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#include "math/fft/fast-fourier-transform.hpp"
namespace FastFourierTransform {
using real = double;
struct C {
real x, y;
C() : x(0), y(0) {}
C(real x, real y) : x(x), y(y) {}
inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); }
inline C conj() const { return C(x, -y); }
};
const real PI = acosl(-1);
int base = 1;
vector< C > rts = { {0, 0},
{1, 0} };
vector< int > rev = {0, 1};
void ensure_base(int nbase) {
if(nbase <= base) return;
rev.resize(1 << nbase);
rts.resize(1 << nbase);
for(int i = 0; i < (1 << nbase); i++) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
while(base < nbase) {
real angle = PI * 2.0 / (1 << (base + 1));
for(int i = 1 << (base - 1); i < (1 << base); i++) {
rts[i << 1] = rts[i];
real angle_i = angle * (2 * i + 1 - (1 << base));
rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
}
++base;
}
}
void fft(vector< C > &a, int n) {
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for(int i = 0; i < n; i++) {
if(i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for(int k = 1; k < n; k <<= 1) {
for(int i = 0; i < n; i += 2 * k) {
for(int j = 0; j < k; j++) {
C z = a[i + j + k] * rts[j + k];
a[i + j + k] = a[i + j] - z;
a[i + j] = a[i + j] + z;
}
}
}
}
vector< int64_t > multiply(const vector< int > &a, const vector< int > &b) {
int need = (int) a.size() + (int) b.size() - 1;
int nbase = 1;
while((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
vector< C > fa(sz);
for(int i = 0; i < sz; i++) {
int x = (i < (int) a.size() ? a[i] : 0);
int y = (i < (int) b.size() ? b[i] : 0);
fa[i] = C(x, y);
}
fft(fa, sz);
C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
for(int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
fa[i] = z;
}
for(int i = 0; i < (sz >> 1); i++) {
C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
fa[i] = A0 + A1 * s;
}
fft(fa, sz >> 1);
vector< int64_t > ret(need);
for(int i = 0; i < need; i++) {
ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
}
return ret;
}
};
#line 1 "math/fft/fast-fourier-transform.hpp"
namespace FastFourierTransform {
using real = double;
struct C {
real x, y;
C() : x(0), y(0) {}
C(real x, real y) : x(x), y(y) {}
inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); }
inline C conj() const { return C(x, -y); }
};
const real PI = acosl(-1);
int base = 1;
vector< C > rts = { {0, 0},
{1, 0} };
vector< int > rev = {0, 1};
void ensure_base(int nbase) {
if(nbase <= base) return;
rev.resize(1 << nbase);
rts.resize(1 << nbase);
for(int i = 0; i < (1 << nbase); i++) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
while(base < nbase) {
real angle = PI * 2.0 / (1 << (base + 1));
for(int i = 1 << (base - 1); i < (1 << base); i++) {
rts[i << 1] = rts[i];
real angle_i = angle * (2 * i + 1 - (1 << base));
rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
}
++base;
}
}
void fft(vector< C > &a, int n) {
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for(int i = 0; i < n; i++) {
if(i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for(int k = 1; k < n; k <<= 1) {
for(int i = 0; i < n; i += 2 * k) {
for(int j = 0; j < k; j++) {
C z = a[i + j + k] * rts[j + k];
a[i + j + k] = a[i + j] - z;
a[i + j] = a[i + j] + z;
}
}
}
}
vector< int64_t > multiply(const vector< int > &a, const vector< int > &b) {
int need = (int) a.size() + (int) b.size() - 1;
int nbase = 1;
while((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
vector< C > fa(sz);
for(int i = 0; i < sz; i++) {
int x = (i < (int) a.size() ? a[i] : 0);
int y = (i < (int) b.size() ? b[i] : 0);
fa[i] = C(x, y);
}
fft(fa, sz);
C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
for(int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
fa[i] = z;
}
for(int i = 0; i < (sz >> 1); i++) {
C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
fa[i] = A0 + A1 * s;
}
fft(fa, sz >> 1);
vector< int64_t > ret(need);
for(int i = 0; i < need; i++) {
ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
}
return ret;
}
};