#include<cassert>
#include <bits/stdc++.h>
#ifndef IO_HPP
#define IO_HPP
#include<iostream>
#include<vector>
#include<queue>
#include<stack>
#include<array>
#include<map>
#include<unordered_map>
#include<set>
#include<unordered_set>
using namespace std;
template<typename T1,typename T2>istream &operator>>(istream&,pair<T1,T2>&);
template<typename...Args>istream &operator>>(istream&,tuple<Args...>&a);
template<typename T>istream &operator>>(istream&is,vector<T>&a);
template<typename T,size_t N>istream &operator>>(istream&is,array<T,N>&a);
template<typename T1,typename T2>
istream &operator>>(istream&is,pair<T1,T2>&a){
  is>>a.first>>a.second;
  return is;
}
template<size_t pos,typename...Args>
void read_tuple(istream&is,tuple<Args...>&a){
  if constexpr(pos<tuple_size<tuple<Args...>>::value){
    is>>get<pos>(a);
    read_tuple<pos+1>(is,a);
  }
}
template<typename...Args>
istream &operator>>(istream&is,tuple<Args...>&a){
  read_tuple<0>(is,a);
  return is;
}
template<typename T>
istream &operator>>(istream&is,vector<T>&a){
  for(T&x:a)is>>x;
  return is;
}
template<typename T,size_t N>
istream &operator>>(istream&is,array<T,N>&a){
  for(T&x:a)is>>x;
  return is;
}
template<typename T1,typename T2>ostream &operator<<(ostream&os,const pair<T1,T2>&);
template<typename...Args>ostream &operator<<(ostream&os,const tuple<Args...>&);
template<typename T>ostream &operator<<(ostream&os,const vector<T>&);
template<typename T,typename Seq,typename Comp>ostream &operator<<(ostream&os,priority_queue<T,Seq,Comp>);
template<typename T>ostream &operator<<(ostream&os,queue<T>);
template<typename T>ostream &operator<<(ostream&os,deque<T>);
template<typename T>ostream &operator<<(ostream&os,stack<T>);
template<typename T,size_t N>ostream &operator<<(ostream&os,const array<T,N>&);
template<typename Key,typename Val,typename Comp>ostream &operator<<(ostream&os,const map<Key,Val,Comp>&);
template<typename Key,typename Val,typename Hash>ostream &operator<<(ostream&os,const unordered_map<Key,Val,Hash>&);
template<typename T,typename Comp>ostream &operator<<(ostream&os,const set<T,Comp>&);
template<typename T,typename Comp>ostream &operator<<(ostream&os,const multiset<T,Comp>&);
template<typename T,typename Hash>ostream &operator<<(ostream&os,const unordered_set<T,Hash>&);
template<typename T1,typename T2>
ostream &operator<<(ostream&os,const pair<T1,T2>&a){
  os<<a.first<<' '<<a.second;
  return os;
}
template<size_t pos,typename...Args>
void write_tuple(ostream&os,const tuple<Args...>&a){
  if constexpr(pos<tuple_size<tuple<Args...>>::value){
    if constexpr(pos>0)os<<' ';
    os<<get<pos>(a);
    write_tuple<pos+1>(os,a);
  }
}
template<typename...Args>
ostream &operator<<(ostream&os,const tuple<Args...>&a){
  write_tuple<0>(os,a);
  return os;
}
template<typename T>
ostream &operator<<(ostream&os,const vector<T>&a){
  os<<'{';
  for(int i=0;i<(int)a.size();i++){
    os<<a[i];
    if(i+1!=a.size())os<<',';
  }
  os<<'}';
  return os;
}
template<typename T,typename Seq,typename Comp>
ostream &operator<<(ostream&os,priority_queue<T,Seq,Comp>a){
  os<<'{';
  if(!a.empty()){
    os<<a.top();a.pop();
    while(!a.empty()){
      os<<',';
      os<<a.top();
      a.pop();
    }
  }
  os<<'}';
  return os;
}
template<typename T>
ostream &operator<<(ostream&os,queue<T>a){
  os<<'{';
  if(!a.empty()){
    os<<a.front();a.pop();
    while(!a.empty()){
      os<<',';
      os<<a.front();
      a.pop();
    }
  }
  os<<'}';
  return os;
}
template<typename T>
ostream &operator<<(ostream&os,deque<T>a){
  os<<'{';
  if(!a.empty()){
    os<<a.front();a.pop_front();
    while(!a.empty()){
      os<<',';
      os<<a.front();
      a.pop_front();
    }
  }
  os<<'}';
  return os;
}
template<typename T>
ostream &operator<<(ostream&os,stack<T>a){
  os<<'{';
  if(!a.empty()){
    os<<a.top();a.pop();
    while(!a.empty()){
      os<<',';
      os<<a.top();
      a.pop();
    }
  }
  os<<'}';
  return os;
}
template<typename T,size_t N>
ostream &operator<<(ostream&os,const array<T,N>&a){
  os<<'{';
  for(int i=0;i<(int)a.size();i++){
    os<<a[i];
    if(i+1!=a.size())os<<',';
  }
  os<<'}';
  return os;
}
template<typename Key,typename Val,typename Comp>
ostream &operator<<(ostream&os,const map<Key,Val,Comp>&a){
  if(a.empty()){
    os<<"{}";
    return os;
  }
  auto itr=a.begin();
  os<<"{["<<itr->first<<","<<itr->second<<']';
  while(++itr!=a.end())os<<",["<<itr->first<<','<<itr->second<<']';
  os<<'}';
  return os;
}
template<typename Key,typename Val,typename Hash>
ostream &operator<<(ostream&os,const unordered_map<Key,Val,Hash>&a){
  if(a.empty()){
    os<<"{}";
    return os;
  }
  auto itr=a.begin();
  os<<"{["<<itr->first<<","<<itr->second<<']';
  while(++itr!=a.end())os<<",["<<itr->first<<','<<itr->second<<']';
  os<<'}';
  return os;
}
template<typename T,typename Comp>
ostream &operator<<(ostream&os,const set<T,Comp>&a){
  if(a.empty()){
    os<<"{}";
    return os;
  }
  auto itr=a.begin();
  os<<'{'<<*itr;
  while(++itr!=a.end())os<<','<<*itr;
  os<<'}';
  return os;
}
template<typename T,typename Comp>
ostream &operator<<(ostream&os,const multiset<T,Comp>&a){
  if(a.empty()){
    os<<"{}";
    return os;
  }
  auto itr=a.begin();
  os<<'{'<<*itr;
  while(++itr!=a.end())os<<','<<*itr;
  os<<'}';
  return os;
}
template<typename T,typename Hash>
ostream &operator<<(ostream&os,const unordered_set<T,Hash>&a){
  if(a.empty()){
    os<<"{}";
    return os;
  }
  auto itr=a.begin();
  os<<'{'<<*itr;
  while(++itr!=a.end())os<<','<<*itr;
  os<<'}';
  return os;
}
#endif
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>void operator++(pair<T1,T2>&a,int){a.first++,a.second++;}
template<typename T1,typename T2>void operator--(pair<T1,T2>&a,int){a.first--,a.second--;}
template<typename T>void operator++(vector<T>&a,int){for(auto &i:a)i++;}
template<typename T>void operator--(vector<T>&a,int){for(auto &i:a)i--;}
using vref=typename vector<bool>::reference;
vref operator|=(vref a,bool b){a=a|b;return a;}
vref operator&=(vref a,bool b){a=a&b;return a;}
vref operator^=(vref a,bool b){a=a^b;return a;}
#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)
#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>
#include<variant>
struct BarrettReduction{
private:
  using i64=long long;
  using u64=unsigned long long;
  using u32=unsigned int;
  using u128=__uint128_t;
  u32 m;
  u64 im;
public:
  BarrettReduction():m(0),im(0){}
  BarrettReduction(u32 n):m(n),im(u64(-1)/n+1){}
  inline i64 quo(u64 x)const{
    if(m==1)return x;
    u64 y=u64((u128(x)*im)>>64);
    u32 r=x-y*m;
    return m<=r?y-1:y;
  }
  inline u32 rem(u64 x)const{
    if(m==1)return 0;
    u64 y=u64((u128(x)*im)>>64);
    u32 r=x-y*m;
    return m<=r?r+m:r;
  }
  inline std::pair<u64,u32>quo_rem(u64 x)const{
    if(m==0)return std::make_pair(x,0);
    u64 y=u64((u128(x)*im)>>64);
    u32 r=x-y*m;
    return m<=r?std::make_pair(y-1,r+m):std::make_pair(y,r);
  }
  inline u32 pow(u32 a,u64 p)const{
    u32 res=m!=1;
    while(p){
      if(p&1)res=rem(u64(res)*a);
      a=rem(u64(a)*a);
      p>>=1;
    }
    return res;
  }
};
constexpr std::pair<long long,long long>ext_gcd(long long a,long long b){
  if(b==0)return std::make_pair(1,0);
  auto [x,y]=ext_gcd(b,a%b);
  std::swap(x,y);
  return std::make_pair(x,y-a/b*x);
}
template<std::signed_integral T>
constexpr std::pair<T,T> inv_mod(T a,T b){
  a%=b;
  if(a<0)a+=b;
  if(a==0)return std::make_pair(b,0);
  T s=b,t=a;
  T m0=0,m1=1;
  while(t){
    T u=s/t;
    s-=t*u;
    m0-=m1*u;
    std::swap(s,t);
    std::swap(m0,m1);
  }
  if(m0<0)m0+=b/s;
  return std::make_pair(s,m0);
}
template<typename T,int id>
struct arbitrary_modint{
  using value_type=std::make_unsigned_t<T>;
  using mul_type=std::conditional_t<(std::numeric_limits<T>::digits<=32),uint64_t,__uint128_t>;
private:
  using mint=arbitrary_modint;
  value_type v;
  static value_type umod;
  static std::conditional_t<(std::numeric_limits<T>::digits<=32),BarrettReduction,std::monostate>br;
  mint sqrt_impl()const{
    if(this->val()<=1)return *this;
    if(umod%8==1){
      mint b=2;
      while(b.pow((umod-1)/2).val()==1)b++;
      value_type m2=umod-1;
      int 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((value_type(1))<<(e-j-1));
        x*=z;
        z*=z;
        y*=z;
        e=j;
      }
      return x;
    }
    else if(umod%8==5){
      mint res=this->pow((umod+3)/8);
      if((res*res).val()==this->val())return res;
      else return res*mint(2).pow((umod-1)/4);
    }
    else return this->pow((umod+1)/4);
  }
public:
  arbitrary_modint():v(0){}
  template<typename U,std::enable_if_t<std::signed_integral<U>||std::is_same_v<U,__int128_t>,std::nullptr_t> =nullptr>
  arbitrary_modint(U x){
    x%=std::make_signed_t<value_type>(umod);
    v=x>=0?x:x+umod;
  }
  template<typename U,std::enable_if_t<std::unsigned_integral<U>||std::is_same_v<U,__uint128_t>,std::nullptr_t> =nullptr>
  arbitrary_modint(U x):v(x%umod){}
  static void set_mod(T m){
    assert(1<=m);
    umod=m;
    if constexpr(std::numeric_limits<T>::digits<=32)br=BarrettReduction(umod);
  }
  static void read_mod(){
    std::cin>>umod;
    if constexpr(std::numeric_limits<T>::digits<=32)br=BarrettReduction(umod);
  }
  static T mod(){return umod;}
  static mint raw(T x){
    mint res;
    res.v=x;
    return res;
  }
  inline T val()const{return v;}
  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){
    mul_type v=mul_type(this->v)*mul_type(b.v);
    if constexpr(std::numeric_limits<T>::digits<=32)this->v=br.rem(v);
    else this->v=v%umod;
    return *this;
  }
  inline mint &operator/=(const mint&b){return *this*=b.inv();}
  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;}
  auto operator<=>(const mint&)const=default;
  inline mint operator++(int){
    mint res=*this;
    this->v++;
    if(this->v==umod)this->v=0;
    return res;
  }
  inline mint operator--(int){
    mint res=*this;
    if(this->v==0)this->v=umod;
    this->v--;
    return res;
  }
  template<std::integral U>
  mint pow(U k)const{
    if constexpr(std::is_signed_v<U>){
      assert(0<=k);
    }
    mint res=1,a(*this);
    while(k){
      if(k&1)res*=a;
      a*=a;
      k>>=1;
    }
    return res;
  }
  mint inv()const{
    mint res;
    auto [g,x]=inv_mod<std::make_signed_t<value_type>>(this->v,umod);
    assert(g==1);
    res.v=x;
    return res;
  }
  std::optional<mint>sqrt()const{
    if(this->val()<=1||this->pow((umod-1)/2)==1)return std::make_optional(this->sqrt_impl());
    else return std::nullopt;
  }
  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<typename T,int id>typename arbitrary_modint<T,id>::value_type arbitrary_modint<T,id>::umod=2;
template<typename T,int id>std::conditional_t<(std::numeric_limits<T>::digits<=32),BarrettReduction,std::monostate>arbitrary_modint<T,id>::br;
template<typename T,int id>
struct std::hash<arbitrary_modint<T,id>>{
  std::size_t operator()(arbitrary_modint<T,id>x)const{
    return std::hash<unsigned int>()(x.val());
  }
};
#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/b-(a%b&&(a^b)<0);}
template<std::integral T>
constexpr T safe_ceil(T a,T b){return a/b+(a%b&&(a^b)>0);}
template<typename T,typename U,typename Func>
T semigroup_pow(T a,U k,const Func&f){
  static_assert(std::is_convertible_v<Func,std::function<T(T,T)>>);
  assert(k>0);
  if(k<=8){
    if(k==1)return a;
    else if(k==2)return f(a,a);
    else if(k==3)return f(a,f(a,a));
    else if(k==4){
      a=f(a,a);
      return f(a,a);
    }
    else if(k==5){
      T a2=f(a,a);
      return f(a,f(a2,a2));
    }
    else if(k==6){
      a=f(a,f(a,a));
      return f(a,a);
    }
    else if(k==7){
      T a3=f(a,f(a,a));
      return f(a,f(a3,a3));
    }
    else{
      a=f(a,a);
      a=f(a,a);
      return f(a,a);
    }
  }
  int m=msb(k)/3;
  T a2=f(a,a);
  T a3=f(a,a2);
  T a5=f(a2,a3);
  T a7=f(a2,a5);
  T res=[&]()->T {
    switch(k>>(m*3)&7){
      case 1:return a;
      case 2:return a2;
      case 3:return a3;
      case 4:return f(a2,a2);
      case 5:return a5;
      case 6:return f(a3,a3);
      case 7:return a7;
    }
    __builtin_unreachable();
  }();
  while(m--){
    int x=k>>(m*3)&7;
    if(x==4){
      res=f(res,res);
      res=f(res,a);
      res=f(res,res);
      res=f(res,res);
    }
    else if(x==6){
      res=f(res,res);
      res=f(res,res);
      res=f(res,a3);
      res=f(res,res);
    }
    else{
      res=f(res,res);
      res=f(res,res);
      res=f(res,res);
      if(x==1)res=f(res,a);
      else if(x==2)res=f(res,a2);
      else if(x==3)res=f(res,a3);
      else if(x==5)res=f(res,a5);
      else if(x==7)res=f(res,a7);
    }
  }
  return 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 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);
}
template<typename T,int p1,int p2,int p3>
T crt3(int a1,int a2,int a3){
  static_assert(p1<p2&&p2<p3);
  static constexpr long long x=pow_mod<int>(p1,p2-2,p2);
  static constexpr long long y=pow_mod<int>((long long)p1*p2%p3,p3-2,p3);
  long long c=(a2-a1+p2)*x%p2;
  long long c2=a1+c*p1;
  c=(a3-c2%p3+p3)*y%p3;
  return T(c2)+T(c)*T(p1)*T(p2);
}
#include<initializer_list>
constexpr bool isprime_constexpr(unsigned long long n){
  if(n==998244353)return true;
  if(n==1000000007)return true;
  if(n<64)return 2891462833508853932ll>>n&1;
  if(n%2==0)return false;
  unsigned long long d=n-1;
  int s=0;
  while(!(d&1))d>>=1,s++;
  int q=63;
  while(!(d>>q))q--;
  unsigned long long r=n;
  for(int i=0;i<5;i++)r*=2-r*n;
  auto redc=[&r,&n](__uint128_t x)->unsigned long long {
    x=(x+__uint128_t((unsigned long long)x*-r)*n)>>64;
    return x>=n?x-n:x;
  };
  __uint128_t r2=-__uint128_t(n)%n;
  unsigned long long one=redc(r2);
  for(unsigned long long base:{2,325,9375,28178,450775,9780504,1795265022}){
    if(base%n==0)continue;
    unsigned long long a=base=redc((base%n)*r2);
    for(int i=q-1;i>=0;i--){
      a=redc(__uint128_t(a)*a);
      if(d>>i&1)a=redc(__uint128_t(a)*base);
    }
    if(a==one)continue;
    for(int i=1;a!=n-one;i++){
      if(i>=s)return false;
      a=redc(__uint128_t(a)*a);
    }
  }
  return true;
}
constexpr int factorize_constexpr(unsigned long long n,unsigned long long*a){
  if(n<=1)return 0;
  int ptr=0;
  while(n%2==0){
    a[ptr++]=2;
    n/=2;
  }
  while(n>1){
    if(isprime_constexpr(n)){
      a[ptr++]=n;
      break;
    }
    unsigned long long pf=n;
    unsigned long long x=0,y=0;
    int c=1;
    do{
      x=((__uint128_t)x*x+c)%pf;
      y=((__uint128_t)y*y+c)%pf;
      y=((__uint128_t)y*y+c)%pf;
      unsigned long long g=std::gcd(x>y?x-y:y-x,pf);
      if(g==pf){
        c++;
        x=y=0;
        continue;
      }
      if(g>1)pf=g;
    }while(!isprime_constexpr(pf));
    while(n%pf==0){
      a[ptr++]=pf;
      n/=pf;
    }
  }
  return ptr;
}
namespace Random{
constexpr unsigned long long to_seed(const char*s){
  unsigned long long h=14695981039346656037ULL;
  while(*s){
    h^=static_cast<unsigned char>(*s++);
    h*=1099511628211ULL;
  }
  return h;
}
constexpr unsigned long long constexpr_random_seed=(to_seed(__TIME__)*0x9e3779b97f4a7c15ULL)^to_seed(__DATE__);
constexpr unsigned long long next_value(unsigned long long n){
  n^=n<<13;
  n^=n>>7;
  n^=n<<17;
  return n;
}
}
constexpr unsigned long long primitive_root_constexpr(unsigned long long x){
  if(!isprime_constexpr(x))throw "not prime";
  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;
  unsigned long long a[64]={};
  int ptr=factorize_constexpr(x-1,a);
  for(unsigned long long v=Random::constexpr_random_seed;;){
    unsigned long long g=v%(x-1)+1;
    bool ok=true;
    for(int i=0;i<ptr;i++){
      if(i>0&&a[i-1]==a[i])continue;
      if(pow_mod<unsigned long long>(g,(x-1)/a[i],x)==1){
        ok=false;
        break;
      }
    }
    if(ok)return g;
    v=Random::next_value(v);
  }
}
template<auto m>
struct ntt_root{
  static_assert(1<=m&&m<(1ull<<63));
  static_assert(isprime_constexpr(m));
  using value_type=std::conditional_t<((m>>31)==0),uint32_t,uint64_t>;
  using mul_type=std::conditional_t<((m>>31)==0),uint64_t,__uint128_t>;
  static constexpr int rank2=lsb(m-1);
  static constexpr value_type g=primitive_root_constexpr(m);
  std::array<value_type,rank2+1>root,invroot;
  std::array<value_type,std::max(0,rank2-1)>rate2,invrate2;
  std::array<value_type,std::max(0,rank2-2)>rate3,invrate3;
  constexpr ntt_root(){
    root[rank2]=pow_mod<value_type>(g,m>>rank2,m);
    invroot[rank2]=pow_mod<value_type>(root[rank2],m-2,m);
    for(int i=rank2-1;i>=0;i--){
      root[i]=(mul_type)root[i+1]*root[i+1]%m;
      invroot[i]=(mul_type)invroot[i+1]*invroot[i+1]%m;
    }
    value_type prod=1,invprod=1;
    for(int i=0;i<rank2-1;i++){
      rate2[i]=(mul_type)root[i+2]*prod%m;
      invrate2[i]=(mul_type)invroot[i+2]*invprod%m;
      prod=(mul_type)prod*invroot[i+2]%m;
      invprod=(mul_type)invprod*root[i+2]%m;
    }
    prod=invprod=1;
    for(int i=0;i<rank2-2;i++){
      rate3[i]=(mul_type)root[i+3]*prod%m;
      invrate3[i]=(mul_type)invroot[i+3]*invprod%m;
      prod=(mul_type)prod*invroot[i+3]%m;
      invprod=(mul_type)invprod*root[i+3]%m;
    }
  }
};
template<typename T>
void dft(std::vector<T>&a){
  using value_type=typename T::value_type;
  using mul_type=typename T::mul_type;
  #ifdef NTT_SIMD
  if constexpr(std::numeric_limits<value_type>::digits<=32){
    if((int)a.size()>=32){
      dft_simd(a);
      return;
    }
  }
  #endif
  static constexpr ntt_root<T::mod()>r;
  static constexpr mul_type mod2=(mul_type)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 mul_type rot1=rot.val(),rot2=(rot*rot).val(),rot3=(rot*T::raw(rot2)).val();
        int of=s<<(h-len);
        for(int i=0;i<p;i++){
          const mul_type a0=a[i+of].val(),a1=(mul_type)a[i+of+p].val()*rot1,a2=(mul_type)a[i+of+p*2].val()*rot2,a3=(mul_type)a[i+of+p*3].val()*rot3;
          const mul_type m=(mul_type)T(a1+mod2-a3).val()*imag.val();
          const mul_type 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){
  using value_type=typename T::value_type;
  using mul_type=typename T::mul_type;
  #ifdef NTT_SIMD
  if constexpr(std::numeric_limits<value_type>::digits<=32){
    if((int)a.size()>=32){
      idft_simd(a);
      return;
    }
  }
  #endif
  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 mul_type rot1=rot.val(),rot2=(rot*rot).val(),rot3=(rot*T::raw(rot2)).val();
        int of=s<<(h-len+2);
        for(int i=0;i<p;i++){
          const mul_type 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 mul_type 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){
    if(n==0||m==0)return {};
    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<auto m>
struct modint{
  static_assert(1<=m&&m<(1ull<<63));
  using value_type=std::conditional_t<((m>>31)==0),uint32_t,uint64_t>;
  using mul_type=std::conditional_t<((m>>31)==0),uint64_t,__uint128_t>;
private:
  value_type v;
  static constexpr value_type umod=m;
  constexpr modint sqrt_impl()const{
    if(this->val()<=1)return *this;
    if(umod%8==1){
      modint b=2;
      while(b.pow((umod-1)/2).val()==1)b++;
      value_type m2=umod-1;
      int e=0;
      while(m2%2==0)m2>>=1,e++;
      modint x=this->pow((m2-1)/2);
      modint y=(*this)*x*x;
      x*=*this;
      modint z=b.pow(m2);
      while(y.val()!=1){
        int j=0;
        modint t=y;
        while(t.val()!=1)t*=t,j++;
        z=z.pow((value_type(1))<<(e-j-1));
        x*=z;
        z*=z;
        y*=z;
        e=j;
      }
      return x;
    }
    else if(umod%8==5){
      modint res=this->pow((umod+3)/8);
      if((res*res).val()==this->val())return res;
      else return res*modint(2).pow((umod-1)/4);
    }
    else return this->pow((umod+1)/4);
  }
  template<typename U,std::enable_if_t<std::unsigned_integral<U>||std::is_same_v<U,__uint128_t>,std::nullptr_t> =nullptr>
  static constexpr value_type take_mod(U x){
    if constexpr(std::numeric_limits<U>::max()<umod)return x;
    if constexpr(umod==(1ull<<61)-1){
      value_type res=(x>>61)+(x&umod);
      if(res>=umod)res-=umod;
      return res;
    }
    if constexpr(umod==(1ull<<61)-(1ull<<24)+1){
      value_type high=x>>61,low=x&((1ull<<61)-1);
      mul_type t=low+(mul_type(high)<<24)-high;
      high=t>>61,low=t&((1ull<<61)-1);
      low=low+(mul_type(high)<<24)-high;
      if(low>=umod)low-=umod;
      return low;
    }
    return x%umod;
  }
public:
  constexpr modint():v(0){}
  template<typename U,std::enable_if_t<std::signed_integral<U>||std::is_same_v<U,__int128_t>,std::nullptr_t> =nullptr>
  constexpr modint(U x){
    x%=std::make_signed_t<value_type>(umod);
    v=x>=0?x:x+umod;
  }
  template<typename U,std::enable_if_t<std::unsigned_integral<U>||std::is_same_v<U,__uint128_t>,std::nullptr_t> =nullptr>
  constexpr modint(U x):v(take_mod<U>(x)){}
  static constexpr value_type mod(){return umod;}
  static constexpr modint zero(){return raw(0);}
  static constexpr modint one(){
    if constexpr(m==1)return raw(0);
    else return raw(1);
  }
  template<typename U>
  static constexpr modint raw(U x){
    modint res;
    res.v=x;
    return res;
  }
  constexpr std::make_signed_t<value_type> val()const{return v;}
  constexpr modint &operator+=(const modint&b){
    this->v+=b.v;
    if(this->v>=umod)this->v-=umod;
    return *this;
  }
  constexpr modint &operator-=(const modint&b){
    this->v-=b.v;
    if(this->v>=umod)this->v+=umod;
    return *this;
  }
  constexpr modint &operator*=(const modint&b){
    this->v=take_mod(mul_type(this->v)*mul_type(b.v));
    return *this;
  }
  constexpr modint &operator/=(const modint&b){return *this*=b.inv();}
  constexpr modint operator+()const{return *this;}
  constexpr modint operator-()const{return modint()-*this;}
  friend constexpr modint operator+(const modint&a,const modint&b){return modint(a)+=b;}
  friend constexpr modint operator-(const modint&a,const modint&b){return modint(a)-=b;}
  friend constexpr modint operator*(const modint&a,const modint&b){return modint(a)*=b;}
  friend constexpr modint operator/(const modint&a,const modint&b){return modint(a)/=b;}
  constexpr auto operator<=>(const modint&)const=default;
  constexpr modint operator++(int){
    modint res=*this;
    this->v++;
    if(this->v==umod)this->v=0;
    return res;
  }
  constexpr modint operator--(int){
    modint res=*this;
    if(this->v==0)this->v=umod;
    this->v--;
    return res;
  }
  template<std::integral U>
  constexpr modint pow(U k)const{
    if constexpr(std::is_signed_v<U>){
      assert(0<=k);
    }
    modint res=1,a(*this);
    while(k){
      if(k&1)res*=a;
      a*=a;
      k>>=1;
    }
    return res;
  }
  constexpr modint inv()const{
    if constexpr(isprime_constexpr(umod)){
      if(std::is_constant_evaluated()){
        if(v==0){
          throw "no inverse";
        }
      }
      else assert(v!=0);
      return pow(umod-2);
    }
    else{
      modint res;
      auto [g,x]=inv_mod<std::make_signed_t<value_type>>(this->v,umod);
      if(std::is_constant_evaluated()){
        if(g!=1){
          throw "no inverse";
        }
      }
      else assert(g==1);
      res.v=x;
      return res;
    }
  }
  std::optional<modint>sqrt()const{
    if(this->val()<=1||this->pow((umod-1)/2)==1)return std::make_optional(this->sqrt_impl());
    else return std::nullopt;
  }
  friend std::istream &operator>>(std::istream&is,modint&b){
    long long a;
    is>>a;
    b=modint(a);
    return is;
  }
  friend std::ostream &operator<<(std::ostream&os,const modint&b){
    os<<b.val();
    return os;
  }
};
template<auto m>
struct std::hash<modint<m>>{
  std::size_t operator()(modint<m>x)const{
    return std::hash<typename modint<m>::value_type>(x.val());
  }
};
using mint998=modint<998244353>;
using mint107=modint<1000000007>;
using mint61=modint<2305843009213693951>;
using mint6124=modint<2305843009196916737>;
template<typename T>
std::vector<T>arbitrary_mod_convolution(const std::vector<T>&a,const std::vector<T>&b){
  if(a.empty()||b.empty())return std::vector<T>{};
  if(std::min(a.size(),b.size())<60){
    std::vector<T>ret(a.size()+b.size()-1,0);
    for(int i=0;i<a.size();i++)for(int j=0;j<b.size();j++)ret[i+j]+=a[i]*b[j];
    return ret;
  }
  using mint1=modint<167772161>;
  using mint2=modint<469762049>;
  using mint3=modint<998244353>;
  std::vector<mint1>a1(a.size()),b1(b.size());
  std::vector<mint2>a2(a.size()),b2(b.size());
  std::vector<mint3>a3(a.size()),b3(b.size());
  for(int i=0;i<a.size();i++){
    a1[i]=a[i].val();
    a2[i]=a[i].val();
    a3[i]=a[i].val();
  }
  for(int i=0;i<b.size();i++){
    b1[i]=b[i].val();
    b2[i]=b[i].val();
    b3[i]=b[i].val();
  }
  a1=ntt_convolution(a1,b1),a2=ntt_convolution(a2,b2),a3=ntt_convolution(a3,b3);
  std::vector<T>ret(a.size()+b.size()-1);
  for(int i=0;i<ret.size();i++)ret[i]=crt3<T,mint1::mod(),mint2::mod(),mint3::mod()>(a1[i].val(),a2[i].val(),a3[i].val());
  return ret;
}
using mint=arbitrary_modint<int,1>;
void SOLVE(){
  int n;
  ll m;
  cin>>n>>m;
  mint::read_mod();
  vector<mint>a(n+1);
  rep(i,1,n+1)cin>>a[i];
  a=semigroup_pow(a,m,[&](vector<mint>a,vector<mint>b){
    vector<mint>ab=arbitrary_mod_convolution(a,b);
    vector<mint>c(n+1);
    rep(i,1,n+1){
      c[i]=a[i]+b[i];
      rep(j,1,i)c[i]+=ab[j]*c[i-j];
    }
    return c;
  });
  rep(i,1,n+1)cout<<a[i]<<" \n"[i==n];
}