結果
| 問題 |
No.2166 Paint and Fill
|
| コンテスト | |
| ユーザー |
 Nachia
|
| 提出日時 | 2022-12-11 23:26:30 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 4,051 ms / 10,000 ms |
| コード長 | 46,145 bytes |
| コンパイル時間 | 3,277 ms |
| コンパイル使用メモリ | 142,804 KB |
| 最終ジャッジ日時 | 2025-02-09 09:48:45 |
|
ジャッジサーバーID （参考情報） |
judge4 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 38 |
ソースコード
#line 2 "nachia\\fps\\formal-power-series-struct.hpp"
#include <vector>
#include <algorithm>
#include <string>
#include <cassert>
#include <iostream>
#line 3 "nachia\\math-modulo\\modulo-primitive-root.hpp"
#include <utility>
namespace nachia{
template<unsigned int MOD>
struct PrimitiveRoot{
static constexpr unsigned long long powm(unsigned long long a, unsigned long long i) {
unsigned long long res = 1, aa = a;
while(i){
if(i & 1) res = res * aa % MOD;
aa = aa * aa % MOD;
i /= 2;
}
return res;
}
static constexpr bool ExamineVal(unsigned int g){
unsigned int t = MOD - 1;
for(unsigned long long d=2; d*d<=t; d++) if(t % d == 0){
if(powm(g, (MOD - 1) / d) == 1) return false;
while(t % d == 0) t /= d;
}
if(t != 1) if(powm(g, (MOD - 1) / t) == 1) return false;
return true;
}
static constexpr unsigned int GetVal(){
for(unsigned int x=2; x<MOD; x++) if(ExamineVal(x)) return x;
return 0;
}
static const unsigned int val = GetVal();
};
}
#line 3 "nachia\\math\\combination.hpp"
namespace nachia{
template<class Modint>
class Comb{
private:
std::vector<Modint> F;
std::vector<Modint> iF;
public:
void extend(int newN){
int prevN = (int)F.size() - 1;
if(prevN >= newN) return;
F.resize(newN+1);
iF.resize(newN+1);
for(int i=prevN+1; i<=newN; i++) F[i] = F[i-1] * Modint::raw(i);
iF[newN] = F[newN].inv();
for(int i=newN; i>prevN; i--) iF[i-1] = iF[i] * Modint::raw(i);
}
Comb(int n = 1){
F.assign(2, Modint(1));
iF.assign(2, Modint(1));
extend(n);
}
Modint factorial(int n) const { return F[n]; }
Modint invFactorial(int n) const { return iF[n]; }
Modint invOf(int n) const { return iF[n] * F[n-1]; }
Modint comb(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return F[n] * iF[r] * iF[n-r];
}
Modint invComb(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return iF[n] * F[r] * F[n-r];
}
Modint perm(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return F[n] * iF[n-r];
}
Modint invPerm(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return iF[n] * F[n-r];
}
Modint operator()(int n, int r) const { return comb(n,r); }
};
} // namespace nachia
#line 1 "nachia\\fps\\ntt-acl.hpp"
#line 2 "nachia\\fps\\ntt-interface.hpp"
namespace nachia {
template<class mint>
struct NttInterface{
template<class Iter>
void Butterfly(Iter, int) const {}
template<class Iter>
void IButterfly(Iter, int) const {}
template<class Iter>
void BitReversal(Iter a, int N) const {
for(int i=0, j=0; j<N; j++){
if(i < j) std::swap(a[i], a[j]);
for(int k = N>>1; k > (i^=k); k>>=1);
}
}
};
} // namespace nachia
#line 1 "nachia\\misc\\bit-operations.hpp"
#line 4 "nachia\\misc\\bit-operations.hpp"
namespace nachia{
int Popcount(unsigned long long c) noexcept {
#ifdef __GNUC__
return __builtin_popcountll(c);
#else
c = (c & (~0ull/3)) + ((c >> 1) & (~0ull/3));
c = (c & (~0ull/5)) + ((c >> 2) & (~0ull/5));
c = (c & (~0ull/17)) + ((c >> 4) & (~0ull/17));
c = (c * (~0ull/257)) >> 56;
return c;
#endif
}
// please ensure x != 0
int MsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
return 63 - __builtin_clzll(x);
#else
int res = 0;
for(int d=32; d>=0; d>>=1) if(x >> d){ res |= d; x >>= d; }
return res;
#endif
}
// please ensure x != 0
int LsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
return __builtin_ctzll(x);
#else
return msb_idx(x & -x);
#endif
}
}
#line 5 "nachia\\fps\\ntt-acl.hpp"
#include <iterator>
#line 8 "nachia\\fps\\ntt-acl.hpp"
#include <array>
namespace nachia{
constexpr int bsf_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
template <class mint>
struct NttFromAcl : NttInterface<mint> {
using u32 = unsigned int;
using u64 = unsigned long long;
static int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (u32)(n)) x++;
return x;
}
struct fft_info {
static constexpr u32 g = nachia::PrimitiveRoot<mint::mod()>::val;
static constexpr int rank2 = bsf_constexpr(mint::mod()-1);
std::array<mint, rank2+1> root;
std::array<mint, rank2+1> iroot;
std::array<mint, std::max(0, rank2-1)> rate2;
std::array<mint, std::max(0, rank2-1)> irate2;
std::array<mint, std::max(0, rank2-2)> rate3;
std::array<mint, std::max(0, rank2-2)> irate3;
fft_info(){
root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for(int i=rank2-1; i>=0; i--){
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];
}
}
};
template<class RandomAccessIterator>
void Butterfly(RandomAccessIterator a, int n) const {
int h = ceil_pow2(n);
static const fft_info info;
int len = 0;
while(len < h){
if(h-len == 1){
int p = 1 << (h-len-1);
mint rot = 1;
for(int s=0; s<(1<<len); s++){
int offset = s << (h-len);
for(int i=0; i<p; i++){
auto l = a[i+offset];
auto r = a[i+offset+p] * rot;
a[i+offset] = l+r;
a[i+offset+p] = l-r;
}
if(s+1 != (1<<len)) rot *= info.rate2[LsbIndex(~(u32)(s))];
}
len++;
} else {
int p = 1 << (h-len-2);
mint rot = 1, imag = info.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++){
auto mod2 = 1ULL * mint::mod() * mint::mod();
auto a0 = 1ULL * a[i+offset].val();
auto a1 = 1ULL * a[i+offset+p].val() * rot.val();
auto a2 = 1ULL * a[i+offset+2*p].val() * rot2.val();
auto a3 = 1ULL * a[i+offset+3*p].val() * rot3.val();
auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto 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);
}
if(s+1 != (1<<len)) rot *= info.rate3[LsbIndex(~(u32)(s))];
}
len += 2;
}
}
}
template<class RandomAccessIterator>
void IButterfly(RandomAccessIterator a, int n) const {
int h = ceil_pow2(n);
static const fft_info info;
constexpr int MOD = mint::mod();
int len = h;
while(len){
if(len == 1){
int p = 1 << (h-len);
mint irot = 1;
for(int s=0; s<(1<<(len-1)); s++){
int offset = s << (h-len+1);
for(int i=0; i<p; i++){
auto l = a[i+offset];
auto r = a[i+offset+p];
a[i+offset] = l+r;
a[i+offset+p] = (u64)(MOD + l.val() - r.val()) * irot.val();
}
if(s+1 != (1<<(len-1))) irot *= info.irate2[LsbIndex(~(u32)(s))];
}
len--;
} else {
int p = 1 << (h-len);
mint irot = 1, iimag = info.iroot[2];
for(int s=0; s<(1<<(len-2)); s++){
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h-len+2);
for(int i=0; i<p; i++){
auto a0 = 1ULL * a[i+offset+0*p].val();
auto a1 = 1ULL * a[i+offset+1*p].val();
auto a2 = 1ULL * a[i+offset+2*p].val();
auto a3 = 1ULL * a[i+offset+3*p].val();
auto a2na3iimag = 1ULL * mint((MOD + a2 - a3) * iimag.val()).val();
a[i+offset] = a0 + a1 + a2 + a3;
a[i+offset+1*p] = (a0 + (MOD - a1) + a2na3iimag) * irot.val();
a[i+offset+2*p] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val();
a[i+offset+3*p] = (a0 + (MOD - a1) + (MOD - a2na3iimag)) * irot3.val();
}
if(s+1 != (1<<(len-2))) irot *= info.irate3[LsbIndex(~(u32)(s))];
}
len -= 2;
}
}
}
};
} // namespace nachia
#line 10 "nachia\\fps\\formal-power-series-struct.hpp"
namespace nachia {
template<class Elem, class NttInst = NttFromAcl<Elem>>
struct FormalPowerSeriesNTT {
public:
using MyType = FormalPowerSeriesNTT;
static constexpr unsigned int MOD = Elem::mod();
static const NttInst nttInst;
private:
using u32 = unsigned int;
static const u32 zeta = nachia::PrimitiveRoot<MOD>::GetVal();
static Elem ZeroElem() noexcept { return Elem(0); }
static Elem OneElem() noexcept { return Elem(1); }
static Comb<Elem> comb;
std::vector<Elem> a;
public:
unsigned int size() const noexcept { return a.size(); }
Elem& operator[](unsigned int x) noexcept { return a[x]; }
const Elem& operator[](unsigned int x) const noexcept { return a[x]; }
Elem get_coeff(unsigned int x) const{ return (x < size()) ? a[x] : ZeroElem(); }
static Comb<Elem>& GetComb() { return comb; }
MyType& removeLeadingZeros(){
unsigned int newsz = size();
while(newsz && a[newsz-1].val() == 0) newsz--;
a.resize(newsz);
if(a.capacity() / 4 > newsz) a.shrink_to_fit();
return *this;
}
FormalPowerSeriesNTT(){ a = { }; }
FormalPowerSeriesNTT(unsigned int new_size) : a(new_size, ZeroElem()) {}
FormalPowerSeriesNTT(std::vector<Elem>&& src) : a(std::move(src)) {}
FormalPowerSeriesNTT(const std::vector<Elem>& src) : a(src) {}
MyType& ntt() {
int N = 1; while (N < (int)size()) N *= 2;
a.resize(N, ZeroElem());
nttInst.Butterfly(a.begin(), N);
return *this;
}
MyType& intt() {
nttInst.IButterfly(a.begin(), a.size());
Elem invN = Elem(size()).inv();
for(unsigned int i=0; i<size(); i++) a[i] *= invN;
return *this;
}
// returns [ a[l], a[l+1], a[l+2], ... , a[r-1] ]
// a[i] = 0 ( i < 0 OR size() <= i )
MyType getSlice(int l, int r) const {
if(l >= r) return MyType();
MyType res(r - l);
for(int i=l; i<r; i++) res[i-l] = (0 <= i && i < (int)size()) ? a[i] : ZeroElem();
return res;
}
// upper < 0 -> upper = lower
MyType& capSize(int lower, int upper = -1) {
if(upper < 0) upper = lower;
if(upper <= (int)size()) a.resize(upper);
if((int)size() <= lower) a.resize(lower, ZeroElem());
return *this;
}
static MyType convolution(const MyType& a, const MyType& b){
if(a.size() <= 30 || b.size() <= 30){
if(a.size() > 30) return convolution(b,a);
if(a.size() == 0 || b.size() == 0) return std::vector<Elem>{};
std::vector<Elem> res(a.size() + b.size() - 1);
for(std::size_t i=0; i<a.size(); i++){
for(std::size_t j=0; j<b.size(); j++) res[i+j] += a[i] * b[j];
}
return res;
}
int z = a.size() + b.size() - 1;
int Z = 1; while(Z < z) Z *= 2;
MyType ax = a.getSlice(0, Z);
MyType bx = b.getSlice(0, Z);
ax.ntt();
bx.ntt();
for(int i=0; i<Z; i++) ax[i] *= bx[i];
ax.intt();
return ax.getSlice(0, z);
}
static MyType back_half_convolution(unsigned int sz, const MyType& smaller, const MyType& larger){
assert(smaller.size() <= sz);
assert(larger.size() <= sz*2);
if(sz <= 5) return convolution(smaller, larger).getSlice(sz, sz*2);
int z = sz*2;
int Z = 1; while(Z < z) Z *= 2;
MyType ax = smaller.getSlice(0, Z).ntt();
MyType bx = larger.getSlice(0, Z).ntt();
for(int i=0; i<Z; i++) ax[i] *= bx[i];
ax.intt();
return ax.getSlice(sz, sz*2);
}
// 1
// ----- = 1 + f + f^2 + f^3 + ...
// 1-f
MyType power_sum(unsigned int sz){
if (sz == 0) { return {}; }
if (sz == 1) { return MyType(std::vector<Elem>{ OneElem() }); }
if(sz <= 30){
auto a = getSlice(0, sz);
std::vector<Elem> res(sz); res[0] = OneElem();
for(u32 i=1; i<sz; i++) for(u32 j=1; j<=i; j++) res[i] += res[i-j] * a[j];
return res;
}
u32 N = 1; while (N < sz) N *= 2;
u32 hN = N / 2;
MyType hInv = power_sum(hN);
MyType tgA = getSlice(0, N).ntt();
MyType htInv = hInv.getSlice(0, N);
htInv.ntt();
MyType R = MyType(N);
for(u32 i=0; i<N; i++) R[i] = tgA[i] * htInv[i];
R = R.intt().getSlice(hN, N + hN).ntt();
for(u32 i=0; i<N; i++) R[i] *= htInv[i];
R.intt();
hInv.capSize(sz, sz);
for(u32 i=hN; i<sz; i++) hInv[i] = R[i - hN];
return hInv;
}
MyType inv(unsigned int sz){
Elem iA0 = a[0].inv();
MyType xA(std::min(sz, size()));
for(u32 i=0; i<xA.size(); i++) xA[i] = -a[i] * iA0;
xA[0] = 0;
xA = xA.power_sum(sz);
for(u32 i=0; i<xA.size(); i++) xA[i] *= iA0;
return xA;
}
MyType& difference(){
if(size() == 0) return *this;
for(u32 i=0; i+1<size(); i++) a[i] = a[i+1] * Elem(i+1);
capSize(0, size() - 1);
return *this;
}
MyType& integral(){
if(size() == 0){
a.push_back(ZeroElem());
return *this;
}
capSize(size()+1);
comb.extend(size());
for(u32 i=size()-1; i>=1; i--) a[i] = a[i-1] * Elem(comb.invOf(i));
a[0] = ZeroElem();
return *this;
}
MyType copied() const { return MyType(*this); }
MyType log(unsigned int sz){
assert(sz != 0);
assert(a[0].val() == 1);
return convolution(inv(sz), copied().difference()).capSize(sz-1,sz-1).integral();
}
MyType exp(unsigned int sz){
MyType res = MyType(std::vector<Elem>{ OneElem() });
while(res.size() < sz){
auto z = res.size();
auto tmp = res.log(z*2);
tmp[0] = -OneElem();
for(u32 i=0; i<z*2 && i<size(); i++) tmp[i] = a[i] - tmp[i];
tmp = back_half_convolution(z, res, tmp);
res.capSize(std::min(sz, z*2), z);
for(u32 i=z; i<res.size(); i++) res[i] = tmp[i-z];
}
return res;
}
MyType& reverse(){ std::reverse(a.begin(), a.end()); return *this; }
MyType pow(unsigned long long k){
int n = size();
if(k == 0){
auto res = MyType(n);
res[0] = 1;
return res;
}
int ctz = 0;
for(int i=0; i<n; i++) if(a[i].val() == 0) ctz = i+1; else break;
if((unsigned long long)ctz >= (n-1) / k + 1) return MyType(n);
auto res = *this;
for(int i=0; i<n-ctz; i++) res[i] = res[i+ctz];
Elem A0 = res[0], iA0 = A0.inv(), pA0 = A0.pow(k);
for(int i=0; i<n; i++) res[i] *= iA0;
res = res.log(n);
for(int i=0; i<n; i++) res[i] *= k;
res = res.exp(n);
for(int i=0; i<n; i++) res[i] *= pA0;
ctz *= k;
for(int i=n-1; i>=ctz; i--) res[i] = res[i-ctz];
for(int i=0; i<ctz; i++) res[i] = 0;
return res;
}
auto begin(){ return a.begin(); }
auto end(){ return a.end(); }
auto begin() const { return a.begin(); }
auto end() const { return a.end(); }
std::string to_string() const {
std::string res = "[";
for(auto x : a){ res += " "; res += std::to_string(*x); }
res += " ]";
return res;
}
std::vector<Elem> get_vector_moved(){
std::vector<Elem> res = std::move(a);
a.clear();
return std::move(a);
}
MyType ax_plus_b(Elem a, Elem b) const {
auto buf = MyType(size() + 1);
for(u32 i=0; i<size(); i++) buf[i] += this->a[i] * b;
for(u32 i=0; i<size(); i++) buf[i+1] += this->a[i] * a;
return buf;
}
MyType operator+(const MyType& r) const {
auto sz = std::max(this->size(), r.size());
MyType res(sz);
for(u32 i=0; i<this->size(); i++) res[i] += this->operator[](i);
for(u32 i=0; i<r.size(); i++) res[i] += r[i];
return res;
}
MyType operator-(const MyType& r) const {
auto sz = std::max(this->size(), r.size());
MyType res(sz);
for(u32 i=0; i<this->size(); i++) res[i] += this->operator[](i);
for(u32 i=0; i<r.size(); i++) res[i] -= r[i];
return res;
}
MyType operator*(const MyType& r) const { return convolution(*this, r); }
MyType& operator*=(const MyType& r){ (*this) = (*this) * r; return *this; }
MyType& operator*=(Elem m){ for(size_t i=0; i<a.size(); i++) a[i] *= m; return *this; }
MyType operator*(Elem m) const { MyType b = *this; b *= m; return b; }
Elem eval(Elem x) const {
int z = size();
Elem res = 0;
for(int i=z-1; i>=0; i--) res = res * x + a[i];
return res;
}
};
template<class Elem, class NttInst> Comb<Elem> FormalPowerSeriesNTT<Elem, NttInst>::comb;
template<class Elem, class NttInst> const NttInst FormalPowerSeriesNTT<Elem, NttInst>::nttInst;
} // namespace nachia
#line 2 "nachia\\math\\ext-gcd.hpp"
#line 6 "nachia\\math\\ext-gcd.hpp"
namespace nachia{
// ax + by = gcd(a,b)
std::pair<long long, long long> ExtGcd(long long a, long long b){
long long x = 1, y = 0;
while(b){
long long u = a / b;
std::swap(a-=b*u, b);
std::swap(x-=y*u, y);
}
return std::make_pair(x, y);
}
} // namespace nachia
#line 5 "nachia\\math-modulo\\static-modint.hpp"
namespace nachia{
template<unsigned int MOD>
struct StaticModint{
private:
using u64 = unsigned long long;
unsigned int x;
public:
using my_type = StaticModint;
template< class Elem >
static Elem safe_mod(Elem x){
if(x < 0){
if(0 <= x+MOD) return x + MOD;
return MOD - ((-(x+MOD)-1) % MOD + 1);
}
return x % MOD;
}
StaticModint() : x(0){}
StaticModint(const my_type& a) : x(a.x){}
StaticModint& operator=(const my_type&) = default;
template< class Elem >
StaticModint(Elem v) : x(safe_mod(v)){}
unsigned int operator*() const noexcept { return x; }
my_type& operator+=(const my_type& r) noexcept { auto t = x + r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
my_type operator+(const my_type& r) const noexcept { my_type res = *this; return res += r; }
my_type& operator-=(const my_type& r) noexcept { auto t = x + MOD - r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
my_type operator-(const my_type& r) const noexcept { my_type res = *this; return res -= r; }
my_type operator-() const noexcept { my_type res = *this; res.x = ((res.x == 0) ? 0 : (MOD - res.x)); return res; }
my_type& operator*=(const my_type& r)noexcept { x = (u64)x * r.x % MOD; return *this; }
my_type operator*(const my_type& r) const noexcept { my_type res = *this; return res *= r; }
my_type pow(unsigned long long i) const noexcept {
my_type a = *this, res = 1;
while(i){ if(i & 1){ res *= a; } a *= a; i >>= 1; }
return res;
}
my_type inv() const { return my_type(ExtGcd(x, MOD).first); }
unsigned int val() const noexcept { return x; }
static constexpr unsigned int mod() { return MOD; }
static my_type raw(unsigned int val) noexcept { auto res = my_type(); res.x = val; return res; }
my_type& operator/=(const my_type& r){ return operator*=(r.inv()); }
my_type operator/(const my_type& r) const { return operator*(r.inv()); }
};
}
#line 4 "nachia\\fps\\shift-of-sampling-points.hpp"
namespace nachia {
template <class Elem>
std::vector<Elem> ShiftOfSamplingPointsOfPolynomial(std::vector<Elem> points, Elem sh, int count=-1){
using Fps = FormalPowerSeriesNTT<Elem>;
int n = points.size();
int m = (count < 0) ? n : count;
if(m == 0){ return {}; }
if(n == 0){ return std::vector<Elem>(m); }
int z = std::max(n, m);
Fps iF(z);
Fps F(z);
F[0] = 1;
for(int i=1; i<z; i++) F[i] = F[i-1] * Elem::raw(i);
iF[z-1] = F[z-1].inv();
for(int i=z-1; i>=1; i--) iF[i-1] = iF[i] * Elem::raw(i);
Fps P(n);
for(int i=0; i<n; i++) P[i] = points[i] * iF[i];
Fps iFI(n);
for(int i=0; i<n; i++) iFI[i] = (i%2) ? -iF[i] : iF[i];
P = (P * iFI).capSize(n);
Elem q = 1;
for(int i=1; i<n; i++) iFI[i] = iF[i] * (q *= (sh - Elem::raw(i-1)));
for(int i=0; i<n; i++) P[i] *= F[i];
P.reverse();
P = (P * iFI).capSize(n);
P.reverse();
for(int i=0; i<n; i++) P[i] *= iF[i];
P = P * iF;
std::vector<Elem> res(m);
res[0] = P[0];
for(int i=1; i<m; i++) res[i] = P[i] * F[i];
return res;
}
} // namespace nachia
#line 5 "nachia\\linear\\simple-matrix.hpp"
namespace nachia{
template<class Elem>
struct SimpleMatrix{
private:
int h;
int w;
std::vector<Elem> elems;
public:
SimpleMatrix(int new_h=0, int new_w=0){ h = new_h; w = new_w; elems.assign(h * w, 0); }
SimpleMatrix(SimpleMatrix const&) = default;
int numRow() const { return h; }
int numColumn() const { return w; }
int height() const { return numRow(); }
int width() const { return numColumn(); }
typename std::vector<Elem>::iterator operator[](int y){ return elems.begin() + (y*w); }
typename std::vector<Elem>::const_iterator operator[](int y) const { return elems.begin() + (y*w); }
static SimpleMatrix Identity(int idx, Elem One){ auto res = SimpleMatrix(idx, idx); for(int i=0; i<idx; i++) res[i][i] = One; return res; }
void swapColumns(int x1, int x2){
assert(0 <= x1 && x1 < numColumn());
assert(0 <= x2 && x2 < numColumn());
for(int y=0; y<numRow(); y++) std::swap((*this)[y][x1], (*this)[y][x2]);
}
void swapRows(int y1, int y2){
assert(0 <= y1 && y1 < numRow());
assert(0 <= y2 && y2 < numRow());
for(int x=0; x<numColumn(); x++) std::swap((*this)[y1][x], (*this)[y2][x]);
}
SimpleMatrix operator*(const SimpleMatrix& r) const {
assert(width() == r.height());
auto res = SimpleMatrix(h, r.w);
for(int i=0; i<h; i++) for(int j=0; j<w; j++) for(int k=0; k<r.w; k++) res[i][k] = res[i][k] + (*this)[i][j] * r[j][k];
return res;
}
SimpleMatrix pow(unsigned long long i){
auto a = *this;
auto res = Identity(height());
while(i){
if(i % 2 == 1) res = res * a;
a = a * a;
i /= 2;
}
return res;
}
};
} // namespace nachia
#line 5 "nachia\\fps\\p-recursive-matrix-product.hpp"
namespace nachia{
template<class Elem>
SimpleMatrix<Elem> PRecursiveMatrixProduct(
SimpleMatrix<FormalPowerSeriesNTT<Elem>> p,
unsigned long long idx
){
using u64 = unsigned long long;
int h = p.height();
std::vector<std::vector<std::vector<Elem>>> res;
res.resize(h);
for(auto& a : res) a.resize(h);
u64 a = 1, b = 1;
for(int i=0; i<h; i++) for(int j=0; j<h; j++) while(b < p[i][j].size()) b <<= 1;
u64 maxA = 1, maxB = b;
while(maxA * maxB <= idx){ maxB <<= 1; maxA <<= 1; }
for(int i=0; i<h; i++) for(int j=0; j<h; j++){
res[i][j].resize(b);
for(u64 k=0; k<b; k++) res[i][j][k] = p[i][j].eval(Elem(maxA) * Elem(k));
}
auto ExtendA = [&](){
for(int i=0; i<h; i++) for(int j=0; j<h; j++){
auto tmp = ShiftOfSamplingPointsOfPolynomial<Elem>(res[i][j], Elem(b));
std::copy(tmp.begin(), tmp.end(), std::back_inserter(res[i][j]));
}
b *= 2;
};
auto ExtendB = [&](){
std::vector<std::vector<std::vector<Elem>>> resbuf;
resbuf.assign(h, std::vector<std::vector<Elem>>(h, std::vector<Elem>(b)));
for(int i=0; i<h; i++) for(int j=0; j<h; j++){
auto hpq = ShiftOfSamplingPointsOfPolynomial<Elem>(res[i][j], Elem(a) / Elem(maxA));
for(int k=0; k<h; k++){
auto resbeg = res[j][k].begin();
auto destbeg = resbuf[i][k].begin();
for(u64 id=0; id<b; id++) destbeg[id] += hpq[id] * resbeg[id];
}
}
std::swap(res, resbuf);
a *= 2;
};
auto EvalP = [p, h](Elem val) -> SimpleMatrix<Elem> {
SimpleMatrix<Elem> res(h, h);
for(int y=0; y<h; y++) for(int x=0; x<h; x++) res[y][x] = p[y][x].eval(val);
return res;
};
auto EvalL = [&res, h](u64 idx) -> SimpleMatrix<Elem> {
SimpleMatrix<Elem> g(h, h);
for(int y=0; y<h; y++) for(int x=0; x<h; x++) g[y][x] = res[y][x][idx];
return g;
};
while(b < maxB){ ExtendA(); ExtendB(); }
u64 pos = 0;
SimpleMatrix<Elem> ans = SimpleMatrix<Elem>::Identity(h, Elem::raw(1));
while(pos + maxA <= idx){ ans = EvalL(pos / maxA) * ans; pos += maxA; }
while(pos < idx){ ans = EvalP(pos++) * ans; }
return ans;
}
} // namespace nachia
#line 4 "nachia\\fps\\polinomial-division.hpp"
namespace nachia{
// return polynomials have no leading zeros
template<class Elem>
std::pair<FormalPowerSeriesNTT<Elem>, FormalPowerSeriesNTT<Elem>> PolynomialDivision(
FormalPowerSeriesNTT<Elem> A,
FormalPowerSeriesNTT<Elem> D,
bool do_get_remainder = true
){
using Fps = FormalPowerSeriesNTT<Elem>;
auto dsize = D.size();
while(dsize != 0 && D[dsize-1].val() == 0) dsize--;
assert(dsize != 0);
if(A.size() == 0){ return std::make_pair(Fps(), Fps()); }
if(A.size() < dsize){ return std::make_pair(Fps(), std::move(A)); }
std::reverse(D.begin(), D.begin() + dsize);
std::reverse(A.begin(), A.end());
int n = A.size();
unsigned int divSize = n - dsize + 1;
auto invD = D.inv(divSize);
auto tmp = (A.getSlice(0, divSize) * invD).getSlice(0, divSize);
Fps ans1(divSize);
for(unsigned int i=0; i<divSize; i++) ans1[divSize-1-i] = tmp[i];
if(!do_get_remainder || dsize == 1) return std::make_pair(std::move(ans1), Fps());
tmp = tmp * D;
Fps ans2(dsize - 1);
for(unsigned int i=0; i<dsize-1; i++) ans2[i] = A[n-1-i] - tmp[n-1-i];
auto ans2sz = ans2.size();
while(ans2sz != 0 && ans2[ans2sz-1].val() == 0) ans2sz--;
return std::make_pair(std::move(ans1), ans2.getSlice(0, ans2sz));
}
} // namespace nachia
#line 1 "nachia\\atcoder\\static_modint.hpp"
#line 1 "nachia\\atcoder\\internal_modint_base.hpp"
#line 1 "nachia\\atcoder\\internal_type_traits.hpp"
#line 5 "nachia\\atcoder\\internal_type_traits.hpp"
#include <numeric>
#include <type_traits>
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<
std::is_same<T, __int128_t>::value || std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<
std::is_same<T, __uint128_t>::value || std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<
std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<
std::is_integral<T>::value || is_signed_int128<T>::value || is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<
(is_integral<T>::value && std::is_signed<T>::value) || is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<
(is_integral<T>::value && std::is_unsigned<T>::value) || is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<
std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<
is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<
is_integral<T>::value && std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
#line 7 "nachia\\atcoder\\internal_modint_base.hpp"
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
} // namespace atcoder
#line 1 "nachia\\atcoder\\internal_math.hpp"
#line 5 "nachia\\atcoder\\internal_math.hpp"
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m){
x %= m;
if(x < 0) x += m;
return x;
}
// Fast moduler by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
using u64 = unsigned long long;
unsigned int _m;
u64 im;
// @param m `1 <= m`
barrett(unsigned int m) : _m(m), im((u64)(-1) / m + 1){}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
u64 z = a;
z *= b;
#ifdef _MSC_VER
u64 x;
_umul128(z, im, &x);
#else
u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if(_m <= v) v += _m;
return v;
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m){
if(m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1, y = safe_mod(x, m);
while(n){
if(n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n){
if(n <= 1) return false;
if(n == 2 || n == 7 || n == 61) return true;
if(n % 2 == 0) return false;
long long d = n - 1;
while(d % 2 == 0) d /= 2;
for(long long a : {2, 7, 61}){
long long t = d, y = pow_mod_constexpr(a, t, n);
while(t != n - 1 && y != 1 && y != n - 1){
y = y * y % n;
t <<= 1;
}
if(y != n - 1 && t % 2 == 0) return false;
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b){
a = safe_mod(a, b);
if(a == 0) return {b, 0};
long long s = b, t = a, m0 = 0, m1 = 1;
while(t){
long long u = s / t;
s -= t * u;
m0 -= m1 * u;
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
if(m0 < 0) m0 += b / s;
return {s, m0};
}
// @param m must be prime
constexpr int primitive_root_constexpr(int m){
if(m == 2) return 1;
if(m == 167772161) return 3;
if(m == 469762049) return 3;
if(m == 754974721) return 11;
if(m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m-1) / 2;
while(x%2 == 0) x /= 2;
for(int i=3; (long long)(i)*i <= x; i += 2){
if(x % i == 0){
divs[cnt++] = i;
while(x % i == 0) x /= i;
}
}
if(x>1) divs[cnt++] = x;
for(int g=2; ; g++){
bool ok = true;
for(int i=0; i<cnt; i++){
if(pow_mod_constexpr(g, (m-1) / divs[i], m) == 1){
ok = false;
break;
}
}
if(ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
} // namespace internal
} // namespace atcoder
#line 10 "nachia\\atcoder\\static_modint.hpp"
namespace atcoder {
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod(){ return m; }
static mint raw(int v){
mint x;
x.w = v;
return x;
}
static_modint() : w(0){}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v){
long long x = (long long)(v % (long long)(umod()));
if(x < 0) x += umod();
w = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v){
w = (unsigned int)(v % umod());
}
static_modint(bool v){ w = ((unsigned int)(v) % umod()); }
unsigned int val() const { return w; }
mint& operator++(){
w++;
if(w == umod()) w = 0;
return *this;
}
mint& operator--(){
if(w == 0) w = umod();
w--;
return *this;
}
mint operator++(int){
mint result = *this;
++*this;
return result;
}
mint operator--(int){
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs){
w += rhs.w;
if(w >= umod()) w -= umod();
return *this;
}
mint& operator-=(const mint& rhs){
w -= rhs.w;
if(w >= umod()) w += umod();
return *this;
}
mint& operator*=(const mint& rhs){
unsigned long long z = w;
z *= rhs.w;
w = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs){ return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while(n){
if(n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if(prime){
assert(w);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(w, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs){ return mint(lhs) += rhs; }
friend mint operator-(const mint& lhs, const mint& rhs){ return mint(lhs) -= rhs; }
friend mint operator*(const mint& lhs, const mint& rhs){ return mint(lhs) *= rhs; }
friend mint operator/(const mint& lhs, const mint& rhs){ return mint(lhs) /= rhs; }
friend bool operator==(const mint& lhs, const mint& rhs){ return lhs.w == rhs.w; }
friend bool operator!=(const mint& lhs, const mint& rhs){ return lhs.w != rhs.w; }
private:
unsigned int w;
static constexpr unsigned int umod(){ return m; }
static constexpr bool prime = internal::is_prime<m>;
};
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
} // namespace internal
} // namespace atcoder
#line 2 "nachia\\misc\\fastio.hpp"
#include <cstdio>
#include <cctype>
#include <cstdint>
#line 6 "nachia\\misc\\fastio.hpp"
namespace nachia{
struct CInStream{
private:
static const unsigned int INPUT_BUF_SIZE = 1 << 17;
unsigned int p = INPUT_BUF_SIZE;
static char Q[INPUT_BUF_SIZE];
public:
using MyType = CInStream;
char seekChar(){
if(p == INPUT_BUF_SIZE){
size_t len = fread(Q, 1, INPUT_BUF_SIZE, stdin);
if(len != INPUT_BUF_SIZE) Q[len] = '\0';
p = 0;
}
return Q[p];
}
void skipSpace(){ while(isspace(seekChar())) p++; }
uint32_t nextU32(){
skipSpace();
uint32_t buf = 0;
while(true){
char tmp = seekChar();
if('9' < tmp || tmp < '0') break;
buf = buf * 10 + (tmp - '0');
p++;
}
return buf;
}
int32_t nextI32(){
skipSpace();
if(seekChar() == '-'){ p++; return (int32_t)(-nextU32()); }
return (int32_t)nextU32();
}
uint64_t nextU64(){
skipSpace();
uint64_t buf = 0;
while(true){
char tmp = seekChar();
if('9' < tmp || tmp < '0') break;
buf = buf * 10 + (tmp - '0');
p++;
}
return buf;
}
int64_t nextI64(){
skipSpace();
if(seekChar() == '-'){ p++; return (int64_t)(-nextU64()); }
return (int64_t)nextU64();
}
char nextChar(){ skipSpace(); char buf = seekChar(); p++; return buf; }
std::string nextToken(){
skipSpace();
std::string buf;
while(true){
char ch = seekChar();
if(isspace(ch) || ch == '\0') break;
buf.push_back(ch);
p++;
}
return buf;
}
MyType& operator>>(unsigned int& dest){ dest = nextU32(); return *this; }
MyType& operator>>(int& dest){ dest = nextI32(); return *this; }
MyType& operator>>(unsigned long& dest){ dest = nextU64(); return *this; }
MyType& operator>>(long& dest){ dest = nextI64(); return *this; }
MyType& operator>>(unsigned long long& dest){ dest = nextU64(); return *this; }
MyType& operator>>(long long& dest){ dest = nextI64(); return *this; }
MyType& operator>>(std::string& dest){ dest = nextToken(); return *this; }
MyType& operator>>(char& dest){ dest = nextChar(); return *this; }
} cin;
struct FastOutputTable{
char LZ[1000][4] = {};
char NLZ[1000][4] = {};
constexpr FastOutputTable(){
using u32 = uint_fast32_t;
for(u32 d=0; d<1000; d++){
LZ[d][0] = ('0' + d / 100 % 10);
LZ[d][1] = ('0' + d / 10 % 10);
LZ[d][2] = ('0' + d / 1 % 10);
LZ[d][3] = '\0';
}
for(u32 d=0; d<1000; d++){
u32 i = 0;
if(d >= 100) NLZ[d][i++] = ('0' + d / 100 % 10);
if(d >= 10) NLZ[d][i++] = ('0' + d / 10 % 10);
if(d >= 1) NLZ[d][i++] = ('0' + d / 1 % 10);
NLZ[d][i++] = '\0';
}
}
};
struct COutStream{
private:
using u32 = uint32_t;
using u64 = uint64_t;
using MyType = COutStream;
static const u32 OUTPUT_BUF_SIZE = 1 << 17;
static char Q[OUTPUT_BUF_SIZE];
static constexpr FastOutputTable TB = FastOutputTable();
u32 p = 0;
static constexpr u32 P10(u32 d){ return d ? P10(d-1)*10 : 1; }
static constexpr u64 P10L(u32 d){ return d ? P10L(d-1)*10 : 1; }
template<class T, class U> static void Fil(T& m, U& l, U x) noexcept { m = l/x; l -= m*x; }
void next_dig9(u32 x){
u32 y;
Fil(y, x, P10(6));
nextCstr(TB.LZ[y]);
Fil(y, x, P10(3));
nextCstr(TB.LZ[y]); nextCstr(TB.LZ[x]);
}
public:
void nextChar(char c){
Q[p++] = c;
if(p == OUTPUT_BUF_SIZE){ fwrite(Q, p, 1, stdout); p = 0; }
}
void nextEoln(){ nextChar('\n'); }
void nextCstr(const char* s){ while(*s) nextChar(*(s++)); }
void nextU32(uint32_t x){
u32 y = 0;
if(x >= P10(9)){
Fil(y, x, P10(9));
nextCstr(TB.NLZ[y]); next_dig9(x);
}
else if(x >= P10(6)){
Fil(y, x, P10(6));
nextCstr(TB.NLZ[y]);
Fil(y, x, P10(3));
nextCstr(TB.LZ[y]); nextCstr(TB.LZ[x]);
}
else if(x >= P10(3)){
Fil(y, x, P10(3));
nextCstr(TB.NLZ[y]); nextCstr(TB.LZ[x]);
}
else if(x >= 1) nextCstr(TB.NLZ[x]);
else nextChar('0');
}
void nextI32(int32_t x){
if(x >= 0) nextU32(x);
else{ nextChar('-'); nextU32((u32)-x); }
}
void nextU64(uint64_t x){
u32 y = 0;
if(x >= P10L(18)){
Fil(y, x, P10L(18));
nextU32(y);
Fil(y, x, P10L(9));
next_dig9(y); next_dig9(x);
}
else if(x >= P10L(9)){
Fil(y, x, P10L(9));
nextU32(y); next_dig9(x);
}
else nextU32(x);
}
void nextI64(int64_t x){
if(x >= 0) nextU64(x);
else{ nextChar('-'); nextU64((u64)-x); }
}
void writeToFile(bool flush = false){
fwrite(Q, p, 1, stdout);
if(flush) fflush(stdout);
p = 0;
}
COutStream(){ Q[0] = 0; }
~COutStream(){ writeToFile(); }
MyType& operator<<(unsigned int tg){ nextU32(tg); return *this; }
MyType& operator<<(unsigned long tg){ nextU64(tg); return *this; }
MyType& operator<<(unsigned long long tg){ nextU64(tg); return *this; }
MyType& operator<<(int tg){ nextI32(tg); return *this; }
MyType& operator<<(long tg){ nextI64(tg); return *this; }
MyType& operator<<(long long tg){ nextI64(tg); return *this; }
MyType& operator<<(const std::string& tg){ nextCstr(tg.c_str()); return *this; }
MyType& operator<<(const char* tg){ nextCstr(tg); return *this; }
MyType& operator<<(char tg){ nextChar(tg); return *this; }
} cout;
char CInStream::Q[INPUT_BUF_SIZE];
char COutStream::Q[OUTPUT_BUF_SIZE];
} // namespace nachia
#line 5 "Main.cpp"
int main(){
using Modint = atcoder::static_modint<998244353>;
using Polynomial = nachia::FormalPowerSeriesNTT<Modint>;
using PolynomialMat = nachia::SimpleMatrix<Polynomial>;
using nachia::cin, nachia::cout;
auto MatMod = [&](const PolynomialMat& mat, const Polynomial& mod) -> PolynomialMat {
int n = mat.height();
PolynomialMat res(n, n);
for(int i=0; i<n; i++) for(int j=0; j<n; j++) res[i][j] = nachia::PolynomialDivision(mat[i][j], mod).second;
return res;
};
int T; cin >> T;
if(T <= 5){
for(int t=0; t<T; t++){
unsigned long long N, K; cin >> N >> K;
if(K >= 998244353){ cout << "0\n"; continue; }
PolynomialMat M_nX = PolynomialMat(2,2);
M_nX[0][0] = std::vector<Modint>{ Modint(N) * 2 , -Modint(2) }; // 2N - 2k
M_nX[0][1] = std::vector<Modint>{ 0, (Modint(N)*2+1) / 2, -Modint(1) / 2 }; // (2N+1)k/2 - k^2/2
M_nX[1][0] = std::vector<Modint>{ 1 };
M_nX[1][1] = std::vector<Modint>{};
auto ansMat = nachia::PRecursiveMatrixProduct(M_nX, K);
Modint ans = ansMat[0][0];
cout << ans.val() << '\n';
}
}
else{
int MAX_K = 100000;
int MATRIX_QUERY = 1001001001;
std::vector<std::pair<unsigned long long, int>> NK(T);
for(auto& nk : NK) cin >> nk.first >> nk.second;
std::vector<std::pair<int, int>> queries;
for(int k=0; k<MAX_K; k++) queries.emplace_back(k, MATRIX_QUERY);
for(int t=0; t<T; t++) queries.emplace_back(NK[t].second, t);
std::sort(queries.begin(), queries.end());
int segN = 1;
while(segN < (int)queries.size()) segN *= 2;
std::vector<PolynomialMat> FX;
std::vector<Polynomial> KX;
FX.assign(segN*2, PolynomialMat::Identity(2, Polynomial(std::vector<Modint>{1})));
KX.assign(segN*2, Polynomial(std::vector<Modint>{1}));
for(int q=0; q<(int)queries.size(); q++){
if(queries[q].second == MATRIX_QUERY){
int k = queries[q].first;
FX[segN+q][0][0] = std::vector<Modint>{ -Modint(k)*2, Modint(2) }; // 2N - 2k
FX[segN+q][0][1] = std::vector<Modint>{ Modint(k)*(1-k) / 2, Modint(k) }; // Nk + k(1-k)/2
FX[segN+q][1][0] = std::vector<Modint>{ 1 };
FX[segN+q][1][1] = std::vector<Modint>{};
}
else{
unsigned long long N = NK[queries[q].second].first;
KX[segN+q] = Polynomial(std::vector<Modint>{ -Modint(N), 1 }); // x - N
}
}
for(int i=segN-1; i>=1; i--) FX[i] = FX[i*2+1] * FX[i*2];
for(int i=segN-1; i>=1; i--) KX[i] = KX[i*2+1] * KX[i*2];
std::vector<PolynomialMat> FXmodKX(segN*2);
FXmodKX[1] = MatMod(PolynomialMat::Identity(2, Polynomial(std::vector<Modint>{1})), KX[1]);
for(int i=1; i<=segN-1; i++){
FXmodKX[i*2] = MatMod(FXmodKX[i], KX[i*2]);
FXmodKX[i*2+1] = MatMod(FX[i*2] * FXmodKX[i], KX[i*2+1]);
}
std::vector<Modint> ans(T);
for(int q=0; q<(int)queries.size(); q++){
if(queries[q].second != MATRIX_QUERY){
ans[queries[q].second] = FXmodKX[segN+q][0][0].eval(0);
}
}
for(int i=0; i<T; i++) cout << ans[i].val() << '\n';
}
return 0;
}