#define CP_BUNDLED_SOURCE
#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]);}cout<<endl;}
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 ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
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>
vvc<T>trans(const vvc<T>&a){
    assert(a.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;
}
template<class T>
vc<string>trans(const vc<string>&a){
    assert(a.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(vc<T>&a){
    return accumulate(all(a),L(0));
}
template<class T>
vc<T>subset(T S){
    vc<T>ans;
    for(T x=S;x>0;x=(x-1)&S)ans.pb(x);
    ans.pb(0);
    return ans;
}
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));
}
#ifndef COMPRESSER_STRUCT
#define COMPRESSER_STRUCT
template<class T>
struct compresser{
    vc<T>x;
    compresser(int n=0){x.reserve(n);}
    compresser(const vc<T>&xs){
        x=xs;
    }
    void push(T p){built=false;x.pb(p);}
    bool built=false;
    void build(){
        if(!chmax(built,1))return;
        sort(all(x));
        x.erase(unique(all(x)),x.end());
    }
    int find(T v){
        build();
        auto itr=lower_bound(all(x),v)-x.begin();
        if(itr==x.size()||x[itr]!=v)return -1;
        return itr;
    }
    int find_next(T v){
        build();
        return lower_bound(all(x),v)-x.begin();
    }
    int size(){
        build();
        return x.size();
    }
    T operator[](int i)const{
        assert(0<=i&&i<x.size());
        return x[i];
    }
};
#endif
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"
#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);
        dbg("==============="s);
        #else
        cout<<fixed<<setprecision(20);
        #endif
    }
};
inline template_setup template_setup;

#endif
struct barrett{
    using u64=uint64_t;
    using u32=uint32_t;
    using i128=__int128_t;
    u64 m;
    int mod;
    void set(int mod_){
        assert(mod_>0);
        mod=mod_;
        m=(i128(1)<<64)/mod;
    }
    unsigned reduce(uint64_t x){
        assert(mod>0);
        x-=(((i128)x*m)>>64)*mod;
        return x<mod?x:x-mod;
    }
};
template<int id>
struct dynamic_modint{
    using u32=uint32_t;
    using u64=uint64_t;
    u32 val;
    dynamic_modint():val(0){}
    dynamic_modint(ll x){
        ll v=x%get_mod();
        if(v<0)v+=get_mod();
        val=v;
    }
    static dynamic_modint raw(int v){
        assert(v>=0);
        dynamic_modint mi;
        mi.val=v;
        return mi;
    }
    dynamic_modint &operator+=(const dynamic_modint&m){
        if((val+=m.val)>=get_mod())val-=get_mod();
        return *this;
    }
    dynamic_modint &operator-=(const dynamic_modint&m){
        if((val+=(get_mod()-m.val))>=get_mod())val-=get_mod();
        return *this;
    }
    dynamic_modint &operator*=(const dynamic_modint&m){
        val=rem(u64(val)*m.val);
        return *this;
    }
    dynamic_modint &operator/=(const dynamic_modint&m){
        val=rem(u64(val)*m.inv().val);
        return *this;
    }
    dynamic_modint operator-() const{
        return dynamic_modint(val?get_mod()-val:0);
    }
    dynamic_modint operator+() const {
        return *this;
    }
    friend dynamic_modint operator+(dynamic_modint lhs, const dynamic_modint& rhs){
        return lhs+=rhs;
    }
    friend dynamic_modint operator-(dynamic_modint lhs, const dynamic_modint& rhs){
        return lhs-=rhs;
    }
    friend dynamic_modint operator*(dynamic_modint lhs, const dynamic_modint& rhs){
        return lhs*=rhs;
    }
    friend dynamic_modint operator/(dynamic_modint lhs,const dynamic_modint&rhs){
        return lhs/=rhs;
    }
    bool operator==(const dynamic_modint&p) const{
        return p.val==val;
    }
    bool operator!=(const dynamic_modint&p) const{
        return p.val!=val;
    }
    dynamic_modint pow(int64_t n) const{
        dynamic_modint res(1),mul(val);
        while(n){
            if(n%2)res*=mul;
            mul*=mul;
            n/=2;
        }
        return res;
    }

    friend ostream&operator<<(ostream&os,const dynamic_modint&p){
        os<<p.val;
        return os;
    }
    friend istream&operator>>(istream&is,dynamic_modint&p){
        int64_t x;
        is>>x;
        p=dynamic_modint(x);
        return is;
    }
    dynamic_modint inv()const{
        int64_t a=val,b=get_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 dynamic_modint(u);
    }
    inline static u32 rem(u64 x){return barrett_reduction().reduce(x);}
    static inline int &get_mod(){
        static int mod=0;
        return mod;
    }
    static void set_mod(int md){
        assert(0<md&&md<=(1ll<<31)-1);
        get_mod()=md;
        barrett_reduction().set(md);
    }
    static inline barrett&barrett_reduction(){
        static barrett b;
        return b;
    }
};

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(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);
    }

};
template< typename T >
T extgcd(T a, T b, T &x, T &y) {
  T d = a;
  if(b != 0) {
    d = extgcd(b, a % b, y, x);
    y -= (a / b) * x;
  } else {
    x = 1;
    y = 0;
  }
  return d;
}
template<class T>
pair<T,T> inv(T x,T m){
    T a1,a2;
    T res=extgcd<ll>(x,m,a1,a2);
    T md=m/res;
    a1=(a1%md+md)%md;
    return {a1,md};
}
template<class T>
pair<T,T> mod_solve(T a,T b,T m){//return x s.t. ax=b mod m
    a%=m,b%=m;if(a<0)a+=m;if(b<0)b+=m;
    T g=gcd(gcd(a,b),m);
    a/=g,b/=g,m/=g;
    if(gcd(a,m)>1)return {-1,-1};
    return {(inv<ll>(a,m).first*b)%m,inv<ll>(a,m).second};
}

//x^2 ≡ b (mod p)
ll mod_sqrt(ll b,ll p){
    static mt19937 mt(random_device{}());
    b%=p;if(b<0)b+=p;
    if(b==0)return 0;
    if(p==2){
        return b;
    }
    assert(p>=3);
    using mint=dynamic_modint<20260801>;mint::set_mod(p);
    if(mint(b).pow((p-1)/2)==-1){
        return -1;
    }
    if(p%4==3){
        return mint(b).pow((p+1)/4).val;
    }
    ll t=[&](){
        ll w;
        while(1){
            ll t=mt()%p;
            w=t*t-b;
            if(mint(w).pow((p-1)/2)==mint(-1)){
                return t;
            }
        }
        assert(0);
    }();
    mint w=mint(t*t-b);
    using T=pair<mint,mint>;
    auto ml=[&](T a,T b)->T{
        return T{a.first*b.first+a.second*b.second*w,a.second*b.first+a.first*b.second};
    };
    ll e=(p+1)/2;
    T ans={1,0};
    T gy={t,1};
    while(e){
        if(e%2)ans=ml(ans,gy);
        gy=ml(gy,gy);
        e/=2;
    }
    return ans.first.val;
}
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){
        if((val+=m.val)>=mod)val-=mod;
        return *this;
    }
    static_modint &operator-=(const static_modint&m){
        if((val+=(mod-m.val))>=mod)val-=mod;
        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);
    }
};
#include<type_traits>
struct unweighted{
    unweighted()=default;
    unweighted(int){}
    operator int()const{return 1;}
};
template<class T=unweighted>
struct edge{
    int from,to,id;
    [[no_unique_address]] T cost;
    #ifdef LOCAL
    friend ostream&operator<<(ostream&os,const edge&e){
        return os<<"{from:"<<e.from<<",to:"<<e.to<<",id:"<<e.id<<",cost:"<<e.cost<<"}";
    }
    #endif
};
template<class T,class=void>struct is_edge:false_type{};
template<class T>struct is_edge<T,void_t<decltype(declval<T>().from),decltype(declval<T>().to),decltype(declval<T>().id),decltype(declval<T>().cost)>>:true_type{};
struct empty_storage{};
template<bool is_directed,class T=unweighted>
struct static_graph{
    constexpr static bool directed(){return is_directed;}
    using edge=conditional_t<is_edge<T>::value,T,::edge<T>>;
    using cost_t=decltype(declval<edge>().cost);
private:
    int n,m,added=0;
    mutable bool csr_built=false;
    [[no_unique_address]] mutable conditional_t<is_directed,bool,empty_storage> inv_built{};
    vc<edge>_all_edges;
    mutable vc<int>csr_start;
    mutable vc<edge>csr_edge;
    [[no_unique_address]] mutable conditional_t<is_directed,vc<int>,empty_storage>inv_start;
    [[no_unique_address]] mutable conditional_t<is_directed,vc<edge>,empty_storage>inv_edge;
public:
    static_graph(int n):n(n),m(-1){assert(n>=0);csr_start.resize(n+1);}
    static_graph(int n,int m):n(n),m(m){assert(n>=0&&m>=0);csr_start.resize(n+1);_all_edges.reserve(m);}
    void resize(int size){
        assert(n<=size);
        assert(!csr_built);
        n=size;
        csr_start.resize(n+1);
    }
    void add_edge(const edge&e){
        assert(0<=e.from&&e.from<n&&0<=e.to&&e.to<n);
        assert(m==-1||added<m);
        _all_edges.push_back(e);
        csr_built=false;
        if constexpr(is_directed)inv_built=false;
        if(++added==m)build();
    }
    void add_edge(int a,int b,cost_t cost=1,int id=-1){
        assert(0<=a&&a<n&&0<=b&&b<n);
        assert(m==-1||added<m);
        if(id==-1)id=(int)_all_edges.size();
        _all_edges.push_back({a,b,id,cost});
        csr_built=false;
        if constexpr(is_directed)inv_built=false;
        if(++added==m)build();
    }
    template<int substract=0>
    void input(int edge_count){
        assert(edge_count>=0);
        rep(i,edge_count){
            INT(a,b);
            a-=substract;
            b-=substract;
            add_edge(a,b);
        }
    }
    void build()const{
        if(csr_built)return;
        csr_built=true;
        csr_start.assign(n+1,0);
        for(auto&e:_all_edges){
            csr_start[e.from]++;
            if constexpr(!is_directed)csr_start[e.to]++;
        }
        rep(i,n)csr_start[i+1]+=csr_start[i];
        csr_edge.resize(csr_start[n]);
        for(auto it=_all_edges.rbegin();it!=_all_edges.rend();++it){
            auto&e=*it;
            csr_edge[--csr_start[e.from]]=e;
            if constexpr(!is_directed)csr_edge[--csr_start[e.to]]={e.to,e.from,e.id,e.cost};
        }
    }
    void build_inv()const{
        if constexpr(!is_directed){
            build();
            return;
        }else{
            if(inv_built)return;
            inv_built=true;
            inv_start.assign(n+1,0);
            for(auto&e:_all_edges){
                inv_start[e.to]++;
            }
            rep(i,n)inv_start[i+1]+=inv_start[i];
            inv_edge.resize(inv_start[n]);
            for(auto it=_all_edges.rbegin();it!=_all_edges.rend();++it){
                auto&e=*it;
                inv_edge[--inv_start[e.to]]={e.to,e.from,e.id,e.cost};
            }
        }
    }
    const vc<edge>&all_edges()const{return _all_edges;}
    int edge_size()const{return (int)_all_edges.size();}
    edge get_edge(int id)const{
        assert(0<=id&&id<edge_size());
        return _all_edges[id];
    }
    int out_deg(int u)const{assert(0<=u&&u<n);build();return csr_start[u+1]-csr_start[u];}
    int in_deg(int u)const{
        assert(0<=u&&u<n);
        if constexpr(!is_directed){
            return out_deg(u);
        }else{
            build_inv();
            return inv_start[u+1]-inv_start[u];
        }
    }
    int deg(int u)const{return out_deg(u);}
    template<class E>
    struct span{
        E*l; E* r;
        E*begin()const{return l;}
        E*end()const{return r;}
        int size()const{return r-l;}
        E&back()const{return r[-1];}
        E&operator[](int i){return l[i];}
        const E&operator[](int i)const{return l[i];}
    };
    auto operator[](int u){
        assert(0<=u&&u<n);build();
        return span<edge>{csr_edge.data()+csr_start[u],csr_edge.data()+csr_start[u+1]};
    }
    auto operator[](int u)const{
        assert(0<=u&&u<n);build();
        return span<const edge>{csr_edge.data()+csr_start[u],csr_edge.data()+csr_start[u+1]};
    }
    auto inv(int u){
        assert(0<=u&&u<n);
        if constexpr(!is_directed){
            return (*this)[u];
        }else{
            build_inv();
            return span<edge>{inv_edge.data()+inv_start[u],inv_edge.data()+inv_start[u+1]};
        }
    }
    auto inv(int u)const{
        assert(0<=u&&u<n);
        if constexpr(!is_directed){
            return (*this)[u];
        }else{
            build_inv();
            return span<const edge>{inv_edge.data()+inv_start[u],inv_edge.data()+inv_start[u+1]};
        }
    }
    int size()const{return n;}
    template<class F>vvc<F>adj()const{
        vvc<F>res(n,vc<F>(n));
        for(auto&e:all_edges()){
            res[e.from][e.to]=e.cost;
            if(directed()==false)res[e.to][e.from]=e.cost;
        }
        return res;
    }
    void clear(){
        added=0;
        csr_built=false;
        if constexpr(is_directed)inv_built=false;
        _all_edges.clear();_all_edges.shrink_to_fit();
        csr_start.assign(n+1,0);csr_start.shrink_to_fit();
        csr_edge.clear();csr_edge.shrink_to_fit();
        if constexpr(is_directed){
            inv_start.clear();inv_start.shrink_to_fit();
            inv_edge.clear();inv_edge.shrink_to_fit();
        }
    }
    template<class F>
    void sort(int i,F f){
        assert(0<=i&&i<n);
        build();
        sort(csr_edge.begin()+csr_start[i],csr_edge.begin()+csr_start[i+1],f);
    }
    template<class F>
    void sort_inv(int i,F f){
        assert(0<=i&&i<n);
        if constexpr(!is_directed){
            build();
            sort(csr_edge.begin()+csr_start[i],csr_edge.begin()+csr_start[i+1],f);
            return;
        }
        build_inv();
        sort(inv_edge.begin()+inv_start[i],inv_edge.begin()+inv_start[i+1],f);
    }
    template<class F>
    static_graph<is_directed,T>extract(F f)const{
        static_graph<is_directed,T>res(n);
        for(auto&e:_all_edges)if(f(e))res.add_edge(e);
        return res;
    }
    template<class F>
    static_graph<1,T>reorder(F f)const{
        static_graph<1,T>res(n);
        for(auto&e:_all_edges){
            if(f(e))res.add_edge(e);
            else res.add_edge({e.to,e.from,e.id,e.cost});
        }
        return res;
    }
};
template<class T=unweighted>
struct tree{
    using Graph=static_graph<0,T>;
    using edge=typename Graph::edge;
    using cost_t=typename Graph::cost_t;
    mutable Graph g;
    mutable int built_hld=-1;
    int n;
    mutable vc<int>in,out,head,size_,par,depth,ord;
    tree(int n):n(n),g(n,n-1){}
    void add_edge(const edge&e){
        g.add_edge(e);
    }
    void add_edge(int a,int b,cost_t cost=1,int id=-1){
        g.add_edge(a,b,cost,id);
    }
    bool is_to_par(auto&e)const{
        return par[e.from]==e.to;
    }
    int size()const{
        return n;
    }
    auto operator[](int u)const{
        assert(0<=u&&u<n);
        return g[u];
    }
    auto operator[](int u){
        assert(0<=u&&u<n);
        return g[u];
    }
    template<int substract>
    void input(){
        rep(i,n-1){
            INT(a,b);
            a-=substract;
            b-=substract;
            g.add_edge(a,b);
        }
    }
    template<int disjoint=0>
    void build(int root=0)const{
        assert(0<=root&&root<n);
        if(built_hld!=-1){
            return;
        }
        built_hld=disjoint;
        in.resize(n);
        out.resize(n);
        head.resize(n);
        size_.assign(n,1);
        par.resize(n);
        depth.resize(n);
        ord.assign(n*(disjoint+1),-1);
        par[root]=-1;
        depth[root]=0;
        head[root]=root;
        int timer=0;
        auto dfs=[&](auto&dfs,int u,int v,int d)->void{
            depth[u]=d++;
            auto s=g[u];
            int heavy=-1,par_id=-1;
            rep(i,s.size()){
                auto&e=s[i];
                if(e.to==v){
                    par_id=i;
                    continue;
                }
                par[e.to]=u;
                dfs(dfs,e.to,u,d);
                size_[u]+=size_[e.to];
                if(heavy==-1||size_[s[heavy].to]<size_[e.to])heavy=i;
            }
            if(heavy!=-1&&heavy!=0){
                swap(s[heavy],s[0]);
                if(par_id==0)par_id=heavy;
                else if(par_id==heavy)par_id=0;
            }
            if(par_id!=-1&&par_id+1!=s.size()){
                swap(s[par_id],s[s.size()-1]);
            }
        };dfs(dfs,root,-1,0);
        {auto dfs=[&](auto&dfs,int u)->void{
            in[u]=timer;
            ord[timer++]=u;
            auto s=g[u];
            bool first=true;
            rep(i,s.size()){
                auto&e=s[i];
                if(e.to==par[u])continue;
                head[e.to]=first?head[u]:e.to;
                first=false;
                dfs(dfs,e.to);
            }
            if constexpr(disjoint)out[u]=timer++;
            else out[u]=timer;
        };dfs(dfs,root);}
        rep(u,n){
            auto s=g[u];
            int ce=s.size()-(par[u]!=-1);
            if(ce>1)sort(s.begin()+1,s.begin()+ce,[&](const edge&a,const edge&b){return in[a.to]<in[b.to];});
        }
    }
    auto heavy_edge(int u)const{
        assert(0<=u&&u<n);
        build();
        auto s=g[u];
        int sz=s.size(),ce=(par[u]==-1?sz:sz-1);
        if(ce<=0)return typename Graph::template span<const edge>{s.l,s.l};
        return typename Graph::template span<const edge>{s.l,s.l+1};
    }
    auto light_edges(int u)const{
        assert(0<=u&&u<n);
        build();
        auto s=g[u];
        int sz=s.size(),ce=(par[u]==-1?sz:sz-1);
        if(ce<=1)return typename Graph::template span<const edge>{s.l,s.l};
        return typename Graph::template span<const edge>{s.l+1,s.l+ce};
    }
    int lca(int a,int b)const{
        assert(0<=a&&a<n&&0<=b&&b<n);
        build();
        auto&h=head;
        auto&d=depth;
        auto&p=par;
        while(1){
            if(h[a]==h[b]){
                if(d[a]<d[b])return a;
                return b;
            }
            if(d[h[a]]<d[h[b]])swap(a,b);
            a=p[h[a]];
        }
    }
    int dist(int a,int b)const{
        assert(0<=a&&a<n&&0<=b&&b<n);
        build();
        return depth[a]+depth[b]-2*depth[lca(a,b)];
    }
    int jumpup(int a,int k)const{
        assert(0<=a&&a<n);
        build();
        if(k<0||depth[a]<k)return -1;
        auto&I=in;
        auto&H=head;
        auto&P=par;
        auto&O=ord;
        while(k>0){ 
            int x=I[a]-I[H[a]];
            if(x>=k){
                return O[I[a]-k];
            }
            k-=x+1;
            a=P[H[a]];
        }
        return a;
    }
    int jump(int s,int t,int k)const{
        assert(0<=s&&s<n&&0<=t&&t<n);
        build();
        int L=lca(s,t);
        int D=depth[s]+depth[t]-2*depth[L];
        if(depth[s]-depth[L]>=k){
            return jumpup(s,k);
        }else{
            return jumpup(t,D-k);
        }
    }   
    int to(int x,int y)const{
        assert(0<=x&&x<n&&0<=y&&y<n&&x!=y);
        build();
        if(in[y]<in[x]||out[x]<=in[y])return g[x].size()-1;
        auto s=g[x];
        int ce=s.size()-(par[x]!=-1);
        if(in[s[0].to]<=in[y]&&in[y]<out[s[0].to])return 0;
        int l=1,r=ce;
        while(l<r){
            int m=(l+r)/2;
            if(in[s[m].to]<=in[y])l=m+1;
            else r=m;
        }
        return l-1;
    }
    vc<pair<int,int>>query(int s,int t,bool edge=false)const{
        assert(0<=s&&s<n&&0<=t&&t<n);
        build();
        auto&h=head;
        auto&d=depth;
        auto&I=in;
        auto&P=par;
        vc<pair<int,int>>rs,rt;
        while(h[s]!=h[t]){
            if(d[h[s]]>d[h[t]]){
                rs.push_back({I[s],I[h[s]]});
                s=P[h[s]];
            }else{
                rt.push_back({I[h[t]],I[t]});
                t=P[h[t]];
            }
        }
        if(s!=t||!edge){
            if(d[s]>d[t]){
                rs.push_back({I[s],I[t]+edge});
            }else{
                rt.push_back({I[s]+edge,I[t]});
            }
        }
        rs.reserve(rs.size()+rt.size());
        for(auto it=rt.rbegin();it!=rt.rend();++it)rs.push_back(*it);
        return rs;
    }
    pair<int,int>get_diameter()const{
        assert(n>0);
        auto find_farthest=[&](int from)->int{
            vc<int>d(n),p(n,-2),st(1,from);
            p[from]=-1;
            int far=from;
            while(!st.empty()){
                int u=st.back();
                st.pop_back();
                if(d[far]<d[u])far=u;
                for(auto&e:g[u])if(e.to!=p[u]){
                    p[e.to]=u;
                    d[e.to]=d[u]+1;
                    st.push_back(e.to);
                }
            }
            return far;
        };
        int v1=find_farthest(0);
        int v2=find_farthest(v1);
        return {v1,v2};
    }
};
using mint=static_modint<998244353>;
using B=binom<mint>;
void solve(){
    INT(n,m);
    vc<mint>b(n);rep(i,n)cin>>b[i];
    static_graph<0>g(n);
    g.input<1>(m);
    vc<mint>val(n);
    vc<int>col(n,-1);
    queue<int>que;
    que.push(0);col[0]=0;
    tree<unweighted>t(n);
    while(que.size()){
        auto p=que.front();que.pop();
        for(auto&e:g[p]){
            if(col[e.to]==-1){
                col[e.to]=col[p]^1;
                t.add_edge(e);
                que.push(e.to);
            }
        }
    }
    vc<static_graph<0>::edge>paredge(n);
    vc<mint>ans(m);
    auto dfs=[&](auto&dfs,int u,int v)->void{
        for(auto&e:t[u]){
            if(e.to==v)continue;
        
            paredge[e.to]=e;
            dfs(dfs,e.to,u);
            mint add=b[e.to]-val[e.to];
            val[e.to]+=add;
            ans[e.id]=add;
            val[u]+=add;
            dbg(u,e.to,add);
        }
    };
    dfs(dfs,0,-1);
    if(val==b){
        rep(i,m)cout<<ans[i]<<" \n"[i+1==m];
        return;
    }
    vc<mint>coef(n);
    vc<mint>ecoef(m);
    rep(i,n){
        for(auto&e:g[i]){
            if(col[i]!=col[e.to])continue;
            auto modify=[&](auto&modify,ll u)->void{
                if(u==0)return;
                ecoef[paredge[u].id]-=coef[u];
                coef[paredge[u].from]-=coef[u];
                coef[u]=0;
                modify(modify,paredge[u].from);
            };
            ecoef[e.id]+=1;
            coef[e.to]+=1,coef[i]+=1;
            modify(modify,e.to);
            modify(modify,i);
            mint nx=((b[0]-val[0])/coef[0]);
            dbg(nx);
            rep(i,m){
                ans[i]+=ecoef[i]*nx;
            }
            rep(i,n)for(auto&e:g[i])if(i<e.to){
                val[i]+=ecoef[e.id]*nx;
                val[e.to]+=ecoef[e.id]*nx;
            }
            dbg(val);
            dbg(b);
            assert(val==b);
            rep(i,m)cout<<ans[i]<<" \n"[i+1==m];
            return;
        }
    } 
    return PRT(-1);
}
signed main(){
    int t=1;
    // cin >> t;
    while(t--)solve();
}