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| # include <iostream>
# include <algorithm>
# include <vector>
# include <cassert>
using namespace std;
namespace lyre {
constexpr int P(998'244'353);
constexpr int pow(int a, int n) {
auto r(1);
while (n) {
r = 1l * r * (n & 1 ? a : 1) % P;
n >>= 1, a = 1l * a * a % P;
}
return r;
}
constexpr int inv(int a)
{ return pow(a, P - 2); }
class poly : public vector<int> {
public:
using vector<int>::vector;
vector<int> trans;
inline void expand(size_t n = 0) {
if (!n) n = size();
if (n != (n & -n)) n = 1 << (__lg(n) + 1);
if (n > size()) resize(n);
}
inline void shrink() {
while (!empty() && !back()) pop_back();
}
inline int& operator[](size_t n) {
if (n >= size()) resize(n + 1);
return at(n);
}
inline int operator[](size_t n) const {
if (n >= size()) return 0;
return at(n);
}
poly operator+(poly const &g) const {
auto f(*this);
f.resize(max(size(), g.size()));
for (auto i(0u); i < f.size(); ++i)
f[i] = (f[i] + g[i]) % P;
return f;
}
poly operator-(poly const &g) const {
auto f(*this);
f.resize(max(size(), g.size()));
for (auto i(0u); i < f.size(); ++i)
f[i] = (f[i] + P - g[i]) % P;
return f;
}
poly operator*(int const x) const {
auto f(*this);
for (auto i(0u); i < f.size(); ++i)
f[i] = 1l * f[i] * x % P;
return f;
}
friend poly operator*(int const x, poly const &o) {
auto f(o);
for (auto i(0u); i < f.size(); ++i)
f[i] = 1l * f[i] * x % P;
return f;
}
void init(size_t n) {
assert(n == (n & -n));
if (trans.size() == n) return;
trans = vector<int>(n);
for (auto i(0u); i < n; ++i)
trans[i] = (trans[i >> 1] >> 1) + (i & 1) * (n >> 1);
}
void dif() {
int const n(size());
assert(n == (n & -n));
init(n);
for (int i(0); i < n; ++i)
if (i < trans[i]) std::swap(at(i), at(trans[i]));
for (int mid(1); mid < n; mid *= 2) {
auto w1(lyre::pow(3, (P - 1) / (mid * 2)));
for (int i(0); i < n; i += mid * 2) {
auto wk(1);
for (int j(0), k(mid); j < mid; ++j, ++k) {
auto x(at(i + j));
auto y(1l * wk * at(i + k) % P);
at(i + j) = (x + y) % P;
at(i + k) = (x + P - y) % P;
wk = 1l * wk * w1 % P;
}
}
}
}
void dit() {
int const n(size());
assert(n == (n & -n));
init(n);
for (int i(0); i < n; ++i)
if (i < trans[i]) std::swap(at(i), at(trans[i]));
for (int mid(1); mid < n; mid *= 2) {
auto w1(lyre::pow(lyre::inv(3), (P - 1) / (mid * 2)));
for (int i(0); i < n; i += mid * 2) {
auto wk(1);
for (int j(0), k(mid); j < mid; ++j, ++k) {
auto x(at(i + j));
auto y(1l * wk * at(i + k) % P);
at(i + j) = (x + y) % P;
at(i + k) = (x + P - y) % P;
wk = 1l * wk * w1 % P;
}
}
}
for (int i(0), v(lyre::inv(n)); i < n; ++i)
at(i) = 1l * at(i) * v % P;
}
poly operator*(poly const &o) const {
auto f(*this), g(o);
poly h(f.size() + g.size() - 1);
h.expand();
f.resize(h.size()), f.dif();
g.resize(h.size()), g.dif();
for (auto i(0u); i < h.size(); ++i)
h[i] = 1l * f[i] * g[i] % P;
h.dit();
return h;
}
poly inv(size_t n) const {
poly f(1), g(1);
g[0] = lyre::inv(f[0] = at(0));
for (size_t i(1); i < n; i *= 2) {
f.resize(i * 2);
for (auto j(i); j < min(i * 2, n); ++j)
f[j] = operator[](j);
f.expand(i * 3), f.dif();
g.expand(i * 3), g.dif();
for (size_t j(0); j < g.size(); ++j)
g[j] = (2 + P - 1l * f[j] * g[j] % P) * g[j] % P;
f.dit(), f.resize(i * 2);
g.dit(), g.resize(i * 2);
}
g.resize(n);
return g;
}
poly derivative() const {
poly f;
f.reserve(size());
for (auto i(1u); i < size(); ++i)
f.emplace_back(1l * i * at(i) % P);
return f;
}
poly integral() const {
poly f(1);
f.reserve(size());
for (auto i(0u); i < size(); ++i)
f.emplace_back(1l * at(i) * lyre::inv(i + 1) % P);
return f;
}
poly ln(size_t n) const {
auto f(derivative()), g(inv(n));
auto h((f * g).integral());
h.resize(n);
return h;
}
poly exp(size_t n) const {
poly f(1), g(1), I(1);
I[0] = g[0] = 1;
for (size_t i(1); i < n; i *= 2) {
f.resize(i * 2);
for (auto j(i); j < min(i * 2, n); ++j)
f[j] = operator[](j);
g = g * (I - g.ln(i * 2) + f);
g.resize(i * 2);
}
g.resize(n);
return g;
}
poly pow(size_t n, int k) const {
auto f(*this);
auto ord(0), p(0);
reverse(f.begin(), f.end());
while (!f.empty() && !f.back())
++ord, f.pop_back();
reverse(f.begin(), f.end());
if (f.empty() || 1ul * ord * k > n)
return poly(n);
p = f[0], f = f * lyre::inv(p);
p = lyre::pow(p, k);
auto g((f.ln(n) * k).exp(n) * p);
reverse(g.begin(), g.end());
g.insert(g.end(), ord * k, 0);
reverse(g.begin(), g.end());
g.resize(n);
return g;
}
void print(size_t n = 0) const {
if (!n) n = size();
cerr << "[ ";
for (auto i(0u); i < n; ++i)
cerr << operator[](i) << (i + 1 < n ? ", " : " ]");
cerr << endl;
}
};
}
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