結果
| 問題 |
No.1303 Inconvenient Kingdom
|
| ユーザー |
 drken1215
|
| 提出日時 | 2025-09-12 04:25:00 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 51,612 bytes |
| コンパイル時間 | 14,690 ms |
| コンパイル使用メモリ | 312,768 KB |
| 実行使用メモリ | 19,904 KB |
| 最終ジャッジ日時 | 2025-09-12 04:26:25 |
| 合計ジャッジ時間 | 84,901 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 16 WA * 1 TLE * 17 |
ソースコード
//
// 除算なし行列式, O(N^4)
//
// reference:
// https://noshi91.hatenablog.com/entry/2020/11/28/115621
//
// verified:
// yukicoder No.1303 Inconvenient Kingdom
// https://yukicoder.me/problems/no/1303
//
#include <bits/stdc++.h>
using namespace std;
//------------------------------//
// Utility
//------------------------------//
template<class S, class T> inline bool chmax(S &a, T b) { return (a < b ? a = b, 1 : 0); }
template<class S, class T> inline bool chmin(S &a, T b) { return (a > b ? a = b, 1 : 0); }
using pint = pair<int, int>;
using pll = pair<long long, long long>;
using tint = array<int, 3>;
using tll = array<long long, 3>;
using fint = array<int, 4>;
using fll = array<long long, 4>;
using qint = array<int, 5>;
using qll = array<long long, 5>;
using vint = vector<int>;
using vll = vector<long long>;
using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T>
using min_priority_queue = priority_queue<T, vector<T>, greater<T>>;
#define COUT(x) cout << #x << " = " << (x) << " (L" << __LINE__ << ")" << endl
// debug stream
template<class T1, class T2> ostream& operator << (ostream &s, pair<T1,T2> P)
{ return s << '<' << P.first << ", " << P.second << '>'; }
template<class T> ostream& operator << (ostream &s, array<T, 3> P)
{ return s << '<' << P[0] << ", " << P[1] << ", " << P[2] << '>'; }
template<class T> ostream& operator << (ostream &s, array<T, 4> P)
{ return s << '<' << P[0] << ", " << P[1] << ", " << P[2] << ", " << P[3] << '>'; }
template<class T> ostream& operator << (ostream &s, vector<T> P)
{ for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; }
template<class T> ostream& operator << (ostream &s, deque<T> P)
{ for (int i = 0; i < P.size(); ++i) { if (i > 0) { s << " "; } s << P[i]; } return s; }
template<class T> ostream& operator << (ostream &s, vector<vector<T> > P)
{ for (int i = 0; i < P.size(); ++i) { s << endl << P[i]; } return s << endl; }
template<class T> ostream& operator << (ostream &s, set<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T> ostream& operator << (ostream &s, multiset<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T> ostream& operator << (ostream &s, unordered_set<T> P)
{ for (auto it : P) { s << "<" << it << "> "; } return s; }
template<class T1, class T2> ostream& operator << (ostream &s, map<T1,T2> P)
{ for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; }
template<class T1, class T2> ostream& operator << (ostream &s, unordered_map<T1,T2> P)
{ for (auto it : P) { s << "<" << it.first << "->" << it.second << "> "; } return s; }
// min non-negative i such that n <= 2^i
int ceil_pow2(int n) {
int i = 0;
while ((1U << i) < (unsigned int)(n)) i++;
return i;
}
// min non-negative i such that (x & (1 << i)) != 0
int bsf(int x) { return __builtin_ctz(x); }
int bsf(unsigned int x) { return __builtin_ctz(x); }
int bsf(long long x) { return __builtin_ctzll(x); }
int bsf(unsigned long long x) { return __builtin_ctzll(x); }
// Union-Find
struct UnionFind {
// core member
vector<int> par, nex;
// constructor
UnionFind() { }
UnionFind(int N) : par(N, -1), nex(N) {
init(N);
}
void init(int N) {
par.assign(N, -1);
nex.resize(N);
for (int i = 0; i < N; ++i) nex[i] = i;
}
// core methods
int root(int x) {
if (par[x] < 0) return x;
else return par[x] = root(par[x]);
}
bool same(int x, int y) {
return root(x) == root(y);
}
bool merge(int x, int y, bool merge_technique = true) {
x = root(x), y = root(y);
if (x == y) return false;
if (merge_technique) if (par[x] > par[y]) swap(x, y); // merge technique
par[x] += par[y];
par[y] = x;
swap(nex[x], nex[y]);
return true;
}
int size(int x) {
return -par[root(x)];
}
// get group
vector<int> group(int x) {
vector<int> res({x});
while (nex[res.back()] != x) res.push_back(nex[res.back()]);
return res;
}
vector<vector<int>> groups() {
vector<vector<int>> member(par.size());
for (int v = 0; v < (int)par.size(); ++v) {
member[root(v)].push_back(v);
}
vector<vector<int>> res;
for (int v = 0; v < (int)par.size(); ++v) {
if (!member[v].empty()) res.push_back(member[v]);
}
return res;
}
// debug
friend ostream& operator << (ostream &s, UnionFind uf) {
const vector<vector<int>> &gs = uf.groups();
for (const vector<int> &g : gs) {
s << "group: ";
for (int v : g) s << v << " ";
s << endl;
}
return s;
}
};
//------------------------------//
// General Matrix
//------------------------------//
// modint matrix
template<class T> struct GeneralMatrix {
// inner value
int H, W;
T ZERO = T();
T ONE;
vector<vector<T>> val;
// constructors
GeneralMatrix() : H(0), W(0) {}
GeneralMatrix(int h, int w, const T &ZERO, const T &ONE)
: H(h), W(w), ZERO(ZERO), ONE(ONE), val(h, vector<T>(w, ZERO)) {}
GeneralMatrix(const GeneralMatrix &mat)
: H(mat.H), W(mat.W), ZERO(mat.ZERO), ONE(mat.ONE), val(mat.val) {}
void init(int h, int w, const T &x) {
H = h, W = w;
val.assign(h, vector<T>(w, x));
}
void resize(int h, int w) {
H = h, W = w;
val.resize(h);
for (int i = 0; i < h; ++i) val[i].resize(w);
}
// getter and debugger
constexpr int height() const { return H; }
constexpr int width() const { return W; }
constexpr bool empty() const { return height() == 0; }
vector<T>& operator [] (int i) { return val[i]; }
constexpr const vector<T>& operator [] (int i) const { return val[i]; }
friend constexpr ostream& operator << (ostream &os, const GeneralMatrix<T> &mat) {
for (int i = 0; i < mat.height(); ++i) {
for (int j = 0; j < mat.width(); ++j) {
if (j) os << ' ';
os << mat.val[i][j];
}
os << '\n';
}
return os;
}
// comparison operators
constexpr bool operator == (const GeneralMatrix &r) const {
return this->val == r.val;
}
constexpr bool operator != (const GeneralMatrix &r) const {
return this->val != r.val;
}
// arithmetic operators
constexpr GeneralMatrix& operator += (const GeneralMatrix &r) {
assert(height() == r.height());
assert(width() == r.width());
for (int i = 0; i < height(); ++i)
for (int j = 0; j < width(); ++j)
val[i][j] = val[i][j] + r.val[i][j];
return *this;
}
constexpr GeneralMatrix& operator -= (const GeneralMatrix &r) {
assert(height() == r.height());
assert(width() == r.width());
for (int i = 0; i < height(); ++i)
for (int j = 0; j < width(); ++j)
val[i][j] = val[i][j] - r.val[i][j];
return *this;
}
constexpr GeneralMatrix& operator *= (const T &v) {
for (int i = 0; i < height(); ++i)
for (int j = 0; j < width(); ++j)
val[i][j] = val[i][j] * v;
return *this;
}
constexpr GeneralMatrix& operator *= (const GeneralMatrix &r) {
assert(width() == r.height());
GeneralMatrix<T> res(height(), r.width());
for (int i = 0; i < height(); ++i)
for (int j = 0; j < r.width(); ++j)
for (int k = 0; k < width(); ++k)
res[i][j] = res[i][j] + val[i][k] * r.val[k][j];
return (*this) = res;
}
constexpr GeneralMatrix operator + () const { return GeneralMatrix(*this); }
constexpr GeneralMatrix operator - () const { return GeneralMatrix(*this) = -GeneralMatrix(*this); }
constexpr GeneralMatrix operator + (const GeneralMatrix &r) const { return GeneralMatrix(*this) += r; }
constexpr GeneralMatrix operator - (const GeneralMatrix &r) const { return GeneralMatrix(*this) -= r; }
constexpr GeneralMatrix operator * (const T &v) const { return GeneralMatrix(*this) *= v; }
constexpr GeneralMatrix operator * (const GeneralMatrix &r) const { return GeneralMatrix(*this) *= r; }
constexpr vector<T> operator * (const vector<T> &v) const {
assert(width() == v.size());
vector<T> res(height(), ZERO);
for (int i = 0; i < height(); i++)
for (int j = 0; j < width(); j++)
res[i] += val[i][j] * v[j];
return res;
}
// transpose
constexpr GeneralMatrix trans() const {
GeneralMatrix<T> res(width(), height());
for (int row = 0; row < width(); row++)
for (int col = 0; col < height(); col++)
res[row][col] = val[col][row];
return res;
}
friend constexpr GeneralMatrix<T> trans(const GeneralMatrix<T> &mat) {
return mat.trans();
}
// pow
constexpr GeneralMatrix pow(long long n) const {
assert(height() == width());
GeneralMatrix<T> res(height(), width()), mul(*this);
for (int row = 0; row < height(); ++row) res[row][row] = ONE;
while (n > 0) {
if (n & 1) res = res * mul;
mul = mul * mul;
n >>= 1;
}
return res;
}
friend constexpr GeneralMatrix<T> pow(const GeneralMatrix<T> &mat, long long n) {
return mat.pow(n);
}
// determinant (without division, O(N^4))
constexpr T det() const {
assert(height() == width());
if (height() == 0) return ONE;
int N = height();
vector<vector<T>> dp(N + 1, vector<T>(N + 1, ZERO));
for (int i = 0; i <= N; i++) dp[i][i] = ONE;
for (int step = 0; step < N; step++) {
vector<vector<T>> nex(N + 1, vector<T>(N + 1, ZERO));
for (int row = 0; row < N; row++) {
for (int col = row; col < N; col++) {
for (int col2 = row + 1; col2 < N; col2++) {
nex[row][col2] += dp[row][col] * (-(*this)[col][col2]);
}
T tmp = dp[row][col] * (*this)[col][row];
for (int col2 = row + 1; col2 <= N; col2++) {
nex[col2][col2] += tmp;
}
}
}
swap(dp, nex);
}
return dp[N][N];
}
friend constexpr T det(const GeneralMatrix<T> &mat) {
return mat.det();
}
// determinant (by Euclidean Algorithm)
constexpr int find_pivot(int cur_rank, int col) const {
int pivot = -1;
for (int row = cur_rank; row < height(); ++row) {
if (val[row][col] != ZERO) {
pivot = row;
break;
}
}
return pivot;
}
constexpr T det_euclid() const {
assert(height() == width());
if (height() == 0) return ONE;
GeneralMatrix<T> A(*this);
int rank = 0;
T res = ONE;
for (int col = 0; col < width(); ++col) {
int pivot = A.find_pivot(rank, col);
if (pivot == -1) return ZERO;
if (pivot != rank) swap(A[pivot], A[rank]), res = -res;
for (int row = rank + 1; row < height(); ++row) {
while (A[row][col] != ZERO) {
swap(A[rank], A[row]), res = -res;
auto quo = A[row][col] / A[rank][col];
for (int col2 = rank; col2 < width(); ++col2) {
A[row][col2] -= A[rank][col2] * quo;
}
}
}
rank++;
}
for (int col = 0; col < height(); ++col) res *= A[col][col];
return res;
}
friend constexpr T det_euclid(const GeneralMatrix<T> &mat) {
return mat.det_euclid();
}
};
//------------------------------//
// mod algorithms
//------------------------------//
// safe mod
template<class T_VAL, class T_MOD>
constexpr T_VAL safe_mod(T_VAL a, T_MOD m) {
assert(m > 0);
a %= m;
if (a < 0) a += m;
return a;
}
// mod pow
template<class T_VAL, class T_MOD>
constexpr T_VAL mod_pow(T_VAL a, T_VAL n, T_MOD m) {
T_VAL res = 1;
while (n > 0) {
if (n % 2 == 1) res = res * a % m;
a = a * a % m;
n >>= 1;
}
return res;
}
// mod inv
template<class T_VAL, class T_MOD>
constexpr T_VAL mod_inv(T_VAL a, T_MOD m) {
T_VAL b = m, u = 1, v = 0;
while (b > 0) {
T_VAL t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
u %= m;
if (u < 0) u += m;
return u;
}
// modint
template<int MOD = 998244353, bool PRIME = true> struct Fp {
// inner value
unsigned int val;
// constructor
constexpr Fp() : val(0) { }
template<std::signed_integral T> constexpr Fp(T v) {
long long tmp = (long long)(v % (long long)(get_umod()));
if (tmp < 0) tmp += get_umod();
val = (unsigned int)(tmp);
}
template<std::unsigned_integral T> constexpr Fp(T v) {
val = (unsigned int)(v % get_umod());
}
constexpr long long get() const { return val; }
constexpr static int get_mod() { return MOD; }
constexpr static unsigned int get_umod() { return MOD; }
// arithmetic operators
constexpr Fp operator + () const { return Fp(*this); }
constexpr Fp operator - () const { return Fp() - Fp(*this); }
constexpr Fp operator + (const Fp &r) const { return Fp(*this) += r; }
constexpr Fp operator - (const Fp &r) const { return Fp(*this) -= r; }
constexpr Fp operator * (const Fp &r) const { return Fp(*this) *= r; }
constexpr Fp operator / (const Fp &r) const { return Fp(*this) /= r; }
constexpr Fp& operator += (const Fp &r) {
val += r.val;
if (val >= get_umod()) val -= get_umod();
return *this;
}
constexpr Fp& operator -= (const Fp &r) {
val -= r.val;
if (val >= get_umod()) val += get_umod();
return *this;
}
constexpr Fp& operator *= (const Fp &r) {
unsigned long long tmp = val;
tmp *= r.val;
val = (unsigned int)(tmp % get_umod());
return *this;
}
constexpr Fp& operator /= (const Fp &r) {
return *this = *this * r.inv();
}
constexpr Fp pow(long long n) const {
assert(n >= 0);
Fp res(1), mul(*this);
while (n) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
constexpr Fp inv() const {
if (PRIME) {
assert(val);
return pow(get_umod() - 2);
} else {
assert(val);
return mod_inv((long long)(val), get_umod());
}
}
// other operators
constexpr bool operator == (const Fp &r) const {
return this->val == r.val;
}
constexpr bool operator != (const Fp &r) const {
return this->val != r.val;
}
constexpr bool operator < (const Fp &r) const {
return this->val < r.val;
}
constexpr bool operator > (const Fp &r) const {
return this->val > r.val;
}
constexpr bool operator <= (const Fp &r) const {
return this->val <= r.val;
}
constexpr bool operator >= (const Fp &r) const {
return this->val >= r.val;
}
constexpr Fp& operator ++ () {
++val;
if (val == get_umod()) val = 0;
return *this;
}
constexpr Fp& operator -- () {
if (val == 0) val = get_umod();
--val;
return *this;
}
constexpr Fp operator ++ (int) {
Fp res = *this;
++*this;
return res;
}
constexpr Fp operator -- (int) {
Fp res = *this;
--*this;
return res;
}
friend constexpr istream& operator >> (istream &is, Fp<MOD> &x) {
long long tmp = 1;
is >> tmp;
tmp = tmp % (long long)(get_umod());
if (tmp < 0) tmp += get_umod();
x.val = (unsigned int)(tmp);
return is;
}
friend constexpr ostream& operator << (ostream &os, const Fp<MOD> &x) {
return os << x.val;
}
friend constexpr Fp<MOD> pow(const Fp<MOD> &r, long long n) {
return r.pow(n);
}
friend constexpr Fp<MOD> inv(const Fp<MOD> &r) {
return r.inv();
}
};
// dynamic modint
struct DynamicModint {
using mint = DynamicModint;
// static menber
static int MOD;
// inner value
unsigned int val;
// constructor
DynamicModint() : val(0) { }
template<std::signed_integral T> DynamicModint(T v) {
long long tmp = (long long)(v % (long long)(get_umod()));
if (tmp < 0) tmp += get_umod();
val = (unsigned int)(tmp);
}
template<std::unsigned_integral T> DynamicModint(T v) {
val = (unsigned int)(v % get_umod());
}
long long get() const { return val; }
static int get_mod() { return MOD; }
static unsigned int get_umod() { return MOD; }
static void set_mod(int mod) { MOD = mod; }
// arithmetic operators
mint operator + () const { return mint(*this); }
mint operator - () const { return mint() - mint(*this); }
mint operator + (const mint &r) const { return mint(*this) += r; }
mint operator - (const mint &r) const { return mint(*this) -= r; }
mint operator * (const mint &r) const { return mint(*this) *= r; }
mint operator / (const mint &r) const { return mint(*this) /= r; }
mint& operator += (const mint &r) {
val += r.val;
if (val >= get_umod()) val -= get_umod();
return *this;
}
mint& operator -= (const mint &r) {
val -= r.val;
if (val >= get_umod()) val += get_umod();
return *this;
}
mint& operator *= (const mint &r) {
unsigned long long tmp = val;
tmp *= r.val;
val = (unsigned int)(tmp % get_umod());
return *this;
}
mint& operator /= (const mint &r) {
return *this = *this * r.inv();
}
mint pow(long long n) const {
assert(n >= 0);
mint res(1), mul(*this);
while (n) {
if (n & 1) res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
mint inv() const {
assert(val);
return mod_inv((long long)(val), get_umod());
}
// other operators
bool operator == (const mint &r) const {
return this->val == r.val;
}
bool operator != (const mint &r) const {
return this->val != r.val;
}
bool operator < (const mint &r) const {
return this->val < r.val;
}
bool operator > (const mint &r) const {
return this->val > r.val;
}
bool operator <= (const mint &r) const {
return this->val <= r.val;
}
bool operator >= (const mint &r) const {
return this->val >= r.val;
}
mint& operator ++ () {
++val;
if (val == get_umod()) val = 0;
return *this;
}
mint& operator -- () {
if (val == 0) val = get_umod();
--val;
return *this;
}
mint operator ++ (int) {
mint res = *this;
++*this;
return res;
}
mint operator -- (int) {
mint res = *this;
--*this;
return res;
}
friend istream& operator >> (istream &is, mint &x) {
long long tmp = 1;
is >> tmp;
tmp = tmp % (long long)(get_umod());
if (tmp < 0) tmp += get_umod();
x.val = (unsigned int)(tmp);
return is;
}
friend ostream& operator << (ostream &os, const mint &x) {
return os << x.val;
}
friend mint pow(const mint &r, long long n) {
return r.pow(n);
}
friend mint inv(const mint &r) {
return r.inv();
}
};
int DynamicModint::MOD;
// Binomial coefficient
template<class mint> struct BiCoef {
vector<mint> fact_, inv_, finv_;
constexpr BiCoef() {}
constexpr BiCoef(int n) : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
init(n);
}
constexpr void init(int n) {
fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
int MOD = fact_[0].get_mod();
for(int i = 2; i < n; i++){
fact_[i] = fact_[i-1] * i;
inv_[i] = -inv_[MOD%i] * (MOD/i);
finv_[i] = finv_[i-1] * inv_[i];
}
}
constexpr mint com(int n, int k) const {
if (n < k || n < 0 || k < 0) return 0;
return fact_[n] * finv_[k] * finv_[n-k];
}
constexpr mint fact(int n) const {
if (n < 0) return 0;
return fact_[n];
}
constexpr mint inv(int n) const {
if (n < 0) return 0;
return inv_[n];
}
constexpr mint finv(int n) const {
if (n < 0) return 0;
return finv_[n];
}
};
//------------------------------//
// NTT
//------------------------------//
// calc primitive root
constexpr int calc_primitive_root(long long m) {
if (m == 1) return -1;
if (m == 2) return 1;
if (m == 998244353) return 3;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 645922817) return 3;
if (m == 897581057) return 3;
long long divs[20] = {};
divs[0] = 2;
long long cnt = 1;
long long x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (long long i = 3; 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 (long long g = 2; ; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (mod_pow(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
// NTT setup
template<class mint, int MOD = mint::get_mod(), int g = calc_primitive_root(mint::get_mod())>
struct ntt_setup {
static constexpr int bsf_constexpr(unsigned int x) {
int i = 0;
while (!(x & (1 << i))) i++;
return i;
};
static constexpr int rank = bsf_constexpr(MOD - 1);
array<mint, rank + 1> root, iroot; // root[i]^(2^i) = 1, root[i] * iroot[i] = 1
array<mint, max(0, rank - 1)> rate2, irate2;
array<mint, max(0, rank - 2)> rate3, irate3;
ntt_setup() {
root[rank] = mint(g).pow((MOD - 1) >> rank);
iroot[rank] = root[rank].inv();
for (int i = rank - 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 < rank - 1; 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 < rank - 2; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
};
// NTT transformation
template<class mint, int MOD = mint::get_mod()>
void ntt_trans(vector<mint> &v) {
int n = (int)v.size();
int h = ceil_pow2(n);
static const ntt_setup<mint> setup;
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 = v[i + offset];
auto r = v[i + offset + p] * rot;
v[i + offset] = l + r;
v[i + offset + p] = l - r;
}
if (s + 1 != (1 << len)) {
rot *= setup.rate2[bsf(~(unsigned int)(s))];
}
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = setup.root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot, rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * MOD * MOD;
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val * rot.val;
auto a2 = 1ULL * v[i + offset + p * 2].val * rot2.val;
auto a3 = 1ULL * v[i + offset + p * 3].val * rot3.val;
auto tmp = 1ULL * mint(a1 + mod2 - a3).val * imag.val;
auto na2 = mod2 - a2;
v[i + offset] = a0 + a2 + a1 + a3;
v[i + offset + p] = a0 + a2 + (mod2 * 2 - (a1 + a3));
v[i + offset + p * 2] = a0 + na2 + tmp;
v[i + offset + p * 3] = a0 + na2 + (mod2 - tmp);
}
if (s + 1 != (1 << len)) {
rot *= setup.rate3[bsf(~(unsigned int)(s))];
}
}
len += 2;
}
}
}
// NTT inv-transformation
template<class mint, int MOD = mint::get_mod()>
void ntt_trans_inv(vector<mint> &v) {
int n = (int)v.size();
int h = ceil_pow2(n);
static const ntt_setup<mint> setup;
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 = v[i + offset];
auto r = v[i + offset + p];
v[i + offset] = l + r;
v[i + offset + p] = (unsigned long long)((long long)(MOD) + l.val - r.val) * irot.val;
}
if (s + 1 != (1 << (len - 1))) {
irot *= setup.irate2[bsf(~(unsigned int)(s))];
}
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = setup.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot, irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val;
auto a2 = 1ULL * v[i + offset + p * 2].val;
auto a3 = 1ULL * v[i + offset + p * 3].val;
auto tmp = 1ULL * mint((MOD + a2 - a3) * iimag.val).val;
v[i + offset] = a0 + a1 + a2 + a3;
v[i + offset + p] = (a0 + (MOD - a1) + tmp) * irot.val;
v[i + offset + p * 2] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val;
v[i + offset + p * 3] = (a0 + (MOD - a1) + (MOD - tmp)) * irot3.val;
}
if (s + 1 != (1 << (len - 2))) {
irot *= setup.irate3[bsf(~(unsigned int)(s))];
}
}
len -= 2;
}
}
mint in = mint(n).inv();
for (int i = 0; i < n; i++) v[i] *= in;
}
// naive convolution
template<class T>
vector<T> sub_convolution_naive(const vector<T> &a, const vector<T> &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m) return {};
vector<T> res(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++) for (int i = 0; i < n; i++) res[i + j] += a[i] * b[j];
} else {
for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
}
return res;
}
// ntt convolution
template<class mint>
vector<mint> sub_convolution_ntt(vector<mint> a, vector<mint> b) {
int MOD = mint::get_mod();
int n = (int)a.size(), m = (int)b.size();
if (!n || !m) return {};
int z = (int)bit_ceil((unsigned int)(n + m - 1));
assert((MOD - 1) % z == 0);
a.resize(z), b.resize(z);
ntt_trans(a), ntt_trans(b);
for (int i = 0; i < z; i++) a[i] *= b[i];
ntt_trans_inv(a);
a.resize(n + m - 1);
return a;
}
// convolution in general mod
template<class mint>
vector<mint> convolution(const vector<mint> &a, const vector<mint> &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m) return {};
if (min(n, m) <= 60) return sub_convolution_naive(std::move(a), std::move(b));
if constexpr (std::is_same_v<mint, Fp<998244353>>) return sub_convolution_ntt(a, b);
static constexpr int MOD0 = 754974721; // 2^24
static constexpr int MOD1 = 167772161; // 2^25
static constexpr int MOD2 = 469762049; // 2^26
using mint0 = Fp<MOD0>;
using mint1 = Fp<MOD1>;
using mint2 = Fp<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
vector<mint0> a0(n, 0), b0(m, 0);
vector<mint1> a1(n, 0), b1(m, 0);
vector<mint2> a2(n, 0), b2(m, 0);
for (int i = 0; i < n; ++i) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
for (int i = 0; i < m; ++i) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
auto c0 = sub_convolution_ntt(std::move(a0), std::move(b0));
auto c1 = sub_convolution_ntt(std::move(a1), std::move(b1));
auto c2 = sub_convolution_ntt(std::move(a2), std::move(b2));
vector<mint> res(n + m - 1);
mint mod0 = MOD0, mod01 = mod0 * MOD1;
for (int i = 0; i < n + m - 1; ++i) {
unsigned int y0 = c0[i].val;
unsigned int y1 = (imod0 * (c1[i] - y0)).val;
unsigned int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val;
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
//------------------------------//
// FPS
//------------------------------//
// mod sqrt
template<class T_VAL, class T_MOD>
T_VAL mod_sqrt(T_VAL a, T_MOD p) {
a = safe_mod(a, p);
if (a <= 1) return a;
using mint = DynamicModint;
mint::set_mod(p);
if (mint(a).pow((p - 1) >> 1) != 1) return T_VAL(-1);
mint b = 1, one = 1;
while (b.pow((p - 1) >> 1) == 1) b++;
T_VAL m = p - 1, e = 0;
while (m % 2 == 0) m >>= 1, e++;
mint x = mint(a).pow((m - 1) >> 1);
mint y = mint(a) * x * x;
x *= a;
mint z = mint(b).pow(m);
while (y != 1) {
T_VAL j = 0;
mint t = y;
while (t != one) {
j++;
t *= t;
}
z = z.pow(T_VAL(1) << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
T_VAL res = x.val;
if (res * 2 > p) res = p - res;
return res;
}
// Formal Power Series
template<class mint> struct FPS : vector<mint> {
static const int SPARSE_BOARDER = 60;
using vector<mint>::vector;
// constructor
constexpr FPS(mint r) : vector<mint>{r} {}
constexpr FPS(const vector<mint> &r) : vector<mint>(r) {}
// core operator
constexpr FPS pre(int siz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), siz));
}
constexpr FPS rev() const {
FPS res = *this;
reverse(begin(res), end(res));
return res;
}
constexpr FPS& normalize() {
while (!this->empty() && this->back() == 0) this->pop_back();
return *this;
}
constexpr mint eval(const mint &v) const {
mint res = 0;
for (int i = (int)this->size()-1; i >= 0; --i) {
res *= v;
res += (*this)[i];
}
return res;
}
constexpr int count_terms() const {
int res = 0;
for (int i = 0; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) res++;
return res;
}
// basic operator
constexpr FPS operator - () const noexcept {
FPS res = (*this);
for (int i = 0; i < (int)res.size(); ++i) res[i] = -res[i];
return res;
}
constexpr FPS operator + (const mint &v) const { return FPS(*this) += v; }
constexpr FPS operator + (const FPS &r) const { return FPS(*this) += r; }
constexpr FPS operator - (const mint &v) const { return FPS(*this) -= v; }
constexpr FPS operator - (const FPS &r) const { return FPS(*this) -= r; }
constexpr FPS operator * (const mint &v) const { return FPS(*this) *= v; }
constexpr FPS operator * (const FPS &r) const { return FPS(*this) *= r; }
constexpr FPS operator / (const mint &v) const { return FPS(*this) /= v; }
constexpr FPS operator / (const FPS &r) const { return FPS(*this) /= r; }
constexpr FPS operator % (const FPS &r) const { return FPS(*this) %= r; }
constexpr FPS operator << (int x) const { return FPS(*this) <<= x; }
constexpr FPS operator >> (int x) const { return FPS(*this) >>= x; }
constexpr FPS& operator += (const mint &v) {
if (this->empty()) this->reserve(1), this->resize(1);
(*this)[0] += v;
return *this;
}
constexpr FPS& operator += (const FPS &r) {
if (r.size() > this->size()) this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] += r[i];
return this->normalize();
}
constexpr FPS& operator -= (const mint &v) {
if (this->empty()) this->reserve(1), this->resize(1);
(*this)[0] -= v;
return *this;
}
constexpr FPS& operator -= (const FPS &r) {
if (r.size() > this->size()) this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] -= r[i];
return this->normalize();
}
constexpr FPS& operator *= (const mint &v) {
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= v;
return *this;
}
constexpr FPS& operator *= (const FPS &r) {
return *this = convolution((*this), r);
}
constexpr FPS& operator /= (const mint &v) {
assert(v != 0);
mint iv = v.inv();
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= iv;
return *this;
}
// division, r must be normalized (r.back() must not be 0)
constexpr FPS& operator /= (const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
if (this->size() < r.size()) {
this->clear();
return *this;
}
int need = (int)this->size() - (int)r.size() + 1;
*this = (rev().pre(need) * r.rev().inv(need)).pre(need).rev();
return *this;
}
constexpr FPS& operator %= (const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
FPS q = (*this) / r;
return *this -= q * r;
}
constexpr FPS& operator <<= (int x) {
FPS res(x, 0);
res.insert(res.end(), begin(*this), end(*this));
return *this = res;
}
constexpr FPS& operator >>= (int x) {
FPS res;
res.insert(res.end(), begin(*this) + x, end(*this));
return *this = res;
}
// advanced operation
// df/dx
constexpr FPS diff() const {
int n = (int)this->size();
if (n <= 0) return FPS();
FPS res(n-1);
for (int i = 1; i < n; ++i) res[i-1] = (*this)[i] * i;
return res;
}
// \int f dx
constexpr FPS integral() const {
int n = (int)this->size();
FPS res(n+1, 0);
for (int i = 0; i < n; ++i) res[i+1] = (*this)[i] / (i+1);
return res;
}
// inv(f), f[0] must not be 0
constexpr FPS inv(int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return inv_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>) return inv_ntt_friendly(deg);
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
FPS res({mint(1) / (*this)[0]});
for (int d = 1; d < deg; d <<= 1) {
res = (res + res - res * res * pre(d << 1)).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS inv_ntt_friendly(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
FPS res(deg);
res[0] = mint(1) / (*this)[0];
for (int d = 1; d < deg; d <<= 1) {
FPS g(d * 2), h(d * 2);
mint iv = mint(d * 2).inv();
for (int i = 0; i < min((int)this->size(), d * 2); i++) g[i] = (*this)[i];
for (int i = 0; i < d; i++) h[i] = res[i];
ntt_trans(g), ntt_trans(h);
for (int i = 0; i < d * 2; i++) g[i] *= h[i];
ntt_trans_inv(g);
for (int i = 0; i < d; i++) g[i] = 0;
ntt_trans(g);
for (int i = 0; i < d * 2; i++) g[i] *= h[i];
ntt_trans_inv(g);
for (int i = d; i < min(deg, d * 2); i++) res[i] = -g[i];
}
return res.pre(deg);
}
constexpr FPS inv_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
vector<mint> res(deg);
res[0] = (*this)[0].inv();
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i) break;
r -= val * res[i - k];
}
res[i] = r * res[0];
}
return res;
}
// log(f) = \int f'/f dx, f[0] must be 1
constexpr FPS log(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (count_terms() <= SPARSE_BOARDER) return log_sparse(deg);
if (deg < 0) deg = (int)this->size();
return ((diff() * inv(deg)).pre(deg - 1)).integral();
}
constexpr FPS log_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg), tmp(deg);
for (int i = 0; i < deg - 1; i++) {
mint r = mint(i + 1) * (*this)[i + 1];
for (auto &&[k, val] : dat) {
if (k > i) break;
r -= val * tmp[i - k];
}
tmp[i] = r;
res[i + 1] = r * bc.inv(i + 1);
}
return res;
}
// exp(f), f[0] must be 0
constexpr FPS exp(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
if (count_terms() <= SPARSE_BOARDER) return exp_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>) return exp_ntt_friendly(deg);
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
FPS res(1, 1);
for (int d = 1; d < deg; d <<= 1) {
res = res * (pre(d << 1) - res.log(d << 1) + 1).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS exp_ntt_friendly(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
FPS fiv;
fiv.reserve(deg + 1);
fiv.emplace_back(mint(0));
fiv.emplace_back(mint(1));
auto inplace_integral = [&](FPS &F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while ((int)fiv.size() <= n) {
int i = fiv.size();
fiv.emplace_back((-fiv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for (int i = 1; i <= n; i++) F[i] *= fiv[i];
};
auto inplace_diff = [](FPS &F) -> void {
if (F.empty()) return;
F.erase(begin(F));
mint coef = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coef;
coef++;
}
};
FPS b{1, (1 < (int)this->size() ? (*this)[1] : 0)}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m <<= 1) {
auto y = b;
y.resize(m * 2);
ntt_trans(y);
z1 = z2;
FPS z(m);
for (int i = 0; i < m; i++) z[i] = y[i] * z1[i];
ntt_trans_inv(z);
fill(begin(z), begin(z) + m / 2, mint(0));
ntt_trans(z);
for (int i = 0; i < m; i++) z[i] *= -z1[i];
ntt_trans_inv(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(m * 2);
ntt_trans(z2);
FPS x(begin(*this), begin(*this) + min((int)this->size(), m));
inplace_diff(x);
x.emplace_back(mint(0));
ntt_trans(x);
for (int i = 0; i < m; i++) x[i] *= y[i];
ntt_trans_inv(x);
x -= b.diff();
x.resize(m * 2);
for (int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = mint(0);
ntt_trans(x);
for (int i = 0; i < m * 2; i++) x[i] *= z2[i];
ntt_trans_inv(x);
x.pop_back();
inplace_integral(x);
for (int i = m; i < min((int)this->size(), m * 2); i++) x[i] += (*this)[i];
fill(begin(x), begin(x) + m, mint(0));
ntt_trans(x);
for (int i = 0; i < m * 2; i++) x[i] *= y[i];
ntt_trans_inv(x);
b.insert(end(b), begin(x) + m, end(x));
}
return FPS(begin(b), begin(b) + deg);
}
constexpr FPS exp_sparse(int deg = -1) const {
if ((int)this->size() == 0) return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i - 1, (*this)[i] * i);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i - 1) break;
r += val * res[i - k - 1];
}
res[i] = r * bc.inv(i);
}
return res;
}
// pow(f) = exp(e * log f)
constexpr FPS pow(long long e, int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return pow_sparse(e, deg);
assert(e >= 0);
if (deg < 0) deg = (int)this->size();
if (deg == 0) return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0) ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e) return FPS(deg, 0);
mint k = (*this)[ord];
FPS res = ((((*this) >> ord) / k).log(deg) * e).exp(deg) * mint(k).pow(e) << (e * ord);
res.resize(deg);
return res;
}
constexpr FPS pow_sparse(long long e, int deg = -1) const {
assert(e >= 0);
if (deg < 0) deg = (int)this->size();
if (deg == 0) return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0) ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e) return FPS(deg, 0);
if ((*this)[0] == 1) return pow_sparse_constant1(e, deg);
auto f = (*this);
rotate(f.begin(), f.begin() + ord, f.end());
mint con = f[0], icon = f[0].inv();
for (int i = 0; i < deg; i++) f[i] *= icon;
auto res = f.pow_sparse_constant1(e, deg);
int ord2 = e * ord;
rotate(res.begin(), res.begin() + (deg - ord2), res.end());
fill(res.begin(), res.begin() + ord2, mint(0));
mint pw = con.pow(e);
for (int i = ord2; i < deg; i++) res[i] *= pw;
return res;
}
constexpr FPS pow_sparse_constant1(mint e, int deg = -1) const {
assert((int)this->size() > 0 && (*this)[0] == 1);
if (deg < 0) deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++) if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 0; i < deg - 1; i++) {
mint &r = res[i + 1];
for (auto &&[k, val] : dat) {
if (k > i + 1) break;
mint t = val * res[i - k + 1];
r += t * (mint(k) * e - mint(i - k + 1));
}
r *= bc.inv(i + 1);
}
return res;
}
// sqrt(f)
constexpr FPS sqrt(int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER) return sqrt_sparse(deg);
if (deg < 0) deg = (int)this->size();
if ((int)this->size() == 0) return FPS(deg, 0);
if ((*this)[0] == mint(0)) {
for (int i = 1; i < (int)this->size(); i++) {
if ((*this)[i] != mint(0)) {
if (i & 1) return FPS();
if (deg - i / 2 <= 0) return FPS(deg, 0);
auto res = ((*this) >> i).sqrt(deg - i / 2);
if (res.empty()) return FPS();
res = res << (i / 2);
if ((int)res.size() < deg) res.resize(deg, mint(0));
return res;
}
}
return FPS(deg, 0);
}
long long sqr = mod_sqrt<long long>((*this)[0].val, mint::get_mod());
if (sqr == -1) return FPS();
assert((*this)[0].val == sqr * sqr % mint::get_mod());
FPS res = {mint(sqr)};
mint iv2 = mint(2).inv();
for (int d = 1; d < deg; d <<= 1) {
res = (res + pre(d << 1) * res.inv(d << 1)).pre(d << 1) * iv2;
}
res.resize(deg);
return res;
}
constexpr FPS sqrt_sparse(int deg) const {
if (deg < 0) deg = (int)this->size();
if ((int)this->size() == 0) return FPS(deg, 0);
if ((*this)[0] == mint(0)) {
for (int i = 1; i < (int)this->size(); i++) {
if ((*this)[i] != mint(0)) {
if (i & 1) return FPS();
if (deg - i / 2 <= 0) return FPS(deg, 0);
auto res = ((*this) >> i).sqrt_sparse(deg - i / 2);
if (res.empty()) return FPS();
res = res << (i / 2);
if ((int)res.size() < deg) res.resize(deg, mint(0));
return res;
}
}
return FPS(deg, 0);
}
mint con = (*this)[0], icon = con.inv();
long long sqr = mod_sqrt<long long>(con.val, mint::get_mod());
if (sqr == -1) return FPS();
assert(con.val == sqr * sqr % mint::get_mod());
auto res = (*this) * icon;
return res.sqrt_sparse_constant1(deg) * sqr;
}
constexpr FPS sqrt_sparse_constant1(int deg) const {
return pow_sparse_constant1(mint(2).inv(), deg);
}
// polynomial taylor shift
constexpr FPS taylor_shift(long long c) const {
int N = (int)this->size() - 1;
BiCoef<mint> bc(N + 1);
FPS<mint> p(N + 1), q(N + 1);
for (int i = 0; i <= N; i++) {
p[i] = (*this)[i] * bc.fact(i);
q[N - i] = mint(c).pow(i) * bc.finv(i);
}
FPS<mint> pq = p * q;
FPS<mint> res(N + 1);
for (int i = 0; i <= N; i++) res[i] = pq[i + N] * bc.finv(i);
return res;
}
// friend operators
friend constexpr FPS diff(const FPS &f) { return f.diff(); }
friend constexpr FPS integral(const FPS &f) { return f.integral(); }
friend constexpr FPS inv(const FPS &f, int deg = -1) { return f.inv(deg); }
friend constexpr FPS log(const FPS &f, int deg = -1) { return f.log(deg); }
friend constexpr FPS exp(const FPS &f, int deg = -1) { return f.exp(deg); }
friend constexpr FPS pow(const FPS &f, long long e, int deg = -1) { return f.pow(e, deg); }
friend constexpr FPS sqrt(const FPS &f, int deg = -1) { return f.sqrt(deg); }
friend constexpr FPS taylor_shift(const FPS &f, long long c) { return f.taylor_shift(c); }
};
//------------------------------//
// Examples
//------------------------------//
// yukicoder No.1303 Inconvenient Kingdom
void yukicoder_1303() {
using mint = Fp<>;
int N, M, u, v;
cin >> N >> M;
vector G(N, vector(N, 0));
vector degs(N, 0);
UnionFind uf(N);
for (int i = 0; i < M; i++) {
cin >> u >> v, u--, v--;
G[u][v]++, G[v][u]++, degs[u]++, degs[v]++;
uf.merge(u, v);
}
auto calc = [&](const vector<int> &group) -> mint {
vector<int> conv(N, -1);
int iter = 0;
for (auto v : group) conv[v] = iter++;
GeneralMatrix<mint> L(iter, iter, mint(0), mint(1));
for (int i = 0; i < iter; i++) L[i][i] = 1;
for (auto v1 : group) {
int i = conv[v1];
int deg = 0;
for (auto v2 : group) {
if (G[v1][v2]) deg += G[v1][v2];
int j = conv[v2];
if (i < iter - 1 && j < iter - 1) L[i][j] = -G[v1][v2];
}
if (i < iter - 1) L[i][i] = deg;
}
return det(L);
};
auto groups = uf.groups();
if (groups.size() > 1) {
mint res = 1;
vector<long long> siz;
for (auto group : groups) siz.push_back(group.size()), res *= calc(group);
sort(siz.begin(), siz.end(), greater<long long>());
long long sum = siz[0] + siz[1], sum2 = sum * sum;
for (int i = 2; i < (int)siz.size(); i++) sum += siz[i], sum2 += siz[i] * siz[i];
long long huben = (sum * sum - sum2);
if (siz[0] == siz[1]) {
long long sumsiz = 0, sumsiz2 = 0;
for (auto s : siz) if (s == siz[0]) sumsiz += s, sumsiz2 += s * s;
long long fac = (sumsiz * sumsiz - sumsiz2) / 2;
res *= fac;
} else {
long long sum_sub = 0;
for (auto s : siz) if (s == siz[1]) sum_sub += s;
long long fac = siz[0] * sum_sub;
res *= fac;
}
cout << huben << endl << res << endl;
} else {
long long huben = 0;
FPS<mint> ZERO, ONE{1};
GeneralMatrix<FPS<mint>> L(N, N, ZERO, ONE);
for (int i = 0; i < N; i++) L[i][i] = ONE;
for (int i = 0; i < N-1; i++) {
L[i][i] = FPS<mint>{mint(degs[i]), mint(N - 1 - degs[i])};
for (int j = 0; j < N-1; j++) {
if (i == j) continue;
if (G[i][j]) L[i][j] = FPS<mint>{mint(-G[i][j])};
else L[i][j] = FPS<mint>{0, mint(-1)};
}
}
FPS<mint> f = det(L);
mint res = f[0] + f[1];
cout << huben << endl << res << endl;
}
}
int main() {
yukicoder_1303();
}