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vector<ll> poly_inv(const vector<ll>& a, int n) {
vector<ll> q = {powMod(a[0], mod-2, mod)};
for (int len = 1; len < n; len *= 2){
vector<ll> a2 = a, q2 = q;
a2.resize(2*len), q2.resize(2*len);
ntt(q2);
for (int _ : {0, 1}) {
ntt(a2);
for (int i = 0; i < 2*len; i++) a2[i] = a2[i]*q2[i] % mod;
ntt(a2, true);
for (int i = 0; i < len; i++) a2[i] = 0;
}
for (int i = len; i < min(n, 2*len); i++) {
q.push_back((mod - a2[i]) % mod);
}}
return q;
}
vector<ll> poly_deriv(vector<ll> a) {
for (int i = 1; i < sz(a); i++)
a[i-1] = a[i] * i % mod;
a.pop_back();
return a;
}
vector<ll> poly_integr(vector<ll> a) {
if (a.empty()) return {0};
a.push_back(a.back() * powMod(sz(a), mod-2, mod) % mod);
for (int i = sz(a)-2; i > 0; i--)
a[i] = a[i-1] * powMod(i, mod-2, mod) % mod;
a[0] = 0;
return a;
}
vector<ll> poly_log(vector<ll> a, int n) {
a = mul(poly_deriv(a), poly_inv(a, n));
a.resize(n-1);
a = poly_integr(a);
return a;
}
vector<ll> poly_exp(vector<ll> a, int n) {
vector<ll> q = {1};
for (int len = 1; len < n; len *= 2) {
vector<ll> p = poly_log(q, 2*len);
for (int i = 0; i < 2*len; i++)
p[i] = (mod - p[i] + (i < sz(a) ? a[i] : 0)) % mod;
vector<ll> q2 = q;
q2.resize(2*len);
ntt(p), ntt(q2);
for (int i = 0; i < 2*len; i++) p[i] = p[i] * q2[i] % mod;
ntt(p, true);
for (int i = len; i < min(n, 2*len); i++) q.push_back(p[i]);
}
return q;
}
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