1), P.eb(1), reverse(all(P)), P.resize(len(f)), P = fps_inv<mint>(P);
f.resize(n + len(P) - 1), f = middle_product<mint>(f, P), reverse(all(f));
transposed_ntt(f, 1);
vc<mint>& G = f;
FOR_R(d, k) {
vc<mint> nxt_G(n);
int b = 1 << d;
for (int L = 0; L < n; L += 2 * b) {
vc<mint> g1(2 * b), g2(2 * b);
FOR(i, 2 * b) { g1[i] = G[L + i] * F[d][2 * L + 2 * b + i]; }
FOR(i, 2 * b) { g2[i] = G[L + i] * F[d][2 * L + i]; }
ntt_doubling<mint, true>(g1), ntt_doubling<mint, true>(g2);
FOR(i, b) { nxt_G[L + i] = g1[i], nxt_G[L + b + i] = g2[i]; }
}
swap(G, nxt_G);
}
G.resize(len(point));
return G;
}
template <typename mint>
vc<mint> multipoint_eval(vc<mint>& f, vc<mint>& x) {
if (x.empty()) return {};
if (mint::can_ntt()) return multipoint_evaluation_ntt(f, x);
SubproductTree<mint> F(x);
return F.evaluation(f);
}
template <typename mint>
vc<mint> multipoint_interpolate(vc<mint>& x, vc<mint>& y) {
if (x.empty()) return {};
SubproductTree<mint> F(x);
return F.interpolation(y);
}
// calculate f(ar^k) for 0 <= k < m
template <typename mint>
vc<mint> multipoint_eval_on_geom_seq(vc<mint> f, mint a, mint r, int m) {
const int n = len(f);
if (m == 0) return {};
auto eval = [&](mint x) -> mint {
mint fx = 0;
mint pow = 1;
FOR(i, n) fx += f[i] * pow, pow *= x;
return fx;
};
if (r == mint(0)) {
vc<mint> res(m);
FOR(i, 1, m) res[i] = f[0];
res[0] = eval(a);
return res;
}
if (n < 60 || m < 60) {
vc<mint> res(m);
FOR(i, m) res[i] = eval(a), a *= r;
return res;
}
assert(r != mint(0));
// a == 1 に帰着
mint pow_a = 1;
FOR(i, n) f[i] *= pow_a, pow_a *= a;
auto calc = [&](mint r, int m) -> vc<mint> {
// r^{t_i} の計算
vc<mint> res(m);
mint pow = 1;
res[0] = 1;
FOR(i, m - 1) {
res[i + 1] = res[i] * pow;
pow *= r;
}
return res;
};
vc<mint> A = calc(r, n + m - 1), B = calc(r.inverse(), max(n, m));
FOR(i, n) f[i] *= B[i];
f = middle_product(A, f);
FOR(i, m) f[i] *= B[i];
return f;
}
// Y[i] = f(ar^i)
template <typename mint>
vc<mint> multipoint_interpolate_on_geom_seq(vc<mint> Y, mint a, mint r) {
const int n = len(Y);
if (n == 0) return {};
if (n == 1) return {Y[0]};
assert(r != mint(0));
mint ir = r.inverse();
vc<mint> POW(n + n - 1), tPOW(n + n - 1);
POW[0] = tPOW[0] = mint(1);
FOR(i, n + n - 2) POW[i + 1] = POW[i] * r, tPOW[i + 1] = tPOW[i] * POW[i];
vc<mint> iPOW(n + n - 1), itPOW(n + n - 1);
iPOW[0] = itPOW[0] = mint(1);
FOR(i, n) iPOW[i + 1] = iPOW[i] * ir, itPOW[i + 1] = itPOW[i] * iPOW[i];
// prod_[1,i] 1-r^k
vc<mint> S(n);
S[0] = mint(1);
FOR(i, 1, n) S[i] = S[i - 1] * (mint(1) - POW[i]);
vc<mint> iS = all_inverse<mint>(S);
mint sn = S[n - 1] * (mint(1) - POW[n]);
FOR(i, n) {
Y[i] = Y[i] * tPOW[n - 1 - i] * itPOW[n - 1] * iS[i] * iS[n - 1 - i];
if (i % 2 == 1) Y[i] = -Y[i];
}
// sum_i Y[i] / 1-r^ix
FOR(i, n) Y[i] *= itPOW[i];
vc<mint> f = middle_product(tPOW, Y);
FOR(i, n) f[i] *= itPOW[i];
// prod 1-r^ix
vc<mint> g(n);
g[0] = mint(1);
FOR(i, 1, n) {
g[i] = tPOW[i] * sn * iS[i] * iS[n - i];
if (i % 2 == 1) g[i] = -g[i];
}
f = convolution<mint>(f, g);
f.resize(n);
reverse(all(f));
mint ia = a.inverse();
mint pow = 1;
FOR(i, n) f[i] *= pow, pow *= ia;
return f;
}
#line 2 "mod/primitive_root.hpp"
#line 2 "nt/factor.hpp"
#line 2 "random/base.hpp"
u64 RNG_64() {
static u64 x_ = u64(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 2 "mod/mongomery_modint.hpp"
// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
using mint = Mongomery_modint;
inline static U1 m, r, n2;
static constexpr int W = numeric_limits<U1>::digits;
static void set_mod(U1 mod) {
assert(mod & 1 && mod <= U1(1) << (W - 2));
m = mod, n2 = -U2(m) % m, r = m;
FOR(5) r *= 2 - m * r;
r = -r;
assert(r * m == U1(-1));
}
static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }
U1 x;
Mongomery_modint() : x(0) {}
Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
U1 val() const {
U1 y = reduce(x);
return y >= m ? y - m : y;
}
mint &operator+=(mint y) {
x = ((x += y.x) >= m ? x - m : x);
return *this;
}
mint &operator-=(mint y) {
x -= (x >= y.x ? y.x : y.x - m);
return *this;
}
mint &operator*=(mint y) {
x = reduce(U2(x) * y.x);
return *this;
}
mint operator+(mint y) const { return mint(*this) += y; }
mint operator-(mint y) const { return mint(*this) -= y; }
mint operator*(mint y) const { return mint(*this) *= y; }
bool operator==(mint y) const {
return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
}
bool operator!=(mint y) const { return not operator==(y); }
mint pow(ll n) const {
assert(n >= 0);
mint y = 1, z = *this;
for (; n; n >>= 1, z *= z)
if (n & 1) y *= z;
return y;
}
};
template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 3 "nt/primetest.hpp"
bool primetest(const u64 x) {
assert(x < u64(1) << 62);
if (x == 2 or x == 3 or x == 5 or x == 7) return true;
if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
if (x < 121) return x > 1;
const u64 d = (x - 1) >> lowbit(x - 1);
using mint = Mongomery_modint_64<202311020>;
mint::set_mod(x);
const mint one(u64(1)), minus_one(x - 1);
auto ok = [&](u64 a) -> bool {
auto y = mint(a).pow(d);
u64 t = d;
while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1;
if (y != minus_one && t % 2 == 0) return false;
return true;
};
if (x < (u64(1) << 32)) {
for (u64 a: {2, 7, 61})
if (!ok(a)) return false;
} else {
for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (!ok(a)) return false;
}
}
return true;
}
#line 5 "nt/factor.hpp"
template <typename mint>
ll rho(ll n, ll c) {
assert(n > 1);
const mint cc(c);
auto f = [&](mint x) { return x * x + cc; };
mint 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 && g == 1; k += m) {
z = y;
FOR(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;
}
ll find_prime_factor(ll n) {
assert(n > 1);
if (primetest(n)) return n;
FOR(100) {
ll m = 0;
if (n < (1 << 30)) {
using mint = Mongomery_modint_32<20231025>;
mint::set_mod(n);
m = rho<mint>(n, RNG(0, n));
} else {
using mint = Mongomery_modint_64<20231025>;
mint::set_mod(n);
m = rho<mint>(n, RNG(0, n));
}
if (primetest(m)) return m;
n = m;
}
assert(0);
return -1;
}
// ソートしてくれる
vc<pair<ll, int>> factor(ll n) {
assert(n >= 1);
vc<pair<ll, int>> pf;
FOR(p, 2, 100) {
if (p * p > n) break;
if (n % p == 0) {
ll e = 0;
do { n /= p, e += 1; } while (n % p == 0);
pf.eb(p, e);
}
}
while (n > 1) {
ll p = find_prime_factor(n);
ll e = 0;
do { n /= p, e += 1; } while (n % p == 0);
pf.eb(p, e);
}
sort(all(pf));
return pf;
}
vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) {
vc<pair<ll, int>> res;
while (n > 1) {
int p = lpf[n];
int e = 0;
while (n % p == 0) {
n /= p;
++e;
}
res.eb(p, e);
}
return res;
}
#line 2 "mod/mod_pow.hpp"
#line 2 "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); }
};
struct Barrett_64 {
u128 mod, mh, ml;
explicit Barrett_64(u64 mod = 1) : mod(mod) {
u128 m = u128(-1) / mod;
if (m * mod + mod == u128(0)) ++m;
mh = m >> 64;
ml = m & u64(-1);
}
u64 umod() const { return mod; }
u64 modulo(u128 x) {
u128 z = (x & u64(-1)) * ml;
z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64);
z = (x >> 64) * mh + (z >> 64);
x -= z * mod;
return x < mod ? x : x - mod;
}
u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); }
};
#line 5 "mod/mod_pow.hpp"
u32 mod_pow(int a, ll n, int mod) {
assert(n >= 0);
if (mod == 1) return 0;
a = ((a %= mod) < 0 ? a + mod : a);
if ((mod & 1) && (mod < (1 << 30))) {
using mint = Mongomery_modint_32<202311021>;
mint::set_mod(mod);
return mint(a).pow(n).val();
}
Barrett bt(mod);
int r = 1;
while (n) {
if (n & 1) r = bt.mul(r, a);
a = bt.mul(a, a), n >>= 1;
}
return r;
}
u64 mod_pow_64(ll a, ll n, u64 mod) {
assert(n >= 0);
if (mod == 1) return 0;
a = ((a %= mod) < 0 ? a + mod : a);
if ((mod & 1) && (mod < (u64(1) << 62))) {
using mint = Mongomery_modint_64<202311021>;
mint::set_mod(mod);
return mint(a).pow(n).val();
}
Barrett_64 bt(mod);
ll r = 1;
while (n) {
if (n & 1) r = bt.mul(r, a);
a = bt.mul(a, a), n >>= 1;
}
return r;
}
#line 6 "mod/primitive_root.hpp"
// int
int primitive_root(int p) {
auto pf = factor(p - 1);
auto is_ok = [&](int g) -> bool {
for (auto&& [q, e]: pf)
if (mod_pow(g, (p - 1) / q, p) == 1) return false;
return true;
};
while (1) {
int x = RNG(1, p);
if (is_ok(x)) return x;
}
return -1;
}
ll primitive_root_64(ll p) {
auto pf = factor(p - 1);
auto is_ok = [&](ll g) -> bool {
for (auto&& [q, e]: pf)
if (mod_pow_64(g, (p - 1) / q, p) == 1) return false;
return true;
};
while (1) {
ll x = RNG(1, p);
if (is_ok(x)) return x;
}
return -1;
}
// https://codeforces.com/contest/1190/problem/F
ll primitive_root_prime_power_64(ll p, ll e) {
assert(p >= 3);
ll g = primitive_root_64(p);
ll q = p;
ll phi = p - 1;
FOR(e - 1) {
q *= p;
phi *= p;
if (mod_pow_64(g, phi / p, q) == 1) g += q / p;
}
return g;
}
#line 3 "poly/multivar_convolution_cyclic.hpp"
/*
example : ns = (2, 3)
[a0, a1, a2, a3, a4, a5] = [a(0,0), a(1,0), a(0,1), a(1,1), a(0,2), a(1,2)]
[b0, b1, b2, b3, b4, b5] = [b(0,0), b(1,0), b(0,1), b(1,1), b(0,2), b(1,2)]
*/
template <typename mint>
vc<mint> multivar_convolution_cyclic(vc<int> ns, vc<mint> f, vc<mint>& g) {
int p = mint::get_mod();
for (auto&& n: ns) assert((p - 1) % n == 0);
mint r = primitive_root(p);
mint ir = r.inverse();
int K = len(ns);
int N = 1;
for (auto&& n: ns) N *= n;
assert(len(f) == N);
assert(len(g) == N);
vc<mint> root(K), iroot(K);
FOR(k, K) { root[k] = r.pow((p - 1) / ns[k]); }
FOR(k, K) { iroot[k] = ir.pow((p - 1) / ns[k]); }
int step = 1;
FOR(k, K) {
int n = ns[k];
FOR(i, N) if (i % (step * n) < step) {
vc<mint> a(n), b(n);
FOR(j, n) {
a[j] = f[i + step * j];
b[j] = g[i + step * j];
}
a = multipoint_eval_on_geom_seq(a, mint(1), root[k], n);
b = multipoint_eval_on_geom_seq(b, mint(1), root[k], n);
FOR(j, n) {
f[i + step * j] = a[j];
g[i + step * j] = b[j];
}
}
step *= n;
}
FOR(i, N) f[i] *= g[i];
step = 1;
FOR(k, K) {
int n = ns[k];
FOR(i, N) if (i % (step * n) < step) {
vc<mint> a(n);
FOR(j, n) { a[j] = f[i + step * j]; }
a = multipoint_eval_on_geom_seq(a, mint(1), iroot[k], n);
FOR(j, n) { f[i + step * j] = a[j]; }
}
step *= n;
}
mint cf = mint(N).inverse();
for (auto&& x: f) x *= cf;
return f;
}
template <typename mint>
vc<vc<mint>> multivar_convolution_cyclic_2d(vc<vc<mint>>& f, vc<vc<mint>>& g) {
int H = len(f);
int W = len(f[0]);
assert(len(g) == H);
assert(len(g[0]) == W);
vc<mint> F(H * W), G(H * W);
FOR(x, H) FOR(y, W) F[x + H * y] = f[x][y];
FOR(x, H) FOR(y, W) G[x + H * y] = g[x][y];
F = multivar_convolution_cyclic(vc<int>({H, W}), F, G);
vv(mint, h, H, W);
FOR(x, H) FOR(y, W) h[x][y] = F[x + H * y];
return h;
}