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const int mod = 998244353;
using lint = long long;
lint ipow(lint x, lint p){
lint ret = 1, piv = x;
while(p){
if(p & 1) ret = ret * piv % mod;
piv = piv * piv % mod;
p >>= 1;
}
return ret;
}
vector<int> berlekamp_massey(vector<int> x){
vector<int> ls, cur;
int lf, ld;
for(int i=0; i<x.size(); i++){
lint t = 0;
for(int j=0; j<cur.size(); j++){
t = (t + 1ll * x[i-j-1] * cur[j]) % mod;
}
if((t - x[i]) % mod == 0) continue;
if(cur.empty()){
cur.resize(i+1);
lf = i;
ld = (t - x[i]) % mod;
continue;
}
lint k = -(x[i] - t) * ipow(ld, mod - 2) % mod;
vector<int> c(i-lf-1);
c.push_back(k);
for(auto &j : ls) c.push_back(-j * k % mod);
if(c.size() < cur.size()) c.resize(cur.size());
for(int j=0; j<cur.size(); j++){
c[j] = (c[j] + cur[j]) % mod;
}
if(i-lf+(int)ls.size()>=(int)cur.size()){
tie(ls, lf, ld) = make_tuple(cur, i, (t - x[i]) % mod);
}
cur = c;
}
for(auto &i : cur) i = (i % mod + mod) % mod;
return cur;
}
int get_nth(vector<int> rec, vector<int> dp, lint n){
int m = rec.size();
vector<int> s(m), t(m);
s[0] = 1;
if(m != 1) t[1] = 1;
else t[0] = rec[0];
auto mul = [&rec](vector<int> v, vector<int> w){
int m = v.size();
vector<int> t(2 * m);
for(int j=0; j<m; j++){
for(int k=0; k<m; k++){
t[j+k] += 1ll * v[j] * w[k] % mod;
if(t[j+k] >= mod) t[j+k] -= mod;
}
}
for(int j=2*m-1; j>=m; j--){
for(int k=1; k<=m; k++){
t[j-k] += 1ll * t[j] * rec[k-1] % mod;
if(t[j-k] >= mod) t[j-k] -= mod;
}
}
t.resize(m);
return t;
};
while(n){
if(n & 1) s = mul(s, t);
t = mul(t, t);
n >>= 1;
}
lint ret = 0;
for(int i=0; i<m; i++) ret += 1ll * s[i] * dp[i] % mod;
return ret % mod;
}
int guess_nth_term(vector<int> x, lint n){
if(n < x.size()) return x[n];
vector<int> v = berlekamp_massey(x);
if(v.empty()) return 0;
return get_nth(v, x, n);
}
struct elem{int x, y, v;}; // A_(x, y) <- v, 0-based. no duplicate please..
vector<int> get_min_poly(int n, vector<elem> M){
// smallest poly P such that A^i = sum_{j < i} {A^j \times P_j}
vector<int> rnd1, rnd2;
mt19937 rng(0x14004);
auto randint = [&rng](int lb, int ub){
return uniform_int_distribution<int>(lb, ub)(rng);
};
for(int i=0; i<n; i++){
rnd1.push_back(randint(1, mod - 1));
rnd2.push_back(randint(1, mod - 1));
}
vector<int> gobs;
for(int i=0; i<2*n+2; i++){
int tmp = 0;
for(int j=0; j<n; j++){
tmp += 1ll * rnd2[j] * rnd1[j] % mod;
if(tmp >= mod) tmp -= mod;
}
gobs.push_back(tmp);
vector<int> nxt(n);
for(auto &i : M){
nxt[i.x] += 1ll * i.v * rnd1[i.y] % mod;
if(nxt[i.x] >= mod) nxt[i.x] -= mod;
}
rnd1 = nxt;
}
auto sol = berlekamp_massey(gobs);
reverse(sol.begin(), sol.end());
return sol;
}
lint det(int n, vector<elem> M){
vector<int> rnd;
mt19937 rng(0x14004);
auto randint = [&rng](int lb, int ub){
return uniform_int_distribution<int>(lb, ub)(rng);
};
for(int i=0; i<n; i++) rnd.push_back(randint(1, mod - 1));
for(auto &i : M){
i.v = 1ll * i.v * rnd[i.y] % mod;
}
auto sol = get_min_poly(n, M)[0];
if(n % 2 == 0) sol = mod - sol;
for(auto &i : rnd) sol = 1ll * sol * ipow(i, mod - 2) % mod;
return sol;
}
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