結果
| 問題 |
No.3099 Parentheses Decomposition
|
| ユーザー |
 Taiki0715
|
| 提出日時 | 2025-05-17 13:30:23 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 6 ms / 2,000 ms |
| コード長 | 26,617 bytes |
| コンパイル時間 | 6,219 ms |
| コンパイル使用メモリ | 331,340 KB |
| 実行使用メモリ | 7,844 KB |
| 最終ジャッジ日時 | 2025-05-17 13:30:31 |
| 合計ジャッジ時間 | 6,605 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 20 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using ll=long long;
using ull=unsigned long long;
using P=pair<ll,ll>;
template<typename T>using minque=priority_queue<T,vector<T>,greater<T>>;
template<typename T>bool chmax(T &a,const T &b){return (a<b?(a=b,true):false);}
template<typename T>bool chmin(T &a,const T &b){return (a>b?(a=b,true):false);}
template<typename T1,typename T2>istream &operator>>(istream &is,pair<T1,T2>&p){is>>p.first>>p.second;return is;}
template<typename T1,typename T2,typename T3>istream &operator>>(istream &is,tuple<T1,T2,T3>&a){is>>std::get<0>(a)>>std::get<1>(a)>>std::get<2>(a);return is;}
template<typename T,size_t n>istream &operator>>(istream &is,array<T,n>&a){for(auto&i:a)is>>i;return is;}
template<typename T>istream &operator>>(istream &is,vector<T> &a){for(auto &i:a)is>>i;return is;}
template<typename T1,typename T2>void operator++(pair<T1,T2>&a,int n){a.first++,a.second++;}
template<typename T1,typename T2>void operator--(pair<T1,T2>&a,int n){a.first--,a.second--;}
template<typename T>void operator++(vector<T>&a,int n){for(auto &i:a)i++;}
template<typename T>void operator--(vector<T>&a,int n){for(auto &i:a)i--;}
#define overload3(_1,_2,_3,name,...) name
#define rep1(i,n) for(int i=0;i<(int)(n);i++)
#define rep2(i,l,r) for(int i=(int)(l);i<(int)(r);i++)
#define rep(...) overload3(__VA_ARGS__,rep2,rep1)(__VA_ARGS__)
#define reps(i,l,r) rep2(i,l,r)
#define all(x) x.begin(),x.end()
#define pcnt(x) __builtin_popcountll(x)
#define fin(x) return cout<<(x)<<'\n',static_cast<void>(0)
#define yn(x) cout<<((x)?"Yes\n":"No\n")
#define uniq(x) sort(all(x)),x.erase(unique(all(x)),x.end())
template<typename T>
inline int fkey(vector<T>&z,T key){return lower_bound(z.begin(),z.end(),key)-z.begin();}
ll myceil(ll a,ll b){return (a+b-1)/b;}
template<typename T,size_t n,size_t id=0>
auto vec(const int (&d)[n],const T &init=T()){
if constexpr (id<n)return vector(d[id],vec<T,n,id+1>(d,init));
else return init;
}
#ifdef LOCAL
#include<debug.h>
#define SWITCH(a,b) (a)
#else
#define debug(...) static_cast<void>(0)
#define debugg(...) static_cast<void>(0)
#define SWITCH(a,b) (b)
template<typename T1,typename T2>ostream &operator<<(ostream &os,const pair<T1,T2>&p){os<<p.first<<' '<<p.second;return os;}
#endif
struct Timer{
clock_t start;
Timer(){
start=clock();
ios::sync_with_stdio(false);
cin.tie(nullptr);
cout<<fixed<<setprecision(16);
}
inline double now(){return (double)(clock()-start)/1000;}
#ifdef LOCAL
~Timer(){
cerr<<"time:";
cerr<<now();
cerr<<"ms\n";
}
#endif
}timer;
void SOLVE();
int main(){
int testcase=1;
//cin>>testcase;
for(int i=0;i<testcase;i++){
SOLVE();
}
}
#include<type_traits>
#include<optional>
constexpr int carmichael_constexpr(int n){
if(n==998244353)return 998244352;
if(n==1000000007)return 1000000006;
if(n<=1)return n;
int res=1;
int t=0;
while(n%2==0){
n/=2;
t++;
}
if(t==2)res=2;
else if(t>=3)res=1<<(t-2);
for(int i=3;i*i<=n;i++)if(n%i==0){
int c=0;
while(n%i==0){
n/=i;
c++;
}
int prod=i-1;
for(int j=0;j<c-1;j++)prod*=i;
res=std::lcm(res,prod);
}
if(n!=1)res=std::lcm(res,n-1);
return res;
}
template<int m>
struct mod_int{
private:
static constexpr unsigned int umod=static_cast<unsigned int>(m);
static constexpr unsigned int car=carmichael_constexpr(m);
using uint=unsigned int;
using mint=mod_int;
uint v;
static_assert(m<uint(1)<<31);
mint sqrt_impl()const{
if(this->val()<=1)return *this;
if constexpr(m%8==1){
mint b=2;
while(b.pow((m-1)/2).val()==1)b++;
int m2=m-1,e=0;
while(m2%2==0)m2>>=1,e++;
mint x=this->pow((m2-1)/2);
mint y=(*this)*x*x;
x*=*this;
mint z=b.pow(m2);
while(y.val()!=1){
int j=0;
mint t=y;
while(t.val()!=1)t*=t,j++;
z=z.pow(1<<(e-j-1));
x*=z;
z*=z;
y*=z;e=j;
}
return x;
}
else if constexpr(m%8==5){
mint ret=this->pow((m+3)/8);
if((ret*ret).val()==this->val())return ret;
else return ret*mint::raw(2).pow((m-1)/4);
}
else{
return this->pow((m+1)/4);
}
}
public:
using value_type=uint;
mod_int():v(0){}
template<typename T,std::enable_if_t<std::is_signed_v<T>,std::nullptr_t> =nullptr>
mod_int(T a){
a%=m;
if(a<0)v=a+umod;
else v=a;
}
template<typename T,std::enable_if_t<std::is_unsigned_v<T>,std::nullptr_t> =nullptr>
mod_int(T a):v(a%umod){}
static constexpr mint raw(int a){
mint ret;
ret.v=a;
return ret;
}
inline uint val()const{return this->v;}
static constexpr int mod(){return m;}
inline mint &operator+=(const mint &b){
this->v+=b.v;
if(this->v>=umod)this->v-=umod;
return *this;
}
inline mint &operator-=(const mint &b){
this->v-=b.v;
if(this->v>=umod)this->v+=umod;
return *this;
}
inline mint &operator*=(const mint &b){
this->v=((unsigned long long)this->v*b.v)%umod;
return *this;
}
inline mint &operator/=(const mint &b){
*this*=b.inv();
return *this;
}
inline mint operator+()const{return *this;}
inline mint operator-()const{return mint()-*this;}
friend inline mint operator+(const mint &a,const mint &b){return mint(a)+=b;}
friend inline mint operator-(const mint &a,const mint &b){return mint(a)-=b;}
friend inline mint operator*(const mint &a,const mint &b){return mint(a)*=b;}
friend inline mint operator/(const mint &a,const mint &b){return mint(a)/=b;}
friend inline bool operator==(const mint &a,const mint &b){return a.val()==b.val();}
friend inline bool operator!=(const mint &a,const mint &b){return !(a==b);}
inline mint operator++(int){
mint ret=*this;
*this+=mint::raw(1);
return ret;
}
inline mint operator--(int){
mint ret=*this;
*this-=mint::raw(1);
return ret;
}
mint pow(long long n)const{
mint ret=mint::raw(1),a(*this);
while(n){
if(n&1)ret*=a;
a*=a;
n>>=1;
}
return ret;
}
inline mint inv()const{
assert(this->v!=0);
return pow(car-1);
}
std::optional<mint>sqrt()const{
if(this->val()<=1||this->pow((m-1)/2)==1)return std::make_optional(this->sqrt_impl());
else return std::nullopt;
}
static constexpr unsigned int order(){return car;}
friend std::istream &operator>>(std::istream &is,mint &b){
long long a;
is>>a;
b=mint(a);
return is;
}
friend std::ostream &operator<<(std::ostream &os,const mint &b){
os<<b.val();
return os;
}
};
template<int m>
struct std::hash<mod_int<m>>{
std::size_t operator()(mod_int<m>x)const{
return std::hash<unsigned int>()(x.val());
}
};
using mint998=mod_int<998244353>;
using mint107=mod_int<1000000007>;
#include<initializer_list>
template<typename T>
struct F{
private:
static int capacity;
static std::vector<T>fact,factinv,inv;
public:
static void resize(int n){
if(capacity>=n)return;
fact.resize(n+1),factinv.resize(n+1),inv.resize(n+1);
for(int i=capacity+1;i<=n;i++){
fact[i]=fact[i-1]*i;
inv[i]=-inv[T::mod()%i]*(T::mod()/i);
factinv[i]=factinv[i-1]*inv[i];
}
capacity=n;
}
static T C(int n,int k){
if(n<k)return 0;
if(k<0)return 0;
resize(n);
return fact[n]*factinv[k]*factinv[n-k];
}
static T P(int n,int k){
if(n<k)return 0;
if(k<0)return 0;
resize(n);
return fact[n]*factinv[n-k];
}
static T H(int n,int k){
if(n==0&&k==0)return 1;
return C(n+k-1,k);
}
static T factorial(int n){
resize(n);
return fact[n];
}
static T factorial_inv(int n){
resize(n);
return factinv[n];
}
static T inverse(int n){
resize(n);
return inv[n];
}
static T S(long long n,int k){
if(n<0)return 0;
if(n<k)return 0;
T ret=0;
resize(k);
for(int i=0;i<=k;i++){
ret+=fact[k]*factinv[i]*factinv[k-i]*T::raw(i).pow(n)*((k-i)&1?-1:1);
}
return ret*factinv[k];
}
template<typename... INT>
static T O(INT...k){
int n=0;
for(int i:std::initializer_list<int>{k...}){
if(i<0)return 0;
n+=i;
}
resize(n);
T ret=fact[n];
for(int i:std::initializer_list<int>{k...})ret*=factinv[i];
return ret;
}
};
template<typename T>int F<T>::capacity=1;
template<typename T>std::vector<T>F<T>::fact{1,1};
template<typename T>std::vector<T>F<T>::factinv{1,1};
template<typename T>std::vector<T>F<T>::inv{0,1};
template<typename T>
vector<vector<T>>matrix_mul(const vector<vector<T>>&a,const vector<vector<T>>&b){
assert(a[0].size()==b.size());
vector<vector<T>>ret(a.size(),vector<T>(b[0].size(),0));
rep(i,a.size())rep(j,b.size())rep(k,b[0].size())ret[i][k]+=a[i][j]*b[j][k];
return ret;
}
template<typename T>
vector<vector<T>>matrix_pow(vector<vector<T>>a,ll k){
assert(a.size()==a[0].size());
vector<vector<T>>ret(a.size(),vector<T>(a.size(),0));
rep(i,a.size())ret[i][i]=1;
while(k){
if(k&1)ret=matrix_mul(ret,a);
a=matrix_mul(a,a);
k>>=1;
}
return ret;
}
template<typename T>
T matrix_det(vector<vector<T>>a){
if(a.empty())return 1;
assert(a.size()==a[0].size());
int n=a.size();
T ret=1;
rep(i,n){
int id=-1;
reps(j,i,n){
if(a[j][i]!=0){
id=j;
break;
}
}
if(id==-1)return 0;
if(i!=id){
ret*=-1;
swap(a[i],a[id]);
}
ret*=a[i][i];
T inv=a[i][i].inv();
rep(j,n)a[i][j]*=inv;
reps(j,i+1,n){
T x=a[j][i];
rep(k,n)a[j][k]-=a[i][k]*x;
}
}
return ret;
}
template<typename T>
vector<vector<T>>matrix_inv(vector<vector<T>>a){
assert(a.size()==a[0].size());
int n=a.size();
vector<vector<T>>ret(n,vector<T>(n,0));
rep(i,n)ret[i][i]=1;
rep(i,n){
int id=-1;
reps(j,i,n){
if(a[j][i]!=0){
id=j;
break;
}
}
if(id==-1)return vector<vector<T>>(n,vector<T>(n,0));
if(i!=id){
swap(a[i],a[id]);
swap(ret[i],ret[id]);
}
T inv=a[i][i].inv();
rep(j,n){
a[i][j]*=inv;
ret[i][j]*=inv;
}
rep(j,n)if(i!=j){
T x=a[j][i];
rep(k,n){
a[j][k]-=a[i][k]*x;
ret[j][k]-=ret[i][k]*x;
}
}
}
return ret;
}
template<typename T>
std::pair<std::vector<T>,std::vector<std::vector<T>>>matrix_linear_equation(std::vector<std::vector<T>>mat,std::vector<T>c){
int n=mat.size(),m=mat[0].size();
assert(n==c.size());
int y=0;
std::vector<std::pair<int,int>>detv;
std::vector<int>rankv;
for(int d=0;d<m;d++){
int id=-1;
for(int i=y;i<n;i++)if(mat[i][d].val()){
id=i;
break;
}
if(id==-1){
rankv.push_back(d);
continue;
}
for(int j=d;j<n;j++)std::swap(mat[y][j],mat[id][j]);
std::swap(c[y],c[id]);
T inv=mat[y][d].inv();
for(int j=d;j<m;j++)mat[y][j]*=inv;
c[y]*=inv;
for(int i=0;i<n;i++)if(i!=y){
c[i]-=mat[i][d]*c[y];
for(int j=m-1;j>=d;j--)mat[i][j]-=mat[i][d]*mat[y][j];
}
detv.push_back({d,y++});
}
for(int i=y;i<n;i++)if(c[i].val())return std::make_pair(std::vector<T>{},std::vector<std::vector<T>>{});
std::vector<T>ret1(m);
std::vector<std::vector<T>>ret2(rankv.size(),std::vector<T>(m));
for(auto [x,i]:detv)ret1[x]=c[i];
for(int d=0;d<rankv.size();d++){
ret2[d][rankv[d]]=-1;
for(auto [x,i]:detv)ret2[d][x]=mat[i][rankv[d]];
}
return std::make_pair(ret1,ret2);
}
template<typename T>
std::vector<T>inverse_table(long long l,long long r){
assert(l<=r);
if(l==r)return std::vector<T>{};
l=(l-1)%T::mod();
r=(r-1)%T::mod();
if(l<0)l+=T::mod();
if(r<0)r+=T::mod();
if(l>r){
int n=std::max(r,T::mod()-l);
std::vector<T>fact(n+1),factinv(n+1);
fact[0]=1;
for(int i=1;i<=n;i++)fact[i]=fact[i-1]*T::raw(i);
factinv[n]=fact[n].inv();
for(int i=n-1;i>=0;i--)factinv[i]=factinv[i+1]*T::raw(i+1);
std::vector<T>ret;
ret.reserve(r-l+T::mod());
for(int i=l+1;i<T::mod();i++)ret.push_back(-fact[T::mod()-i-1]*factinv[T::mod()-i]);
ret.push_back(0);
for(int i=1;i<=r;i++)ret.push_back(fact[i-1]*factinv[i]);
return ret;
}
else{
int n=r-l;
std::vector<T>ret(n);
ret[0]=1;
for(int i=1;i<n;i++)ret[i]=ret[i-1]*T::raw(l+i);
T inv=(ret[n-1]*(n+l)).inv();
for(int i=n-1;i>=0;i--){
ret[i]*=inv;
inv*=T::raw(i+1+l);
}
return ret;
}
}
#include<concepts>
template<typename T>
constexpr std::enable_if_t<std::numeric_limits<T>::digits<=32,int>msb(T n){return n==0?-1:31-__builtin_clz(n);}
template<typename T>
constexpr std::enable_if_t<(std::numeric_limits<T>::digits>32),int>msb(T n){return n==0?-1:63-__builtin_clzll(n);}
template<typename T>
constexpr std::enable_if_t<std::numeric_limits<T>::digits<=32,int>lsb(T n){return n==0?-1:__builtin_ctz(n);}
template<typename T>
constexpr std::enable_if_t<(std::numeric_limits<T>::digits>32),int>lsb(T n){return n==0?-1:__builtin_ctzll(n);}
template<typename T>
constexpr std::enable_if_t<std::is_integral_v<T>,T>floor_pow2(T n){return n==0?0:T(1)<<msb(n);}
template<typename T>
constexpr std::enable_if_t<std::is_integral_v<T>,T>ceil_pow2(T n){return n<=1?1:T(1)<<(msb(n-1)+1);}
template<std::integral T>
constexpr T safe_div(T a,T b){return a>=0?a/b:-((-a+b-1)/b);}
template<typename T>
constexpr std::enable_if_t<(std::numeric_limits<T>::digits<=32),T>pow_mod(T a,T n,T mod){
using u64=unsigned long long;
u64 res=1;
while(n>0){
if(n&1)res=((u64)res*a)%mod;
a=((u64)a*a)%mod;
n>>=1;
}
return T(res);
}
template<typename T>
constexpr std::enable_if_t<(std::numeric_limits<T>::digits>32),T>pow_mod(T a,T n,T mod){
using u128=__uint128_t;
u128 res=1;
while(n>0){
if(n&1)res=((u128)res*a)%mod;
a=((u128)a*a)%mod;
n>>=1;
}
return T(res);
}
constexpr int primitive_root_constexpr(int x){
if(x==167772161)return 3;
if(x==469762049)return 3;
if(x==754974721)return 11;
if(x==880803841)return 26;
if(x==998244353)return 3;
if(x==2)return 1;
int x2=x;
int p[20]={};
int c=0;
x--;
for(int i=2;i*i<=x;i++){
if(x%i==0){
p[c++]=i;
while(x%i==0)x/=i;
}
}
if(x!=1)p[c++]=x;
x=x2;
for(int g=2;;g++){
bool ok=true;
for(int i=0;i<c;i++)if(pow_mod(g,(x-1)/p[i],x)==1){
ok=false;
break;
}
if(ok)return g;
}
}
template<int m>
struct ntt_root{
static constexpr int rank2=lsb(m-1);
static constexpr int g=primitive_root_constexpr(m);
std::array<int,rank2+1>root,invroot;
std::array<int,std::max(0,rank2-1)>rate2,invrate2;
std::array<int,std::max(0,rank2-2)>rate3,invrate3;
constexpr ntt_root(){
root[rank2]=pow_mod(g,m>>rank2,m);
invroot[rank2]=pow_mod(root[rank2],m-2,m);
for(int i=rank2-1;i>=0;i--){
root[i]=(long long)root[i+1]*root[i+1]%m;
invroot[i]=(long long)invroot[i+1]*invroot[i+1]%m;
}
int prod=1,invprod=1;
for(int i=0;i<rank2-1;i++){
rate2[i]=(long long)root[i+2]*prod%m;
invrate2[i]=(long long)invroot[i+2]*invprod%m;
prod=(long long)prod*invroot[i+2]%m;
invprod=(long long)invprod*root[i+2]%m;
}
prod=invprod=1;
for(int i=0;i<rank2-2;i++){
rate3[i]=(long long)root[i+3]*prod%m;
invrate3[i]=(long long)invroot[i+3]*invprod%m;
prod=(long long)prod*invroot[i+3]%m;
invprod=(long long)invprod*root[i+3]%m;
}
}
};
template<typename T>
void dft(std::vector<T>&a){
static constexpr ntt_root<T::mod()>r;
static constexpr unsigned long long mod2=(unsigned long long)T::mod()*T::mod();
int n=a.size();
int h=lsb(n);
int len=0;
while(len<h){
if(h-len==1){
T rot=T::raw(1);
for(int s=0;s<(1<<len);s++){
int of=s*2;
T u=a[of],v=a[of+1]*rot;
a[of]=u+v;
a[of+1]=u-v;
rot*=T::raw(r.rate2[lsb(~(unsigned int)s)]);
}
len++;
}
else{
int p=1<<(h-len-2);
T rot=T::raw(1),imag=T::raw(r.root[2]);
for(int s=0;s<(1<<len);s++){
const unsigned long long rot1=rot.val(),rot2=rot1*rot1%T::mod(),rot3=rot1*rot2%T::mod();
int of=s<<(h-len);
for(int i=0;i<p;i++){
const unsigned long long a0=a[i+of].val(),a1=(unsigned long long)a[i+of+p].val()*rot1,a2=(unsigned long long)a[i+of+p*2].val()*rot2,a3=(unsigned long long)a[i+of+p*3].val()*rot3;
const unsigned long long m=(unsigned long long)T(a1+mod2-a3).val()*imag.val();
const unsigned long long k=mod2-a2;
a[i+of]=a0+a2+a1+a3;
a[i+of+p]=a0+a2+(mod2*2-a1-a3);
a[i+of+p*2]=a0+k+m;
a[i+of+p*3]=a0+k+(mod2-m);
}
rot*=T::raw(r.rate3[lsb(~(unsigned int)s)]);
}
len+=2;
}
}
}
template<typename T>
void idft(std::vector<T>&a){
static constexpr ntt_root<T::mod()>r;
int n=a.size();
int h=lsb(n);
int len=h;
while(len){
if(len==1){
int p=1<<(h-1);
for(int i=0;i<p;i++){
T u=a[i],v=a[i+p];
a[i]=u+v;
a[i+p]=u-v;
}
len--;
}
else{
int p=1<<(h-len);
T rot=T::raw(1),imag=T::raw(r.invroot[2]);
for(int s=0;s<(1<<(len-2));s++){
const unsigned long long rot1=rot.val(),rot2=rot1*rot1%T::mod(),rot3=rot1*rot2%T::mod();
int of=s<<(h-len+2);
for(int i=0;i<p;i++){
const unsigned long long a0=a[i+of].val(),a1=a[i+of+p].val(),a2=a[i+of+p*2].val(),a3=a[i+of+p*3].val();
const unsigned long long k=T((T::mod()+a2-a3)*imag.val()).val();
a[i+of]=a0+a1+a2+a3;
a[i+of+p]=(a0+T::mod()-a1+k)*rot1;
a[i+of+p*2]=(a0+a1+T::mod()*2-a2-a3)*rot2;
a[i+of+p*3]=(a0+T::mod()*2-a1-k)*rot3;
}
rot*=T::raw(r.invrate3[lsb(~(unsigned int)s)]);
}
len-=2;
}
}
}
template<typename T>
std::vector<T>ntt_convolution(std::vector<T> a,std::vector<T> b){
int n=a.size(),m=b.size(),s=n+m-1;
if(std::min(n,m)<60){
std::vector<T>ret(s,0);
if(n<m)for(int i=0;i<m;i++)for(int j=0;j<n;j++)ret[i+j]+=a[j]*b[i];
else for(int i=0;i<n;i++)for(int j=0;j<m;j++)ret[i+j]+=a[i]*b[j];
return ret;
}
int z=ceil_pow2(s);
a.resize(z,0);
b.resize(z,0);
dft(a),dft(b);
std::vector<T>c(z);
for(int i=0;i<z;i++)c[i]=a[i]*b[i];
idft(c);
T g=T::raw(z).inv();
for(int i=0;i<s;i++)c[i]*=g;
return {c.begin(),c.begin()+s};
}
template<typename M>
struct SWAG{
private:
using S=typename M::S;
std::vector<S>a,b;
std::vector<S>proda,prodb;
void eval(){
proda.resize(a.size()+1,M::e());
prodb.resize(b.size()+1,M::e());
for(int i=1;i<(int)proda.size();i++)proda[i]=M::op(a[i-1],proda[i-1]);
for(int i=1;i<(int)prodb.size();i++)prodb[i]=M::op(prodb[i-1],b[i-1]);
}
public:
SWAG():proda{M::e()},prodb{M::e()}{}
void push_front(S v){
a.emplace_back(move(v));
proda.emplace_back(M::op(a.back(),proda.back()));
}
void push_back(S v){
b.emplace_back(std::move(v));
prodb.emplace_back(M::op(prodb.back(),b.back()));
}
inline S front()const{return a.empty()?b[0]:a.back();}
inline S back()const{return b.empty()?a[0]:b.back();}
void pop_front(){
if(a.empty()){
int mid=b.size()/2;
a=std::vector<S>(b.rbegin()+mid,b.rend());
b=std::vector<S>(b.end()-mid,b.end());
eval();
}
a.pop_back();
proda.pop_back();
}
void pop_back(){
if(b.empty()){
int mid=a.size()/2;
b=vector<S>(a.rbegin()+mid,a.rend());
a=vector<S>(a.end()-mid,a.end());
eval();
}
b.pop_back();
prodb.pop_back();
}
inline S all_prod()const{return M::op(proda.back(),prodb.back());}
inline int size()const{return a.size()+b.size();}
inline bool empty()const{return a.empty()&&b.empty();}
};
template<typename T>
std::vector<T>shift_sample_point(const std::vector<T>&f,T k,int m=-1){
int n=f.size();
if(m==-1)m=n;
int c=k.val();
std::vector<T>a=inverse_table<T>(c-n+1,c+m);
std::vector<T>b(n);
for(int i=0;i<n;i++){
b[i]=F<T>::factorial_inv(i)*F<T>::factorial_inv(n-i-1)*f[i];
if((n-i-1)&1)b[i]=-b[i];
}
int s=ceil_pow2(n+m-1);
a.resize(s);
b.resize(s);
dft(a),dft(b);
for(int i=0;i<s;i++)a[i]*=b[i];
idft(a);
a=std::vector<T>(a.begin()+n-1,a.begin()+n+m-1);
T inv=T::raw(s).inv();
struct M{
using S=T;
static inline S op(const S&x,const S&y){return x*y;}
static inline S e(){return T::raw(1);}
};
SWAG<M>swag;
for(int i=c-n+1;i<=c;i++)swag.push_back(i);
for(int i=0;i<m;i++){
T now=swag.all_prod();
if(now.val())a[i]*=now*inv;
else a[i]=f[(i+c)%T::mod()];
swag.pop_front();
swag.push_back(c+i+1);
}
return a;
}
template<typename T>
vector<vector<T>>poly_matrix_prod(const vector<vector<vector<T>>>&mat,ll k){
int n=mat.size();
auto shift=[](const vector<vector<vector<T>>>&m,T x){
int d=m.size(),n=m[0].size();
vector<vector<vector<T>>>ret(d,vector<vector<T>>(n,vector<T>(n)));
rep(i,n)rep(j,n){
vector<T>g(d);
rep(l,d)g[l]=m[l][i][j];
g=shift_sample_point(g,x);
rep(l,d)ret[l][i][j]=g[l];
}
return ret;
};
auto fx=[](const vector<T>&f,T x)->T {
T ret=0,power=1;
rep(i,f.size()){
ret+=f[i]*power;
power*=x;
}
return ret;
};
int deg=1;
rep(i,n)rep(j,n)chmax<int>(deg,mat[i][j].size()-1);
ll v=1;
while(deg*v*v<k)v*=2;
vector<vector<vector<T>>>G(deg+1,vector<vector<T>>(n,vector<T>(n)));
T inv=T(v).inv();
rep(i,deg+1){
T x=T(v)*i;
rep(j,n)rep(l,n)G[i][j][l]=fx(mat[j][l],x);
}
for(ll w=1;w<v;w<<=1){
T w2=w;
auto g1=shift(G,w2*inv);
auto g2=shift(G,(w2*deg*v+v)*inv);
auto g3=shift(G,(w2*deg*v+v+w2)*inv);
rep(i,w*deg+1){
G[i]=matrix_mul(g1[i],G[i]),g2[i]=matrix_mul(g3[i],g2[i]);
}
copy(g2.begin(),g2.end()-1,back_inserter(G));
}
vector<vector<T>>ret(n,vector<T>(n,0));
rep(i,n)ret[i][i]=1;
ll i=0;
while(i+v<=k)ret=matrix_mul(G[i/v],ret),i+=v;
while(i<k){
vector<vector<T>>mat2(n,vector<T>(n));
rep(j,n)rep(l,n)mat2[j][l]=fx(mat[j][l],i);
ret=matrix_mul(mat2,ret);
i++;
}
return ret;
}
template<typename T>
vector<vector<T>>find_p_recursive(vector<T>a,int d){
int n=a.size();
int k=(n+2)/(d+2)-1;
if(k<=0)return {};
int m=(k+1)*(d+1);
vector<vector<T>>mat(m-1,vector<T>(m));
rep(i,m-1){
rep(j,k+1){
T p=1;
rep(l,d+1){
mat[i][(d+1)*j+l]=p*a[i+j];
p*=i+j;
}
}
}
auto g=matrix_linear_equation(mat,vector<T>(m-1,0)).second;
if(g.empty())return vector<vector<T>>{};
vector<T>c=g[0];
while(all_of(c.end()-d-1,c.end(),[](T x){return x==0;}))c.erase(c.end()-d-1,c.end());
k=c.size()/(d+1);
vector<vector<T>>ret(k);
rep(j,k){
int i=j*(d+1);
vector<T>f(1+d,0),sum(1+d,0);
f[0]=1;
rep(l,d+1){
rep(x,1+d)sum[x]+=f[x]*c[i+l];
for(int x=d;x>=1;x--)f[x]=f[x-1]+f[x]*j;
f[0]*=j;
}
ret[j]=sum;
}
return ret;
}
template<typename T>
T calc_p_recursive_naive(const vector<T>&a,const vector<vector<T>>&coef,ll n){
if(a.size()>n)return a[n];
int r=coef.size()-1;
int d=coef[0].size()-1;
assert(a.size()>=r);
vector<pair<T,T>>que(r);
rep(i,r)que[i]={a[i],1};
int ptr=0;
for(ll i=0;i<=n-r;i++){
T num=0,den=1;
rep(j,r){
T c=0,power=1;
rep(k,d+1){
c+=coef[j][k]*power;
power*=i;
}
int idx=ptr+j;
if(idx>=r)idx-=r;
num=num*que[idx].second+que[idx].first*den*c;
den=den*que[idx].second;
}
T c=0,power=1;
rep(k,d+1){
c+=coef[r][k]*power;
power*=i;
}
que[ptr++]=make_pair(num,-den*c);
if(ptr==r)ptr=0;
}
if(--ptr==-1)ptr=r-1;
return que[ptr].first/que[ptr].second;
}
template<typename T>
T calc_p_recursive_fast(const vector<T>&a,const vector<vector<T>>&coef,ll n){
if(a.size()>n)return a[n];
int deg=coef.size()-1;
vector<vector<vector<T>>>num(deg,vector<vector<T>>(deg));
rep(i,deg){
num[0][i]=coef[deg-i-1];
rep(j,num[0][i].size())num[0][i][j]=-num[0][i][j];
}
reps(i,1,deg)num[i][i-1]=coef[deg];
vector<vector<vector<T>>>den={{coef[deg]}};
vector<vector<T>>a0(deg,vector<T>(deg,0));
rep(i,deg)a0[i][0]=a[deg-i-1];
T ret=matrix_mul(poly_matrix_prod(num,n-deg+1),a0)[0][0];
ret/=poly_matrix_prod(den,n-deg+1)[0][0];
return ret;
}
template<typename T>
T kth_term_p_recursive(vector<T>a,int K){
if(K<a.size())return a[K];
if(all_of(all(a),[](T x){return x.val()==0;}))return 0;
assert(a.size()>=5);
int n=a.size()-2;
for(int d=0;;d++){
int k=(n+2)/(d+2)-1;
if(k<=0)break;
int s=(n+2)/(d+2)*(d+2)-2;
vector<T>prefix(a.begin(),a.begin()+s);
vector<vector<T>>b=find_p_recursive(prefix,d);
if(b.empty())continue;
bool ok=true;
reps(i,s,a.size()){
if(calc_p_recursive_naive(prefix,b,i).val()!=a[i].val()){
ok=false;
break;
}
}
if(ok){
if constexpr(T::mod()==998244353)return calc_p_recursive_fast(prefix,b,K);
else return calc_p_recursive_naive(prefix,b,K);
}
}
cerr<<"not found\n";
__builtin_unreachable();
}
template<typename T>
std::vector<T>enumerate_p_recursive(std::vector<T>a,int K){
if(a.size()>K){
a.resize(K);
return a;
}
int n=(int)a.size()-2;
for(int d=0;;d++){
int k=(n+2)/(d+2)-1;
if(k<=0)break;
int s=(n+2)/(d+2)*(d+2)-2;
std::vector<T>prefix(a.begin(),a.begin()+s);
std::vector<std::vector<T>>b=find_p_recursive(prefix,d);
if(b.empty())continue;
int r=(int)b.size()-1;
int deg=(int)b[0].size()-1;
std::vector<T>res(K);
for(int i=0;i<r;i++)res[i]=prefix[i];
for(int i=0;i<K-r;i++){
T &v=res[i+r];
for(int j=0;j<r;j++){
T c=T::raw(0),power=T::raw(1);
for(int l=0;l<=deg;l++){
c+=b[j][l]*power;
power*=T::raw(i);
}
v+=c*res[i+j];
}
T c=T::raw(0),power=T::raw(1);
for(int l=0;l<=deg;l++){
c+=b[r][l]*power;
power*=T::raw(i);
}
v/=-c;
if(i+r<(int)a.size()&&v!=a[i+r]){
res.clear();
break;
}
}
if(!res.empty())return res;
}
__builtin_unreachable();
}
using mint=mint998;
void SOLVE(){
string s;
cin>>s>>s;
int n=s.size()/2;
if(all_of(s.begin(),s.begin()+n,[&](char c){return c=='(';})){
auto coef=[](int x,int y)->mint {
if(x==0||y==0)return x+y==1;
mint a=F<mint>::C(x-1+y-1,x-1),b=F<mint>::C(x+y-2,x),c=F<mint>::C(x-2+y,x-2),d=F<mint>::C(x-1+y-1,x-1);
return (a*d-b*c)*2;
};
vector<mint>dp(50);
dp[0]=1;
rep(i,1,50){
rep(j,i){
mint c=0;
int k=i-j;
rep(l,k+1)c+=coef(l,k-l);
debug(i,j,c);
dp[i]+=dp[j]*c;
}
}
cout<<kth_term_p_recursive(dp,n)<<endl;
}
else cout<<mint(2).pow(n)<<endl;
}