#define CP_BUNDLED_SOURCE
#ifndef CP_BUNDLE_HEADER_3840A09F237A779E
#define CP_BUNDLE_HEADER_3840A09F237A779E
#ifdef TEMPLATE
#else
#define TEMPLATE
# pragma GCC optimize("O3")
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <string>
#include <cstring>
#include <vector>
#include <list>
#include <queue>
#include <stack>
#include <deque>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <algorithm>
#include <numeric>
#include <cmath>
#include <climits>
#include <cassert>
#include <functional>
#include <iterator>
#include <utility>
#include <complex>
#include <bitset>
#include <chrono>
#include <random>
#include <limits>
#include <optional>
#include <variant>
#include <any>

#include <array>
#include <bit>
#include <compare>
#include <concepts>
#include <numbers>
#include <ranges>
#include <span>
#include <string_view>
#include <tuple>
#include <type_traits>
#include <version>
using namespace std;
using uint=unsigned;
using ll=long long;
using ull=unsigned long long;
using ld=long double;
using pii=pair<int,int>;
using pll=pair<ll,ll>;
using i128=__int128;
using u128=unsigned __int128;
template<class T>using vc=vector<T>;
template<class T>using vvc=vc<vc<T>>;
template<class T>using vvvc=vvc<vc<T>>;
template<class T>using smpq=priority_queue<T,vector<T>,greater<T>>;
template<class T>using bipq=priority_queue<T>;
#define rep(i,n) for(ll i=0;i<(ll)(n);i++)
#define REP(i,j,n) for(ll i=(j);i<(ll)(n);i++)
#define DREP(i,n,m) for(ll i=(n);i>=(m);i--)
#define drep(i,n) for(ll i=((n)-1);i>=0;i--)
#define rall(x) x.rbegin(),x.rend()
#define mp(...) make_pair(__VA_ARGS__)
#define pb push_back
#define fi first
#define se second
#define is insert
#define bg begin()
#define ed end()
#define all(x) x.begin(),x.end()
void scan(int&a) { cin >> a; }
void scan(ll&a) { cin >> a; }
void scan(string&a) { cin >> a; }
void scan(char&a) { cin >> a; }
void scan(uint&a) { cin >> a; }
void scan(ull&a) { cin >> a; }
void scan(bool&a) { cin >> a; }
void scan(ld&a){ cin>> a;}
template<class T> void scan(vector<T>&a) { for(auto&x:a) scan(x); }
void read() {}
template<class Head, class... Tail> void read(Head&head, Tail&... tail) { scan(head); read(tail...); }
#define INT(...) int __VA_ARGS__; read(__VA_ARGS__);
#define LL(...) ll __VA_ARGS__; read(__VA_ARGS__);
#define ULL(...) ull __VA_ARGS__; read(__VA_ARGS__);
#define STR(...) string __VA_ARGS__; read(__VA_ARGS__);
#define VC(type, name, ...) vector<type> name(__VA_ARGS__); read(name);
#define VVC(type, name, size, ...) vector<vector<type>> name(size, vector<type>(__VA_ARGS__)); read(name);
template<class T>void print(T a) { cout << a; }
template<class T> void print(vector<T>a) { for(int i=0;i<(int)a.size();i++){if(i)cout<<" ";print(a[i]);}}
void PRT() { cout <<endl; return ; }
template<class T> void PRT(T a) { print(a); cout <<endl; return; }
template<class Head, class... Tail> void PRT(Head head, Tail ... tail) { print(head); cout << " "; PRT(tail...); return; }
template<class T,class F>
bool chmin(T &x, F y){
    if(x>y){
        x=y;
        return true;
    }
    return false;
}
template<class T, class F>
bool chmax(T &x, F y){
    if(x<y){
        x=y;
        return true;
    }
    return false;
}
template<typename T>
T floor(T a,T b){
    return a/b-(a%b&&((a<0)!=(b<0)));
}
template<typename T>
T ceil(T a,T b){
    return a/b+(a%b&&((a<0)==(b<0)));
}
template<typename T>
T bmod(T x,T y){
    return x-y*floor(x,y);
}
template<typename T>
pair<T,T>divmod(T x,T y){
    T q=floor(x,y);
    return{q,x-q*y};
}
void YesNo(bool b){
    cout<<(b?"Yes":"No")<<endl;
}
void YESNO(bool b){
    cout<<(b?"YES":"NO")<<endl;
}
void AliceBob(bool b){
    cout<<(b?"Alice":"Bob")<<endl;
}
void TakahashiAoki(bool b){
    cout<<(b?"Takahashi":"Aoki")<<endl;
}
void Yes(){
    cout<<"Yes"<<endl;
}
void No(){
    cout<<"No"<<endl;
}
vc<int>stovi(const string&s,const string&S){
    vc<int>v(s.size());
    rep(i,s.size()){
        auto t=S.find(s[i]);
        assert(t!=string::npos);
        v[i]=t;
    }
    return v;
}
template<class T=ll>
T isqrt(T x){
    T F=sqrtl(x);
    while((F+1)*(F+1)<=x)F++;
    while(F*F>x)F--;
    return F;
}
template<class T>
T tri(T x){return x*(x-1)/2;}

//[l,r)
template<class T>T tri(T l,T r){return (r-l)*(l+r-1)/2;}
//n を先頭に持ってくる
template<class T>
vc<T>rot(vc<T>v,int n){
    rotate(v.begin(),v.begin()+n,v.end());
    return v;
}
template<class T>
vc<T>iota(int n){
    vc<T>v(n);rep(i,n)v[i]=i;
    return v;
}
template<class T,int minfirst>
vc<int>argsort(const vc<T>&a){
    auto idx=iota<int>(a.size());
    sort(all(idx),[&](int i,int j){
        return (minfirst?make_pair(a[i],i)<make_pair(a[j],j):make_pair(a[i],i)>make_pair(a[j],j));
    });
    return idx;
}
template<class T>
vvc<T>trans(const vvc<T>&a){
    assert(a.size()&&a[0].size());
    rep(i,a.size())assert(a[i].size()==a[0].size());
    vvc<T>b(a[0].size(),vc<T>(a.size()));
    rep(i,a.size())rep(j,a[0].size())b[j][i]=a[i][j];
    return b;
}
vc<string>trans(const vc<string>&a){
    assert(a.size()&&a[0].size());
    rep(i,a.size())assert(a[i].size()==a[0].size());
    vc<string>b(a[0].size(),string(a.size(),0));
    rep(i,a.size())rep(j,a[0].size())b[j][i]=a[i][j];
    return b;
}
template<class T>
int popcount(T n){
    return __builtin_popcountll(n);
}
template<class T,class L=ll>
L sum(const vc<T>&a){
    return accumulate(all(a),L(0));
}
template<class T>
struct subset_view{
    T s;
    struct iterator{
        T s,x;
        bool done;
        T operator*()const{return x;}
        iterator&operator++(){
            if(x==0)done=true;
            else x=(x-1)&s;
            return*this;
        }
        bool operator!=(const iterator&r)const{return done!=r.done;}
    };
    iterator begin()const{return{s,s,false};}
    iterator end()const{return{s,0,true};}
};
template<class T>
subset_view<T>subset(T s){
    return{s};
}
template<class T>
T max(vc<T>&a){
    return *max_element(all(a));
}
template<class T>
T min(vc<T>&a){
    return *min_element(all(a));
}
template<class T,class L=ll>
vc<L> presum(vc<T> &a){
    vc<L> ret(a.size()+1);
    rep(i,a.size())ret[i+1]=ret[i]+a[i];
    return ret;
}
template<class T, class F>
vc<T> &operator+=(vc<T> &a,F b){
    for (auto&v:a)v += b;
    return a;
}
template<class T, class F>
vc<T> &operator-=(vc<T>&a,F b){
    for (auto&v:a)v-=b;
    return a;
}
template<class T, class F>
vc<T> &operator*=(vc<T>&a,F b){
    for (auto&v:a)v*=b;
    return a;
}
template<class T=ll>
constexpr T pow(T a,T b){
    T res=1;
    while(b){
        if(b&1)res*=a;
        a*=a;
        b/=2;
    }
    return res;
}
constexpr ll ten(ll a){
    return pow<ll>(10,a);
}
template<typename T>constexpr T inf=numeric_limits<T>::max()/2-1;
template<class T>
int tbit(T x){
    using U=make_unsigned_t<T>;
    U y=(U)x;
    return y?(int)bit_width(y)-1:-1;
}
template<class T>
int lbit(T x){
    using U=make_unsigned_t<T>;
    U y=(U)x;
    return y?(int)countr_zero(y):-1;
}
template<class T>
int tbit(T x,int p){
    using U=make_unsigned_t<T>;
    constexpr int W=numeric_limits<U>::digits;
    U y=(U)x;
    if(p<0)return -1;
    if(p>=W-1)return tbit(y);
    return tbit(y&((U(1)<<(p+1))-1));
}
template<class T>
int lbit(T x,int p){
    using U=make_unsigned_t<T>;
    constexpr int W=numeric_limits<U>::digits;
    U y=(U)x;
    if(p<0)return lbit(y);
    if(p>=W)return -1;
    return lbit(y&(~U(0)<<p));
}
istream& operator>>(istream&is,i128&x){
    string s;is>>s;
    x=0;
    int i=0,neg=0;
    if(s[0]=='-')neg=1,i=1;
    for(;i<(int)s.size();i++)x=x*10+s[i]-'0';
    if(neg)x=-x;
    return is;
}
ostream& operator<<(ostream&os,i128 x){
    if(x==0)return os<<0;
    if(x<0)os<<"-";
    u128 y=x<0?-(u128)x:(u128)x;
    string s;
    while(y)s.pb('0'+y%10),y/=10;
    reverse(all(s));
    return os<<s;
}
#ifdef LOCAL
#include "debug/debug.hpp"
inline int dbging=1;
#else
#define dbg(...) ((void)0)
#endif

struct template_setup{
    template_setup(){
        #ifdef LOCAL
        freopen("input.txt","r",stdin);
        freopen("output.txt","w",stdout);
        
        #endif
        cin.tie(0)->sync_with_stdio(0);
        #ifdef LOCAL
        cout<<fixed<<setprecision(6);
        #else
        cout<<fixed<<setprecision(20);
        #endif
    }
};
inline template_setup template_setup;

#if defined(LOCAL)&&!defined(CP_NO_AUTO_ALL)&&!defined(CP_BUNDLED_SOURCE)
#include <all.hpp>
#endif

#endif
#endif
#ifndef CP_BUNDLE_HEADER_8DD6832EF023BDC7
#define CP_BUNDLE_HEADER_8DD6832EF023BDC7
template<uint32_t mod>
struct static_modint{
    static_assert(0<mod&&mod<=(1u<<31)-1);
    using u32=uint32_t;
    using u64=uint64_t;
    u32 val;
    static_modint():val(0){}
    static_modint(ll x){
        ll v=x%mod;
        if(v<0)v+=mod;
        val=v;
    }
    constexpr static uint32_t get_mod(){
        return mod;
    }
    static static_modint raw(int v){
        assert(v>=0);
        static_modint mi;
        mi.val=v;
        return mi;
    }
    static_modint &operator+=(const static_modint&m){
        u32 x=val+m.val-mod;
        val=x+(mod&-(x>>31));
        return *this;
    }
    static_modint &operator-=(const static_modint&m){
        u32 x=val-m.val;
        val=x+(mod&-(x>>31));
        return *this;
    }
    static_modint &operator*=(const static_modint&m){
        val=u64(val)*m.val%mod;
        return *this;
    }
    static_modint &operator/=(const static_modint&m){
        val=u64(val)*m.inv().val%mod;
        return *this;
    }
    static_modint operator-() const{
        return static_modint(mod-val);
    }
    static_modint operator+() const {
        return *this;
    }
    friend static_modint operator+(static_modint lhs, const static_modint& rhs){
        return lhs+=rhs;
    }
    friend static_modint operator-(static_modint lhs, const static_modint& rhs){
        return lhs-=rhs;
    }
    friend static_modint operator*(static_modint lhs, const static_modint& rhs){
        return lhs*=rhs;
    }
    friend static_modint operator/(static_modint lhs,const static_modint&rhs){
        return lhs/=rhs;
    }
    bool operator==(const static_modint&p) const{
        return p.val==val;
    }
    bool operator!=(const static_modint&p) const{
        return p.val!=val;
    }
    static_modint pow(int64_t n) const{
        static_modint res(1),mul(val);
        while(n){
            if(n%2)res*=mul;
            mul*=mul;
            n/=2;
        }
        return res;
    }

    friend ostream&operator<<(ostream&os,const static_modint&p){
        os<<p.val;
        return os;
    }
    friend istream&operator>>(istream&is,static_modint&p){
        int64_t x;
        is>>x;
        p=static_modint(x);
        return is;
    }
    static_modint inv()const{
        int64_t a=val,b=mod,u=1,v=0,t;
        #ifdef LOCAL
        assert(gcd(a,b)==1);
        #endif
        while(b>0){
            t=a/b;
            swap(a-=t*b,b);
            swap(u-=t*v,v);
        }
        return static_modint(u);
    }
};
#endif
#ifndef CP_BUNDLE_HEADER_6661CEF0E4F73CF3
#define CP_BUNDLE_HEADER_6661CEF0E4F73CF3
#include <random>
template<class T>
struct set_treap{
    struct Node{
        Node*lc,*rc;
        T val;
        int pri;
        int size;
        Node*update(){
            size=1;
            if(lc)size+=lc->size;
            if(rc)size+=rc->size;
            return this;
        }
    };
    void build(vc<T>v){
        for(auto&x:v)assert(x>=0);
        sort(all(v));
        v.erase(unique(all(v)),v.end());
        int n=v.size();
        if(n==0){
            root=NULL;
            return;
        }
        vc<Node*>pts(n);
        for(int i=0;i<n;i++){
            pts[i]=new_node();
            pts[i]->val=v[i];
            pts[i]->pri=mt();
        }
        vc<Node*>st;
        for(int i=0;i<n;i++){
            Node*last=NULL;
            while(!st.empty()&&st.back()->pri<pts[i]->pri){
                last=st.back();
                st.pop_back();
            }
            pts[i]->lc=last;
            if(!st.empty())st.back()->rc=pts[i];
            st.push_back(pts[i]);
        }
        root=st[0];
        auto dfs=[&](auto&self,Node*cur)->void{
            if(!cur)return;
            self(self,cur->lc);
            self(self,cur->rc);
            cur->update();
        };
        dfs(dfs,root);
    }
    Node*new_node(){
        Node*nn=new Node();
        nn->pri=mt();
        nn->lc=nn->rc=0;
        nn->size=1;
        return nn;
    }
    static mt19937 mt;
    Node*root;
    set_treap(){
        mt.seed(random_device{}());
        root=NULL;
    }
    Node*merge(Node*l,Node*r){
        if(!l)return r;
        if(!r)return l;
        if(l->pri>r->pri){
            l->rc=merge(l->rc,r);
            return l->update();
        }else{
            r->lc=merge(l,r->lc);
            return r->update();
        }
    }
    pair<Node*,Node*>split(Node*l,T k){
        if(!l)return {NULL,NULL};
        if(l->val<k){
            auto res=split(l->rc,k);
            l->rc=res.first;
            return {l->update(),res.second};
        }else{
            auto res=split(l->lc,k);
            l->lc=res.second;
            return {res.first,l->update()};
        }
    }
    Node*insert(Node*x,T val){
        auto s1=split(x,val);
        auto s2=split(s1.second,val+1);
        if(s2.first==NULL){
            s2.first=new_node();
            s2.first->val=val;
        }
        return merge(s1.first,merge(s2.first,s2.second));
    }
    void insert(T val){
        assert(val>=0);
        root=insert(root,val);
    }
    Node*erase(Node*x,T val){
        auto s1=split(x,val);
        auto s2=split(s1.second,val+1);
        return merge(s1.first,s2.second);
    }
    void erase(T val){
        assert(val>=0);
        root=erase(root,val);
    }
    bool count(T val){
        assert(val>=0);
        auto s1=split(root,val);
        auto s2=split(s1.second,val+1);
        auto res=s2.first!=NULL;
        root=merge(s1.first,merge(s2.first,s2.second));
        return res;
    }
    //min in [val,♾️)
    T next(T val){
        assert(val>=0);
        auto s1=split(root,val);
        auto now=s1.second;
        while(now){
            if(now->lc==NULL)break;
            now=now->lc;
        }
        T ans=numeric_limits<T>::max();
        if(now)chmin(ans,now->val);
        root=merge(s1.first,s1.second);
        if(ans==numeric_limits<T>::max())return -1;
        else return ans;
    }
    //max in (-♾️,val)
    T prev(T val){
        assert(val>=0);
        auto s1=split(root,val);
        auto now=s1.first;
        while(now){
            if(now->rc==NULL)break;
            now=now->rc;
        }
        T ans=-1;
        if(now)chmax(ans,now->val);
        root=merge(s1.first,s1.second);
        return ans;
    }
    //[val,♾️)
    int bigger(T val){
        assert(val>=0);
        int ans=size();
        auto s1=split(root,val);
        if(s1.first)ans-=s1.first->size;
        root=merge(s1.first,s1.second);
        return ans;
    }
    int size(){
        if(root==NULL)return 0;
        return root->size;
    }
    //(-♾️,val)
    int smaller(T val){
        assert(val>=0);
        if(root==NULL)return 0;
        return size()-bigger(val);
    }
    //[l,r)
    int range_count(T l,T r){
        assert(0<=l&&l<=r);
        return -bigger(r)+bigger(l);
    }
    T kth(Node*x,int k){
        assert(x);
        assert(0<=k&&k<x->size);
        if(x->lc){
            if(x->lc->size>k){
                return kth(x->lc,k);
            }
            k-=x->lc->size;
        }
        if(k==0)return x->val;
        --k;
        return kth(x->rc,k);
    }
    T kth(int k){
        assert(k>=0);
        if(k>=size())return -1;
        return kth(root,k);
    }
};
template<class T>
mt19937 set_treap<T>::mt(random_device{}());
#endif
#ifndef CP_BUNDLE_HEADER_ECFFAD09CB9B85DC
#define CP_BUNDLE_HEADER_ECFFAD09CB9B85DC

namespace MATMAT{
template <typename T, typename = void>
struct is_mint : std::false_type {};
template <typename T>
struct is_mint<T, std::void_t<decltype(T::get_mod())>> : std::true_type {};
};
template<class T,int N,int M>
struct fixed_matrix;
template<class T>
struct matrix{
    int n,m;
    vvc<T>a;
    matrix()=default;
    matrix(vvc<T>b):a(b),n(b.size()),m(0){
        if(b.size())m=b[0].size();
        for(auto&row:b)assert((int)row.size()==m);
    }
    matrix(int n,int m):n(n),m(m){
        assert(n>=0&&m>=0);
        a.assign(n,vc<T>(m));
    }
    vc<T>&operator[](int i){
        assert(0<=i&&i<n);
        return a[i];
    }
    const vc<T>&operator[](int i)const{
        assert(0<=i&&i<n);
        return a[i];
    }
    matrix&operator+=(const matrix&mt2){
        assert(mt2.n==n&&mt2.m==m);
        rep(i,n)rep(j,m)a[i][j]+=mt2[i][j];
        return *this;
    }
    matrix&operator-=(const matrix&mt2){
        assert(mt2.n==n&&mt2.m==m);
        rep(i,n)rep(j,m)a[i][j]-=mt2[i][j];
        return *this;
    }
    matrix&operator*=(const matrix&mt2){
        assert(m==mt2.n);
        vc<vc<T>>res(n,vc<T>(mt2.m));
        auto Mt2=mt2.trans();
        if constexpr(MATMAT::is_mint<T>::value){
            rep(i,n)rep(j,mt2.m){
                __int128 tmp{};
                rep(k,m){
                    tmp+=(ll)a[i][k].val*Mt2[j][k].val; 
                }
                res[i][j]=tmp%T::get_mod();
            }
        }else{
            rep(i,n)rep(j,mt2.m)rep(k,m){
                res[i][j]+=a[i][k]*Mt2[j][k];
            }
        }
        a=move(res);
        n=a.size();
        m=(n?a[0].size():mt2.m);
        return *this;
    }
    matrix trans()const{
        matrix res(m,n);
        rep(i,n)rep(j,m)res[j][i]=a[i][j];
        return res;
    }
    int size()const{
        return n;
    }
    int size2()const{
        return m;
    }
    matrix operator-() const{
        vvc<T>b=a;
        for(auto&x:b)for(auto&e:x)e=-e;
        return matrix(b);
    }
    matrix operator+() const {
        return *this;
    }
    friend matrix operator+(matrix lhs, const matrix& rhs){
        return lhs+=rhs;
    }
    friend matrix operator-(matrix lhs, const matrix& rhs){
        return lhs-=rhs;
    }
    friend matrix operator*(matrix lhs, const matrix& rhs){
        return lhs*=rhs;
    }
    matrix&operator*=(const T&v){
        for(auto&row:a)for(auto&e:row)e*=v;
        return *this;
    }
    friend matrix operator*(matrix lhs, const T&rhs){
        return lhs*=rhs;
    }
    friend matrix operator*(const T&lhs, matrix rhs){
        return rhs*=lhs;
    }
    template<int N,int M>
    matrix&operator*=(const fixed_matrix<T,N,M>&mt2);
    static matrix unit(int n){
        assert(n>=0);
        matrix mt(n,n);
        rep(i,n)mt[i][i]=1;
        return mt;
    }
    matrix pow(ll n){
        assert(this->n==this->m);
        assert(n>=0);
        matrix res=unit(this->n);
        matrix mul=a;
        while(n){
            if(n%2)res*=mul;
            mul*=mul;
            n/=2;
        }
        return res;
    }
    T det()const{
        assert(n==m);
        auto A=a;
        T res=1;
        rep(i,n){
            if(A[i][i]==0){
                REP(j,i+1,n){
                    if(A[j][i]!=0){
                        swap(A[i],A[j]);
                        res=-res;
                        break;
                    }
                }
            }
            if(A[i][i]==0)return 0;
            res*=A[i][i];
            T IN=1/A[i][i];
            REP(j,i,n)A[i][j]*=IN;
            REP(j,i+1,n){
                T coef=A[j][i];
                REP(k,i,n){
                    A[j][k]-=A[i][k]*coef;
                }
            }
        } 
        return res;
    }
    int rank()const{
        matrix a;
        if(n>m)a=matrix(*this).trans();
        else a=matrix(*this);
        int N=a.n,M=a.m;
        int hj=0;
        rep(i,N){
            int HI{-1},HJ{-1}; 
            REP(hj2,hj,M){
                REP(hi,i,N){
                    if(a[hi][hj2]!=0){
                        HI=hi,HJ=hj2;
                        break;    
                    }
                }
                if(HI!=-1)break;
            }
            if(HJ==-1){
                return i;
            }
            if(i!=HI)swap(a[i],a[HI]);
            T IN=1/a[i][HJ];
            REP(j,HJ,M)a[i][j]*=IN;
            REP(hi,i+1,N){
                T coef=a[hi][HJ];
                REP(hj2,HJ,M){
                    a[hi][hj2]-=a[i][hj2]*coef; 
                }
            }
            hj=HJ+1;
        }
        return N;
    }
    optional<matrix> inverse()const{
        assert(n==m);
        auto A=a;
        auto res=unit(n);
        int hj=0;
        rep(i,n){
            int HI{-1},HJ{-1}; 
            REP(hj2,hj,n){
                REP(hi,i,n){
                    if(A[hi][hj2]!=0){
                        HI=hi,HJ=hj2;
                        break;    
                    }
                }
                if(HI!=-1)break;
            }
            if(HJ==-1){
                return nullopt;
            }
            if(i!=HI)swap(A[i],A[HI]),swap(res[i],res[HI]);
            T IN=1/A[i][HJ];
            REP(j,0,n)A[i][j]*=IN,res[i][j]*=IN;
            REP(hi,i+1,n){
                T coef=A[hi][HJ];
                rep(hj2,n){
                    A[hi][hj2]-=A[i][hj2]*coef; 
                    res[hi][hj2]-=res[i][hj2]*coef;
                }
            }
            hj=HJ+1;
        }
        drep(i,n){
            DREP(j,i-1,0){
                rep(k,n)res[j][k]-=res[i][k]*A[j][i];
            }
        }
        return res;
    }
};
#endif
#ifndef CP_BUNDLE_HEADER_412FFE9058386D60
#define CP_BUNDLE_HEADER_412FFE9058386D60
template<class,class=void>
struct binom_has_get_mod:false_type{};
template<class mint>
struct binom_has_get_mod<mint,void_t<decltype(mint::get_mod())>>:true_type{};
template<class mint>
struct binom{
private:
    static vector<mint>&fact_table(){static vector<mint>v={1};return v;}
    static vector<mint>&invfact_table(){static vector<mint>v={1};return v;}
    static vector<mint>&invs_table(){static vector<mint>v={0};return v;}
    static int&built_mod(){static int mod=-1;return mod;}
public:
    static void build(int n){
        auto&_fact=fact_table();
        auto&_invfact=invfact_table();
        auto&_invs=invs_table();
        if constexpr(binom_has_get_mod<mint>::value){
            auto mod=mint::get_mod();
            if(built_mod()!=mod){
                _fact={1};
                _invfact={1};
                _invs={0};
                built_mod()=mod;
            }
        }
        if(n<(int)_fact.size())return;
        int old=_fact.size();
        _fact.resize(n+1);
        _invfact.resize(n+1);
        _invs.resize(n+1);
        if constexpr(binom_has_get_mod<mint>::value){
            auto mod=mint::get_mod();
            for(int i=old;i<=n;i++){
                _fact[i]=_fact[i-1]*i;
                if(i==1)_invs[i]=1;
                else _invs[i]=-_invs[mod%i]*(mod/i);
                _invfact[i]=_invfact[i-1]*_invs[i];
            }
        }else{
            for(int i=old;i<=n;i++){
                _fact[i]=_fact[i-1]*i;
                _invs[i]=mint(1)/i;
                _invfact[i]=_invfact[i-1]*_invs[i];
            }
        }
    }
    static mint fact(int i){
        assert(i>=0);
        build(i);
        return fact_table()[i];
    }
    static mint invfact(int i){
        assert(i>=0);
        build(i);
        return invfact_table()[i];
    }
    static mint inv(int i){
        assert(i>0);
        build(i);
        return invs_table()[i];
    }
    static mint C(int a,int b){//aCb
        if(b==0)return 1;
        if(a<0||b<0||a-b<0)return mint(0);
        build(a);
        auto&_fact=fact_table();
        auto&_invfact=invfact_table();
        return _fact[a]*_invfact[b]*_invfact[a-b];
    }
    static mint iC(int a,int b){//1/aCb
        if(b==0)return 1;
        if(a<0||b<0||a-b<0)return mint(0);
        build(a);
        auto&_fact=fact_table();
        auto&_invfact=invfact_table();
        return _fact[b]*_fact[a-b]*_invfact[a];
    }
    static mint P(int a,int b){
        if(a<b||b<0)return 0;
        build(a);
        auto&_fact=fact_table();
        auto&_invfact=invfact_table();
        return _fact[a]*_invfact[a-b];
    }
    static mint H(int a,int b){
        return C(a+b-1,b);
    }

};
#endif
using mint=static_modint<ten(9)+7>;
using B=binom<mint>;
void solve(){
    LL(a,b,n);
    if(n==0)return PRT(2);
    if(n==1)return PRT(2*a);
    matrix<mint>mt(2,2);
    mt[0][0]=2*a;
    mt[0][1]=-a*a+b;
    mt[1][0]=1;
    mt=mt.pow(n-1);
    PRT(mt[0][0]*2*a+mt[0][1]*2);
}
signed main(){
    int t=1;
    // cin >> t;
    while(t--)solve();
}