#include <bits/stdc++.h>
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;
}
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--;}
#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

#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();
  }
}
template<typename T>
std::vector<T> berlekamp_massey(const std::vector<T>&s){
  const int n=s.size();
  std::vector<T>b,c;
  b.reserve(n+1),c.reserve(n+1);
  b.emplace_back(1),c.emplace_back(1);
  T y=1;
  for(int i=1;i<=n;i++){
    int l=c.size(),m=b.size();
    T x=0;
    for(int j=0;j<l;j++)x+=c[j]*s[i-l+j];
    b.emplace_back(0);
    ++m;
    if(x==0)continue;
    T f=x/y;
    if(l<m){
      auto c2=c;
      c.insert(c.begin(),m-l,0);
      for(int j=0;j<m;j++)c[m-1-j]-=f*b[m-1-j];
      b=c2;
      y=x;
    }
    else{
      for(int j=0;j<m;j++)c[l-1-j]-=f*b[m-1-j];
    }
  }
  std::reverse(c.begin(),c.end());
  return c;
}
template<typename T>
std::vector<T>reeds_sloane(const std::vector<T>a,std::pair<int,int>pe){
  int n=a.size();
  std::vector<std::vector<T>>Q(pe.second),B(pe.second);
  std::vector<int>nb(pe.second,-1);
  std::vector<T>tb(pe.second);
  T powp=1;
  for(int j=0;j<pe.second;j++){
    Q[j]={powp};
    powp*=pe.first;
  }
  for(int i=0;i<n;i++){
    std::vector<std::pair<T,int>>tu(pe.second);
    for(int j=0;j<pe.second;j++){
      T x=0;
      for(int k=0;k<(int)Q[j].size();k++){
        x+=Q[j][k]*a[i-k];
      }
      if(x.val()==0)tu[j]=std::make_pair(1,pe.second);
      else{
        auto x2=x.val();
        int ne=tu[j].second;
        while(x2%pe.first==0)x2/=pe.first,ne++;
        tu[j]=std::make_pair(x2,ne);
      }
    }
    std::vector<std::vector<T>>nQ(Q);
    for(int j=0;j<pe.second;j++){
      if(tu[j].second==pe.second)continue;
      int k=pe.second-1-tu[j].second;
      if(nb[k]==-1)nQ[j].resize(i+2);
      else{
        T c=tu[j].first/tb[k];
        if(i+B[k].size()-nb[k]>nQ[j].size())nQ[j].resize(i+B[k].size()-nb[k]);
        for(int l=0;l<(int)B[k].size();l++)nQ[j][i+l-nb[k]]-=c*B[k][l];
      }
    }
    for(int j=0;j<pe.second;j++){
      if(Q[j].size()<nQ[j].size()){
        int k=pe.second-1-tu[j].second;
        B[j]=Q[k];
        nb[j]=i;
        tb[j]=tu[k].first;
      }
    }
    Q=std::move(nQ);
  }
  return Q[0];
}
#include<type_traits>
#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);}
constexpr unsigned long long binary_gcd(unsigned long long a,unsigned long long b){
  if(a==0||b==0||a==b)return a<b?b:a;
  int n=lsb(a),m=lsb(b);
  while(a!=b){
    if(a>b)a=(a-b)>>lsb(a-b);
    else b=(b-a)>>lsb(b-a);
  }
  return a<<(n<m?n:m);
}
#include<initializer_list>
bool isprime(unsigned long long n){
  using u64=unsigned long long;
  if(n<=1)return false;
  if(n%2==0)return n==2;
  u64 d=n-1;
  int s=0;
  while(!(d&1))d>>=1,s++;
  int q=63;
  while(!(d>>q))q--;
  u64 r=n;
  for(int i=0;i<5;i++)r*=2-r*n;
  auto redc=[&r,&n](__uint128_t x)->u64 {
    x=(x+__uint128_t(u64(x)*-r)*n)>>64;
    return x>=n?x-n:x;
  };
  __uint128_t r2=-__uint128_t(n)%n;
  u64 one=redc(r2);
  for(u64 base:{2,325,9375,28178,450775,9780504,1795265022}){
    if(base%n==0)continue;
    u64 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;
}
template<std::integral T>
std::vector<std::pair<T,int>>factorize(T n)noexcept{
  std::vector<unsigned long long>fs;
  auto div=[](unsigned long long x)noexcept->unsigned long long {
    unsigned long long r=x;
    for(int i=0;i<5;i++)r*=2-r*x;
    unsigned long long r2=-__uint128_t(x)%x;
    auto redc=[&r,&x](__uint128_t t)->unsigned long long {
      t=(t+__uint128_t((unsigned long long)t*-r)*x)>>64;
      return t>=x?t-x:t;
    };
    unsigned long long a=0,b=0;
    const unsigned long long one=redc(r2);
    unsigned long long e=one;
    int m=1ll<<((63-__builtin_clzll(x))>>3);
    while(true){
      unsigned long long ca=a,cb=b;
      unsigned long long sk=one;
      for(int i=0;i<m;i++){
        a=redc(__uint128_t(a)*a+e);
        b=redc(__uint128_t(b)*b+e);
        b=redc(__uint128_t(b)*b+e);
        unsigned long long c=redc(a),d=redc(b);
        sk=redc(__uint128_t(sk)*(c>d?c-d:d-c));
      }
      unsigned long long g=binary_gcd(redc(sk),x);
      if(g>1){
        if(g<x)return g;
        for(int i=0;i<m;i++){
          ca=redc(__uint128_t(ca)*ca+e);
          cb=redc(__uint128_t(cb)*cb+e);
          cb=redc(__uint128_t(cb)*cb+e);
          unsigned long long c=redc(ca),d=redc(cb);
          unsigned long long cg=binary_gcd(c>d?c-d:d-c,x);
          if(cg>1){
            if(cg<x)return cg;
            else{
              e+=one;
              a=b=0;
              break;
            }
          }
        }
      }
    }
  };
  static unsigned long long st[64];
  int p=0;
  while(!(n&1)){
    n>>=1;
    fs.push_back(2);
  }
  if(n>1)st[p++]=n;
  while(p){
    unsigned long long now=st[--p];
    if(isprime(now)){
      fs.push_back(now);
      continue;
    }
    unsigned long long d=div(now);
    st[p++]=d;
    now/=d;
    if(now!=1)st[p++]=now;
  }
  std::sort(fs.begin(),fs.end());
  std::vector<std::pair<T,int>>res;
  for(int i=0;i<(int)fs.size();){
    int j=i;
    while(j<(int)fs.size()&&fs[i]==fs[j])j++;
    res.emplace_back(fs[i],j-i);
    i=j;
  }
  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 Ts>
struct CRT{
  using T=std::make_signed_t<Ts>;
  std::vector<std::pair<T,int>>f;
  std::vector<T>pe;
  std::vector<T>invs;
  using T2=std::conditional_t<(std::numeric_limits<T>::digits<=32),int64_t,__int128_t>;
  CRT(){}
  CRT(T n):f(factorize(n)){
    pe.resize(f.size());
    for(int i=0;i<(int)f.size();i++){
      pe[i]=1;
      for(int j=0;j<f[i].second;j++)pe[i]*=f[i].first;
    }
    invs.resize(f.size());
    T prod=1;
    for(int i=0;i<(int)f.size();i++){
      invs[i]=pow_mod(prod%pe[i],pe[i]/f[i].first*(f[i].first-1)-1,pe[i]);
      prod*=pe[i];
    }
  }
  Ts operator()(std::vector<Ts>v){
    assert(v.size()==pe.size());
    T res=0,prod=1;
    for(int i=0;i<(int)pe.size();i++){
      v[i]%=pe[i];
      if constexpr(std::is_signed_v<Ts>){
        if(v[i]<0)v[i]+=pe[i];
      }
      T x=T2(T(v[i])-res)*T2(invs[i])%pe[i];
      res+=x*prod;
      prod*=pe[i];
      if(res<0)res+=prod;
    }
    return res;
  }
};
#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 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());
  }
};
struct is_modint_impl{
  template<typename T>
  static auto check(T&&x)->decltype(x.mod(),std::true_type{});
  template<typename T>
  static auto check(...)->std::false_type;
};
template<typename T>
struct is_modint:public decltype(is_modint_impl::check<T>(std::declval<T>())){};
template<typename T>
inline constexpr bool is_modint_v=is_modint<T>::value;
struct is_dynamic_modint_impl{
  template<typename T>
  static auto check(T&&x)->decltype(x.set_mod((typename T::value_type)0),std::true_type{});
  template<typename T>
  static auto check(...)->std::false_type;
};
template<typename T>
struct is_dynamic_modint:public decltype(is_dynamic_modint_impl::check<T>(std::declval<T>())){};
template<typename T>
inline constexpr bool is_dynamic_modint_v=is_dynamic_modint<T>::value;
template<typename T>
inline constexpr bool is_static_modint_v=is_modint_v<T>&&!is_dynamic_modint_v<T>;
struct is_uso_modint_impl{
  template<typename T>
  static auto check(T&&x)->decltype(x.uso(),std::true_type{});
  template<typename T>
  static auto check(...)->std::false_type;
};
template<typename T>
struct is_uso_modint:public decltype(is_uso_modint_impl::check<T>(std::declval<T>())){};
template<typename T>
inline constexpr bool is_uso_modint_v=is_uso_modint<T>::value;
template<typename T>
std::vector<T>find_linear_recurrence(std::vector<T>a){
  static_assert(is_modint_v<T>);
  if(T::mod()==1)return {0};
  using mint=arbitrary_modint<typename T::value_type,20260702>;
  CRT<typename T::value_type>crt(T::mod());
  std::vector<std::vector<typename T::value_type>>f(crt.f.size());
  int l=0;
  for(int i=0;i<(int)crt.f.size();i++){
    mint::set_mod(crt.pe[i]);
    std::vector<mint>b(a.size());
    for(int j=0;j<(int)a.size();j++)b[j]=a[j].val();
    if(crt.f[i].second==1)b=berlekamp_massey(b);
    else b=reeds_sloane(b,crt.f[i]);
    f[i].resize(b.size());
    for(int j=0;j<(int)b.size();j++)f[i][j]=b[j].val();
    if(l<(int)f[i].size())l=f[i].size();
  }
  std::vector<T>res(l);
  for(int i=0;i<l;i++){
    std::vector<typename T::value_type>now(f.size());
    for(int j=0;j<(int)f.size();j++){
      now[j]=i<(int)f[j].size()?f[j][i]:0;
    }
    res[i]=crt(now);
  }
  return res;
}
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;
}
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;}
  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>;
using mint=modint<2>;
void SOLVE(){
  string t;
  int n;
  cin>>t>>n;
  vector<mint>pre(t.size()*2);
  string a="RSPX";
  string win="XRSP";
  rep(i,t.size()){
    int p=a.find(t[i]);
    pre[i*2]=p&1;
    pre[i*2+1]=p>>1;
  }
  auto coef=find_linear_recurrence(pre);
  pre.resize((t.size()+n)*2);
  rep(i,t.size()*2,pre.size()){
    mint now=0;
    rep(j,1,coef.size())if(i-j>=0)now+=coef[j]*pre[i-j];
    pre[i]=now;
  }
  string ans;
  rep(i,n){
    int id=(pre[t.size()*2+i*2].val())+(pre[t.size()*2+i*2+1].val())*2;
    ans+=win[id];
  }
  cout<<ans<<endl;
}