#ifndef LOCAL
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#endif

#include <bits/stdc++.h>
using namespace std;

using ll=long long;
#define int ll

bool dbg=false;

#define rng(i,a,b) for(int i=int(a);i<int(b);i++)
#define rep(i,b) rng(i,0,b)
#define gnr(i,a,b) for(int i=int(b)-1;i>=int(a);i--)
#define per(i,b) gnr(i,0,b)
#define pb push_back
#define eb emplace_back
#define a first
#define b second
#define bg begin()
#define ed end()
#define all(x) x.bg,x.ed
#define si(x) int(x.size())
#ifdef LOCAL
#define dmp(x) cerr<<__LINE__<<" "<<#x<<" "<<x<<endl
#else
#define dmp(x) void(0)
#endif

template<class t,class u> bool chmax(t&a,u b){if(a<b){a=b;return true;}else return false;}
template<class t,class u> bool chmin(t&a,u b){if(b<a){a=b;return true;}else return false;}

template<class t> using vc=vector<t>;
template<class t> using vvc=vc<vc<t>>;

using pi=pair<int,int>;
using vi=vc<int>;
using vvi=vc<vc<int>>;

template<class t,class u>
ostream& operator<<(ostream& os,const pair<t,u>& p){
	return os<<"{"<<p.a<<","<<p.b<<"}";
}

template<class t> ostream& operator<<(ostream& os,const vc<t>& v){
	os<<"{";
	for(auto e:v)os<<e<<",";
	return os<<"}";
}

#define mp make_pair
#define mt make_tuple
#define one(x) memset(x,-1,sizeof(x))
#define zero(x) memset(x,0,sizeof(x))
#ifdef LOCAL
void dmpr(ostream&os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
	os<<t<<" ";
	dmpr(os,args...);
}
#define dmp2(...) dmpr(cerr,__LINE__,##__VA_ARGS__)
#else
#define dmp2(...) void(0)
#endif

using uint=unsigned;
using ull=unsigned long long;

template<class t,size_t n>
ostream& operator<<(ostream&os,const array<t,n>&a){
	return os<<vc<t>(all(a));
}

ll rand_int(ll l, ll r) { //[l, r]
	//#ifdef LOCAL
	static mt19937_64 gen;
	/*#else
	static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count());
	#endif*/
	return uniform_int_distribution<ll>(l, r)(gen);
}

ll rand_int(ll k){ //[0,k)
	return rand_int(0,k-1);
}
string rand_string(int n,char lw,char up){
	string s(n,'?');
	rep(i,n)s[i]=rand_int(lw,up);
	return s;
}

int current_run_id,run_batch_size=1000;
int calc_random_limit(){
	return current_run_id/run_batch_size+1;
}
template<class t>
void generate_single(t&a){
	a=rand_int(1,calc_random_limit());
}
void generate_single(string&a){
	int n;generate_single(n);
	a=rand_string(n,'a','b');
}
template<class t,class u>
void generate_single(pair<t,u>&a){
	generate_single(a.a);
	generate_single(a.b);
}
//https://trap.jp/post/1224/
template<class... Args>
void input(Args&... a){
	if(dbg){
		(generate_single(a),...);
	}else{
		#ifdef USE_FAST_IO
		sc.read(a...);
		#else
		(cin >> ... >> a);
		#endif
	}
}
#define INT(...) int __VA_ARGS__;input(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;input(__VA_ARGS__)
#define ULL(...) ull __VA_ARGS__;input(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;input(__VA_ARGS__)
#define CHR(...) char __VA_ARGS__;input(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__;input(__VA_ARGS__)
#define LD(...) ld __VA_ARGS__;input(__VA_ARGS__)
#define overload3(a,b,c,d,...) d
#define VI2(name,size) vi name(size);rep(i_##name,size)input(name[i_##name]);
#define VI3(name,size,offset) vi name(size);rep(i_##name,size)input(name[i_##name]),name[i_##name]+=offset;
#define VI(...) overload3(__VA_ARGS__,VI3,VI2)(__VA_ARGS__)
#define VPI(name,size) vc<pi> name(size);rep(i_##name,size)input(name[i_##name]);
#define VVI(name,sizeN,sizeM) vvi name(sizeN,vi(sizeM));\
rep(i_##name,sizeN)rep(j_##name,sizeM)input(name[i_##name][j_##name]);
#define VS(name,size) vc<string> name(size);rep(i_##name,size)input(name[i_##name]);

#define overload5(a,b,c,d,e,f,...) f
#define VVC4(type,name,sizeN,sizeM) vvc<type> name(sizeN,vc<type>(sizeM));
#define VVC5(type,name,sizeN,sizeM,ini) vvc<type> name(sizeN,vc<type>(sizeM,ini));
#define VVC(...) overload5(__VA_ARGS__,VVC5,VVC4)(__VA_ARGS__)

template<class T>
T vvvc(T v){
	return v;
}

template<class T,class...Args>
auto vvvc(int n,T v,Args...args){
	return vector(n,vvvc(v,args...));
}

template<int i,class T>
void print_tuple(ostream&,const T&){
}

template<int i,class T,class H,class ...Args>
void print_tuple(ostream&os,const T&t){
	if(i)os<<",";
	os<<get<i>(t);
	print_tuple<i+1,T,Args...>(os,t);
}

template<class ...Args>
ostream& operator<<(ostream&os,const tuple<Args...>&t){
	os<<"{";
	print_tuple<0,tuple<Args...>,Args...>(os,t);
	return os<<"}";
}

void printsuc(int suc){
	#ifdef USE_FAST_IO
		if(suc==1)pr.write('\n');
		if(suc==2)pr.write(' ');
	#else
		if(suc==1){
			if(dbg)cout<<endl;
			else{
				#ifdef LOCAL
				cout<<endl;
				#else
				cout<<"\n";
				#endif
			}
		}
		if(suc==2)
			cout<<" ";
	#endif
}

template<class t>
void print_single(t x,int suc=1){
	#ifdef USE_FAST_IO
	pr.write(x);
	#else
	cout<<x;
	#endif
	printsuc(suc);
}

template<class t,class u>
void print_single(const pair<t,u>&p,int suc=1){
	print_single(p.a,2);
	print_single(p.b,suc);
}

template<class T>
void print_single(const vector<T>&v,int suc=1){
	rep(i,v.size())
		print_single(v[i],i==int(v.size())-1?3:2);
	printsuc(suc);
}

template<class T,size_t N>
void print_single(const array<T,N>&v,int suc=1){
	rep(i,N)
		print_single(v[i],i==int(N)-1?3:2);
	printsuc(suc);
}

template<class T>
void print(const T&t){
	print_single(t);
}

template<class T,class ...Args>
void print(const T&t,const Args&...args){
	print_single(t,2);
	print(args...);
}

template<class T>
void printvv(const vvc<T>&vs){
	for(const auto&row:vs)print(row);
}

string readString(){
	string s;
	cin>>s;
	return s;
}

template<class T>
T sq(const T& t){
	return t*t;
}

void YES(bool ex=true){
	cout<<"YES\n";
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
void NO(bool ex=true){
	cout<<"NO\n";
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
void Yes(bool ex=true){
	cout<<"Yes\n";
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
void No(bool ex=true){
	cout<<"No\n";
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
//#define CAPITAL
/*
void yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<"\n";
	#else
	cout<<"Yes"<<"\n";
	#endif
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
void no(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<"\n";
	#else
	cout<<"No"<<"\n";
	#endif
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}*/
void possible(bool ex=true){
	#ifdef CAPITAL
	cout<<"POSSIBLE"<<"\n";
	#else
	cout<<"Possible"<<"\n";
	#endif
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}
void impossible(bool ex=true){
	#ifdef CAPITAL
	cout<<"IMPOSSIBLE"<<"\n";
	#else
	cout<<"Impossible"<<"\n";
	#endif
	if(ex)exit(0);
	#ifdef LOCAL
	cout.flush();
	#endif
}

constexpr ll ten(int n){
	return n==0?1:ten(n-1)*10;
}

const ll infLL=LLONG_MAX/3;

#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif

int topbit(signed t){
	return t==0?-1:31-__builtin_clz(t);
}
int topbit(ll t){
	return t==0?-1:63-__builtin_clzll(t);
}
int topbit(ull t){
	return t==0?-1:63-__builtin_clzll(t);
}
int botbit(signed a){
	return a==0?32:__builtin_ctz(a);
}
int botbit(ll a){
	return a==0?64:__builtin_ctzll(a);
}
int botbit(ull a){
	return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
	return __builtin_popcount(t);
}
int popcount(ll t){
	return __builtin_popcountll(t);
}
int popcount(ull t){
	return __builtin_popcountll(t);
}
int bitparity(ll t){
	return __builtin_parityll(t);
}
bool ispow2(int i){
	return i&&(i&-i)==i;
}
ll mask(int i){
	return (ll(1)<<i)-1;
}
ull umask(int i){
	return (ull(1)<<i)-1;
}
ll minp2(ll n){
	if(n<=1)return 1;
	else return ll(1)<<(topbit(n-1)+1);
}

bool inc(int a,int b,int c){
	return a<=b&&b<=c;
}

template<class S> void mkuni(S&v){
	sort(all(v));
	v.erase(unique(all(v)),v.ed);
}

template<class t> bool isuni(vc<t> v){
	int s=si(v);
	mkuni(v);
	return si(v)==s;
}

template<class t>
void myshuffle(vc<t>&a){
	rep(i,si(a))swap(a[i],a[rand_int(0,i)]);
}

template<class S,class u>
int lwb(const S&v,const u&a){
	return lower_bound(all(v),a)-v.bg;
}
template<class t,class u>
bool bis(const vc<t>&v,const u&a){
	return binary_search(all(v),a);
}

//VERIFY: yosupo
//KUPC2017J
//AOJDSL1A
//without rank
struct unionfind{
	vi p,s;
	int c;
	unionfind(int n):p(n,-1),s(n,1),c(n){}
	void clear(){
		fill(all(p),-1);
		fill(all(s),1);
		c=si(p);
	}
	int find(int a){
		return p[a]==-1?a:(p[a]=find(p[a]));
	}
	//set b to a child of a
	bool unite(int a,int b){
		a=find(a);
		b=find(b);
		if(a==b)return false;
		p[b]=a;
		s[a]+=s[b];
		c--;
		return true;
	}
	bool same(int a,int b){
		return find(a)==find(b);
	}
	int sz(int a){
		return s[find(a)];
	}
};

vvc<int> readGraph(int n,int m){
	vvc<int> g(n);
	rep(i,m){
		int a,b;
		cin>>a>>b;
		//sc.read(a,b);
		a--;b--;
		g[a].pb(b);
		g[b].pb(a);
	}
	return g;
}

vvc<int> rand_tree(int n){
	vvc<int> t(n);
	unionfind uf(n);
	while(uf.c>1){
		int a=rand_int(n);
		int b=rand_int(n);
		if(uf.unite(a,b)){
			t[a].pb(b);
			t[b].pb(a);
		}
	}
	return t;
}

vvc<int> readTree(int n){
	if(dbg){
		return rand_tree(n);
	}else{
		return readGraph(n,n-1);
	}
}

vi readRooted(int n){
	assert(!dbg);
	vi par(n,-1);
	rng(i,1,n){
		input(par[i]);
		par[i]--;
		assert(inc(0,par[i],i-1));
	}
	return par;
}

void printTree(const vvc<int> t){
	int n=si(t);
	int degsum=0;
	rep(i,n)degsum+=si(t[i]);
	if(degsum==n-1){
		//directed
		rep(i,si(t))for(auto j:t[i]){
			print(i+1,j+1);
		}
	}else if(degsum==2*(n-1)){
		//undirected
		rep(i,si(t))for(auto j:t[i])if(i<j){
			print(i+1,j+1);
		}
	}else{
		assert(false);
	}
}

template<class t>
vc<t> presum(const vc<t>&a){
	vc<t> s(si(a)+1);
	rep(i,si(a))s[i+1]=s[i]+a[i];
	return s;
}
vc<ll> presum(const vi&a){
	vc<ll> s(si(a)+1);
	rep(i,si(a))s[i+1]=s[i]+a[i];
	return s;
}
//BIT で数列を管理するときに使う (CF850C)
template<class t>
vc<t> predif(vc<t> a){
	gnr(i,1,si(a))a[i]-=a[i-1];
	return a;
}
template<class t>
vvc<ll> imos(const vvc<t>&a){
	int n=si(a),m=si(a[0]);
	vvc<ll> b(n+1,vc<ll>(m+1));
	rep(i,n)rep(j,m)
		b[i+1][j+1]=b[i+1][j]+b[i][j+1]-b[i][j]+a[i][j];
	return b;
}

//verify してないや
void transvvc(int&n,int&m){
	swap(n,m);
}
template<class t,class... Args>
void transvvc(int&n,int&m,vvc<t>&a,Args&...args){
	assert(si(a)==n);
	vvc<t> b(m,vi(n));
	rep(i,n){
		assert(si(a[i])==m);
		rep(j,m)b[j][i]=a[i][j];
	}
	a.swap(b);
	transvvc(n,m,args...);
}
//CF854E
void rotvvc(int&n,int&m){
	swap(n,m);
}
template<class t,class... Args>
void rotvvc(int&n,int&m,vvc<t>&a,Args&...args){
	assert(si(a)==n);
	vvc<t> b(m,vi(n));
	rep(i,n){
		assert(si(a[i])==m);
		rep(j,m)b[m-1-j][i]=a[i][j];
	}
	a.swap(b);
	rotvvc(n,m,args...);
}

//ソートして i 番目が idx[i]
//CF850C
template<class t>
vi sortidx(const vc<t>&a){
	int n=si(a);
	vi idx(n);iota(all(idx),0);
	sort(all(idx),[&](int i,int j){return a[i]<a[j];});
	return idx;
}
//vs[i]=a[idx[i]]
//例えば sortidx で得た idx を使えば単にソート列になって返ってくる
//CF850C
template<class t>
vc<t> a_idx(const vc<t>&a,const vi&idx){
	int n=si(a);
	assert(si(idx)==n);
	vc<t> vs(n);
	rep(i,n)vs[i]=a[idx[i]];
	return vs;
}
//CF850C
vi invperm(const vi&p){
	int n=si(p);
	vi q(n);
	rep(i,n)q[p[i]]=i;
	return q;
}

template<class t,class s=t>
s SUM(const vc<t>&a){
	return accumulate(all(a),s(0));
}
template<class t,size_t K,class s=t>
s SUM(const array<t,K>&a){
	return accumulate(all(a),s(0));
}

template<class t>
t MAX(const vc<t>&a){
	return *max_element(all(a));
}

template<class t>
pair<t,int> MAXi(const vc<t>&a){
	auto itr=max_element(all(a));
	return mp(*itr,itr-a.bg);
}

template<class A>
auto MIN(const A&a){
	return *min_element(all(a));
}

template<class t>
pair<t,int> MINi(const vc<t>&a){
	auto itr=min_element(all(a));
	return mp(*itr,itr-a.bg);
}

vi vid(int n){
	vi res(n);iota(all(res),0);
	return res;
}

template<class S>
void soin(S&s){
	sort(all(s));
}

template<class S,class F>
void soin(S&s,F&&f){
	sort(all(s),forward<F>(f));
}

template<class S>
S soout(S s){
	soin(s);
	return s;
}

template<class S>
void rein(S&s){
	reverse(all(s));
}

template<class S>
S reout(S s){
	rein(s);
	return s;
}

template<class t,class u>
pair<t,u>&operator+=(pair<t,u>&a,pair<t,u> b){
	a.a+=b.a;a.b+=b.b;return a;}
template<class t,class u>
pair<t,u>&operator-=(pair<t,u>&a,pair<t,u> b){
	a.a-=b.a;a.b-=b.b;return a;}
template<class t,class u>
pair<t,u> operator+(pair<t,u> a,pair<t,u> b){return mp(a.a+b.a,a.b+b.b);}
template<class t,class u>
pair<t,u> operator-(pair<t,u> a,pair<t,u> b){return mp(a.a-b.a,a.b-b.b);}
template<class t,class u,class v>
pair<t,u>&operator*=(pair<t,u>&a,v b){
	a.a*=b;a.b*=b;return a;}
template<class t,class u,class v>
pair<t,u> operator*(pair<t,u> a,v b){return a*=b;}
template<class t,class u>
pair<t,u> operator-(pair<t,u> a){return mp(-a.a,-a.b);}
namespace std{
template<class t,class u>
istream&operator>>(istream&is,pair<t,u>&a){
	return is>>a.a>>a.b;
}
}

template<class t>
t gpp(vc<t>&vs){
	assert(si(vs));
	t res=move(vs.back());
	vs.pop_back();
	return res;
}

template<class t,class u>
void pb(vc<t>&a,const vc<u>&b){
	a.insert(a.ed,all(b));
}

template<class t,class...Args>
vc<t> cat(vc<t> a,Args&&...b){
	(pb(a,forward<Args>(b)),...);
	return a;
}

template<class t,class u>
vc<t>& operator+=(vc<t>&a,u x){
	for(auto&v:a)v+=x;
	return a;
}

template<class t,class u>
vc<t> operator+(vc<t> a,u x){
	return a+=x;
}

template<class t>
vc<t>& operator+=(vc<t>&a,const vc<t>&b){
	a.resize(max(si(a),si(b)));
	rep(i,si(b))a[i]+=b[i];
	return a;
}

template<class t>
vc<t> operator+(const vc<t>&a,const vc<t>&b){
	vc<t> c(max(si(a),si(b)));
	rep(i,si(a))c[i]+=a[i];
	rep(i,si(b))c[i]+=b[i];
	return c;
}

template<class t,class u>
vc<t>& operator-=(vc<t>&a,u x){
	for(auto&v:a)v-=x;
	return a;
}
template<class t,class u>
vc<t> operator-(vc<t> a,u x){
	return a-=x;
}

template<class t,class u>
vc<t>& operator*=(vc<t>&a,u x){
	for(auto&v:a)v*=x;
	return a;
}
template<class t,class u>
vc<t> operator*(vc<t> a,u x){
	return a*=x;
}

template<class t,class u>
vc<t>& operator/=(vc<t>&a,u x){
	for(auto&v:a)v/=x;
	return a;
}
template<class t,class u>
vc<t> operator/(vc<t> a,u x){
	return a/=x;
}

template<class t>
vc<t>& operator<<=(vc<t>&a,int k){
	assert(k>=0);
	a.insert(a.bg,k,t(0));
	return a;
}
template<class t>
vc<t> operator<<(vc<t> a,int k){
	return a<<=k;
}

template<class t>
vc<t>& operator>>=(vc<t>&a,int k){
	if(si(a)<=k)a.clear();
	else a.erase(a.bg,a.bg+k);
	return a;
}
template<class t>
vc<t> operator>>(vc<t> a,int k){
	return a>>=k;
}

template<class t,class u>
void remval(vc<t>&a,const u&v){
	a.erase(remove(all(a),v),a.ed);
}
//消した要素の個数を返してくれる
//UCUP 2-8-F
template<class t,class F>
int remif(vc<t>&a,F f){
	auto itr=remove_if(all(a),f);
	int res=a.ed-itr;
	a.erase(itr,a.ed);
	return res;
}
template<class t>
void rempos(vc<t>&a,int i){
	assert(inc(0,i,si(a)-1));
	a.erase(a.bg+i);
}

template<class VS,class u>
void fila(VS&vs,const u&a){
	fill(all(vs),a);
}

template<class t,class u>
int findid(const vc<t>&vs,const u&a){
	auto itr=find(all(vs),a);
	if(itr==vs.ed)return -1;
	else return itr-vs.bg;
}

template<class t>
void rtt(vc<t>&vs,int i){
	rotate(vs.bg,vs.bg+i,vs.ed);
}

//Multiuni2023-8 C
//f(lw)=false,...,f(n-1)=false,f(n)=true,...,f(up)=true,
//のときに n を返す
template<class F>
int find_min_true(int lw,int up,F f){
	while(up-lw>1){
		const int mid=(lw+up)/2;
		if(f(mid))up=mid;
		else lw=mid;
	}
	return up;
}
//f(lw)=true,f(up)=false
template<class F>
int find_max_true(int lw,int up,F f){
	while(up-lw>1){
		const int mid=(lw+up)/2;
		if(f(mid))lw=mid;
		else up=mid;
	}
	return lw;
}

template<class t> using pqmin=priority_queue<t,vc<t>,greater<t>>;
template<class t> using pqmax=priority_queue<t>;
using T=tuple<int,int,int>;

//mint107 は verify してねえ
//#define DYNAMIC_MOD

struct modinfo{uint mod,root;
#ifdef DYNAMIC_MOD
constexpr modinfo(uint m,uint r):mod(m),root(r),im(0){set_mod(m);}
ull im;
constexpr void set_mod(uint m){
	mod=m;
	im=ull(-1)/m+1;
}
uint product(uint a,uint b)const{
	ull z=ull(a)*b;
	uint x=((unsigned __int128)z*im)>>64;
	uint v=uint(z)-x*mod;
	return v<mod?v:v+mod;
}
#endif
};
template<modinfo const&ref>
struct modular{
	static constexpr uint const &mod=ref.mod;
	static modular root(){return modular(ref.root);}
	uint v;
	//modular(initializer_list<uint>ls):v(*ls.bg){}
	modular(ll vv=0){s(vv%mod+mod);}
	modular& s(uint vv){
		v=vv<mod?vv:vv-mod;
		return *this;
	}
	modular operator-()const{return modular()-*this;}
	modular& operator+=(const modular&rhs){return s(v+rhs.v);}
	modular&operator-=(const modular&rhs){return s(v+mod-rhs.v);}
	modular&operator*=(const modular&rhs){
		#ifndef DYNAMIC_MOD
		v=ull(v)*rhs.v%mod;
		#else
		v=ref.product(v,rhs.v);
		#endif
		return *this;
	}
	modular&operator/=(const modular&rhs){return *this*=rhs.inv();}
	modular operator+(const modular&rhs)const{return modular(*this)+=rhs;}
	modular operator-(const modular&rhs)const{return modular(*this)-=rhs;}
	modular operator*(const modular&rhs)const{return modular(*this)*=rhs;}
	modular operator/(const modular&rhs)const{return modular(*this)/=rhs;}
	modular pow(ll n)const{
		if(n<0)return inv().pow(-n);
		modular res(1),x(*this);
		while(n){
			if(n&1)res*=x;
			x*=x;
			n>>=1;
		}
		return res;
	}
	modular inv()const{return pow(mod-2);}
	/*modular inv()const{
		int x,y;
		int g=extgcd<ll>(v,mod,x,y);
		assert(g==1);
		if(x<0)x+=mod;
		return modular(x);
	}*/
	friend modular operator+(ll x,const modular&y){
		return modular(x)+y;
	}
	friend modular operator-(ll x,const modular&y){
		return modular(x)-y;
	}
	friend modular operator*(ll x,const modular&y){
		return modular(x)*y;
	}
	friend modular operator/(ll x,const modular&y){
		return modular(x)/y;
	}
	friend ostream& operator<<(ostream&os,const modular&m){
		return os<<m.v;
	}
	friend istream& operator>>(istream&is,modular&m){
		ll x;is>>x;
		m=modular(x);
		return is;
	}
	bool operator<(const modular&r)const{return v<r.v;}
	bool operator==(const modular&r)const{return v==r.v;}
	bool operator!=(const modular&r)const{return v!=r.v;}
	explicit operator bool()const{
		return v;
	}
};

#define USE_GOOD_MOD

//size of input must be a power of 2
//output of forward fmt is bit-reversed
//output elements are in the range [0,mod*4)
//input of inverse fmt should be bit-reversed
template<class mint>
void inplace_fmt(const int n,mint*const f,bool inv){
	static constexpr uint mod=mint::mod;
	static constexpr uint mod2=mod*2;
	static constexpr int L=30;
	static mint g[L],ig[L],p2[L];
	if(g[0].v==0){
		rep(i,L){
			mint w=-mint::root().pow(((mod-1)>>(i+2))*3);
			g[i]=w;
			ig[i]=w.inv();
			p2[i]=mint(1<<i).inv();
		}
	}
	if(!inv){
		int b=n;
		if(b>>=1){//input:[0,mod)
			rep(i,b){
				uint x=f[i+b].v;
				f[i+b].v=f[i].v+mod-x;
				f[i].v+=x;
			}
		}
		if(b>>=1){//input:[0,mod*2)
			mint p=1;
			for(int i=0,k=0;i<n;i+=b*2){
				rng(j,i,i+b){
					uint x=(f[j+b]*p).v;
					f[j+b].v=f[j].v+mod-x;
					f[j].v+=x;
				}
				p*=g[__builtin_ctz(++k)];
			}
		}
		while(b){
			if(b>>=1){//input:[0,mod*3)
				mint p=1;
				for(int i=0,k=0;i<n;i+=b*2){
					rng(j,i,i+b){
						uint x=(f[j+b]*p).v;
						f[j+b].v=f[j].v+mod-x;
						f[j].v+=x;
					}
					p*=g[__builtin_ctz(++k)];
				}
			}
			if(b>>=1){//input:[0,mod*4)
				mint p=1;
				for(int i=0,k=0;i<n;i+=b*2){
					rng(j,i,i+b){
						uint x=(f[j+b]*p).v;
						f[j].v=(f[j].v<mod2?f[j].v:f[j].v-mod2);
						f[j+b].v=f[j].v+mod-x;
						f[j].v+=x;
					}
					p*=g[__builtin_ctz(++k)];
				}
			}
		}
	}else{
		int b=1;
		if(b<n/2){//input:[0,mod)
			mint p=1;
			for(int i=0,k=0;i<n;i+=b*2){
				rng(j,i,i+b){
					ull x=f[j].v+mod-f[j+b].v;
					f[j].v+=f[j+b].v;
					f[j+b].v=x*p.v%mod;
				}
				p*=ig[__builtin_ctz(++k)];
			}
			b<<=1;
		}
		for(;b<n/2;b<<=1){
			mint p=1;
			for(int i=0,k=0;i<n;i+=b*2){
				rng(j,i,i+b/2){//input:[0,mod*2)
					ull x=f[j].v+mod2-f[j+b].v;
					f[j].v+=f[j+b].v;
					f[j].v=(f[j].v)<mod2?f[j].v:f[j].v-mod2;
					f[j+b].v=x*p.v%mod;
				}
				rng(j,i+b/2,i+b){//input:[0,mod)
					ull x=f[j].v+mod-f[j+b].v;
					f[j].v+=f[j+b].v;
					f[j+b].v=x*p.v%mod;
				}
				p*=ig[__builtin_ctz(++k)];
			}
		}
		if(b<n){//input:[0,mod*2)
			rep(i,b){
				uint x=f[i+b].v;
				f[i+b].v=f[i].v+mod2-x;
				f[i].v+=x;
			}
		}
		mint z=p2[__lg(n)];
		rep(i,n)f[i]*=z;
	}
}

template<class mint>
void inplace_fmt(vector<mint>&f,bool inv){
	inplace_fmt(si(f),f.data(),inv);
}

//size of input must be a power of 2
//output elements are in the range [0,mod*4)
template<class mint>
void half_fmt(const int n,mint*const f){
	static constexpr uint mod=mint::mod;
	static constexpr uint mod2=mod*2;
	static const int L=30;
	static mint g[L],h[L];
	if(g[0].v==0){
		rep(i,L){
			g[i]=-mint::root().pow(((mod-1)>>(i+2))*3);
			h[i]=mint::root().pow((mod-1)>>(i+2));
		}
	}
	int b=n;
	int lv=0;
	if(b>>=1){//input:[0,mod)
		mint p=h[lv++];
		for(int i=0,k=0;i<n;i+=b*2){
			rng(j,i,i+b){
				uint x=(f[j+b]*p).v;
				f[j+b].v=f[j].v+mod-x;
				f[j].v+=x;
			}
			p*=g[__builtin_ctz(++k)];
		}
	}
	if(b>>=1){//input:[0,mod*2)
		mint p=h[lv++];
		for(int i=0,k=0;i<n;i+=b*2){
			rng(j,i,i+b){
				uint x=(f[j+b]*p).v;
				f[j+b].v=f[j].v+mod-x;
				f[j].v+=x;
			}
			p*=g[__builtin_ctz(++k)];
		}
	}
	while(b){
		if(b>>=1){//input:[0,mod*3)
			mint p=h[lv++];
			for(int i=0,k=0;i<n;i+=b*2){
				rng(j,i,i+b){
					uint x=(f[j+b]*p).v;
					f[j+b].v=f[j].v+mod-x;
					f[j].v+=x;
				}
				p*=g[__builtin_ctz(++k)];
			}
		}
		if(b>>=1){//input:[0,mod*4)
			mint p=h[lv++];
			for(int i=0,k=0;i<n;i+=b*2){
				rng(j,i,i+b){
					uint x=(f[j+b]*p).v;
					f[j].v=(f[j].v<mod2?f[j].v:f[j].v-mod2);
					f[j+b].v=f[j].v+mod-x;
					f[j].v+=x;
				}
				p*=g[__builtin_ctz(++k)];
			}
		}
	}
}

template<class mint>
void half_fmt(vector<mint>&f){
	half_fmt(si(f),f.data());
}

#ifdef USE_GOOD_MOD

template<class mint>
vc<mint> multiply(vc<mint> x,const vc<mint>&y,bool same=false){
	if(x.empty()||y.empty())return {};
	int n=si(x)+si(y)-1;
	int s=1;
	while(s<n)s*=2;
	x.resize(s);inplace_fmt(x,false);
	if(!same){
		static vc<mint> z;
		z.clear();z.resize(s);
		rep(i,si(y))z[i]=y[i];
		inplace_fmt(z,false);
		rep(i,s)x[i]*=z[i];
	}else{
		rep(i,s)x[i]*=x[i];
	}
	inplace_fmt(x,true);x.resize(n);
	return x;
}
template<class mint>
vc<mint> multiply_givenlength(vc<mint> x,const vc<mint>&y,bool same=false){
	int s=si(x);
	assert(ispow2(s));
	assert(si(y));
	x.resize(s);inplace_fmt(x,false);
	if(!same){
		static vc<mint> z;
		z.clear();z.resize(s);
		rep(i,si(y))z[i]=y[i];
		inplace_fmt(z,false);
		rep(i,s)x[i]*=z[i];
	}else{
		rep(i,s)x[i]*=x[i];
	}
	inplace_fmt(x,true);
	return x;
}

#else

//59501818244292734739283969-1=5.95*10^25 までの値を正しく計算
//最終的な列の大きさが 2^24 までなら動く
//最終的な列の大きさが 2^20 以下のときは，下の 3 つの素数を使ったほうが速い（は？）
//VERIFY: yosupo
//Yukicoder No980 (same=true)
namespace arbitrary_convolution{
	constexpr modinfo base0{167772161,3};//2^25 * 5 + 1
	constexpr modinfo base1{469762049,3};//2^26 * 7 + 1
	constexpr modinfo base2{754974721,11};//2^24 * 45 + 1
	//extern constexpr modinfo base0{1045430273,3};//2^20 * 997 + 1
	//extern constexpr modinfo base1{1051721729,6};//2^20 * 1003 + 1
	//extern constexpr modinfo base2{1053818881,7};//2^20 * 1005 + 1
	using mint0=modular<base0>;
	using mint1=modular<base1>;
	using mint2=modular<base2>;
	template<class t,class mint>
	vc<t> sub(const vc<mint>&x,const vc<mint>&y,bool same=false){
		int n=si(x)+si(y)-1;
		int s=1;
		while(s<n)s*=2;
		vc<t> z(s);rep(i,si(x))z[i]=x[i].v;
		inplace_fmt(z,false);
		if(!same){
			vc<t> w(s);rep(i,si(y))w[i]=y[i].v;
			inplace_fmt(w,false);
			rep(i,s)z[i]*=w[i];
		}else{
			rep(i,s)z[i]*=z[i];
		}
		inplace_fmt(z,true);z.resize(n);
		return z;
	}
	template<class mint>
	vc<mint> multiply(const vc<mint>&x,const vc<mint>&y,bool same=false){
		auto d0=sub<mint0>(x,y,same);
		auto d1=sub<mint1>(x,y,same);
		auto d2=sub<mint2>(x,y,same);
		int n=si(d0);
		vc<mint> res(n);
		static const mint1 r01=mint1(mint0::mod).inv();
		static const mint2 r02=mint2(mint0::mod).inv();
		static const mint2 r12=mint2(mint1::mod).inv();
		static const mint2 r02r12=r02*r12;
		static const mint w1=mint(mint0::mod);
		static const mint w2=w1*mint(mint1::mod);
		rep(i,n){
			ull a=d0[i].v;
			ull b=(d1[i].v+mint1::mod-a)*r01.v%mint1::mod;
			ull c=((d2[i].v+mint2::mod-a)*r02r12.v+(mint2::mod-b)*r12.v)%mint2::mod;
			res[i].v=(a+b*w1.v+c*w2.v)%mint::mod;
		}
		return res;
	}
	template<class t,class mint>
	vc<t>&sub_givenlength(const vc<mint>&x,const vc<mint>&y,bool same=false){
		int s=si(x);
		assert(ispow2(s));
		assert(si(y)==s);
		static vc<t> z;
		z.clear();z.resize(s);
		rep(i,si(x))z[i]=x[i].v;
		inplace_fmt(z,false);
		if(!same){
			static vc<t> w;
			w.clear();w.resize(s);
			rep(i,si(y))w[i]=y[i].v;
			inplace_fmt(w,false);
			rep(i,s)z[i]*=w[i];
		}else{
			rep(i,s)z[i]*=z[i];
		}
		inplace_fmt(z,true);
		return z;
	}
	template<class mint>
	vc<mint> multiply_givenlength(vc<mint> x,const vc<mint>&y,bool same=false){
		auto&d0=sub_givenlength<mint0>(x,y,same);
		auto&d1=sub_givenlength<mint1>(x,y,same);
		auto&d2=sub_givenlength<mint2>(x,y,same);
		int n=si(d0);
		x.resize(n);
		static const mint1 r01=mint1(mint0::mod).inv();
		static const mint2 r02=mint2(mint0::mod).inv();
		static const mint2 r12=mint2(mint1::mod).inv();
		static const mint2 r02r12=r02*r12;
		static const mint w1=mint(mint0::mod);
		static const mint w2=w1*mint(mint1::mod);
		rep(i,n){
			ull a=d0[i].v;
			ull b=(d1[i].v+mint1::mod-a)*r01.v%mint1::mod;
			ull c=((d2[i].v+mint2::mod-a)*r02r12.v+(mint2::mod-b)*r12.v)%mint2::mod;
			x[i].v=(a+b*w1.v+c*w2.v)%mint::mod;
		}
		return x;
	}
}
using arbitrary_convolution::multiply;
using arbitrary_convolution::multiply_givenlength;

#endif

//UTPC2021 C
namespace integer_convolution{
	extern constexpr modinfo base0{1045430273,3};//2^20 * 997 + 1
	extern constexpr modinfo base1{1051721729,6};//2^20 * 1003 + 1
	//extern constexpr modinfo base0{469762049,3};//2^26 * 7 + 1
	//extern constexpr modinfo base1{754974721,11};//2^24 * 45 + 1
	using mint0=modular<base0>;
	using mint1=modular<base1>;
	template<class t>
	vc<t> sub(const vi&x,const vi&y,bool same=false){
		int n=si(x)+si(y)-1;
		int s=1;
		while(s<n)s*=2;
		vc<t> z(s);rep(i,si(x))z[i]=x[i];
		inplace_fmt(z,false);
		if(!same){
			vc<t> w(s);rep(i,si(y))w[i]=y[i];
			inplace_fmt(w,false);
			rep(i,s)z[i]*=w[i];
		}else{
			rep(i,s)z[i]*=z[i];
		}
		inplace_fmt(z,true);z.resize(n);
		return z;
	}
	vi multiply(const vi&x,const vi&y,bool same=false){
		auto d0=sub<mint0>(x,y,same);
		auto d1=sub<mint1>(x,y,same);
		const mint1 r=mint1(mint0::mod).inv();
		const ll v=ll(mint0::mod)*mint1::mod;
		int n=si(d0);
		vi res(n);
		rep(i,n){
			res[i]=d0[i].v+(r*(d1[i]-d0[i].v)).v*(ull)mint0::mod;
			if(res[i]>v/2)res[i]-=v;
		}
		return res;
	}
}

//最大で 1<<mx のサイズの fft が登場！
template<class mint>
vc<mint> large_convolution(const vc<mint>&a,const vc<mint>&b,int mx){
	int n=si(a),m=si(b);
	vc<mint> c(n+m-1);
	int len=1<<(mx-1);
	for(int i=0;i<n;i+=len){
		for(int j=0;j<n;j+=len){
			int x=min(len,n-i),y=min(len,m-j);
			auto d=multiply(vc<mint>(a.bg+i,a.bg+i+x),vc<mint>(b.bg+j,b.bg+j+y));
			rep(k,si(d))
				c[i+j+k]+=d[k];
		}
	}
	return c;
}

//input A: N 次,B ?,M
//output D: M 次多項式
//C を M 次多項式として
//[x^N] A*B*C = [x^M] D*C
//となるような D を返す
//CF796F
template<class mint>
vc<mint> transpose_advance(const vc<mint>&a,const vc<mint>&b,int m){
	int n=si(a)-1;
	auto d=multiply(a,b);
	vc<mint> res(m+1);
	if(n>=m){
		rep(i,m+1)res[i]=d[i+n-m];
	}else{
		rng(i,m-n,m+1)res[i]=d[i+n-m];
	}
	return res;
}

//Yukicoder 2166
template<class mint>
void chmult(vc<mint>&x,const vc<mint>&y,int s){
	x=multiply(move(x),y);
	x.resize(s);
}

//Poly というのは常にサイズ 1 以上であることにしよう
//low のあたりをかならずサイズ s のものを返すようにいじった
//その影響で何かが起きているかも知れないし，起きていないかも知れない
template<class mint>
struct Poly:public vc<mint>{
	/*Poly(){}
	//Poly(const vc<mint>&val):vc<mint>(val){}
	//Poly(vc<mint>&&val):vc<mint>(val){}
	explicit Poly(int n):vc<mint>(n){}
	explicit Poly(int n,mint v):vc<mint>(n,v){}
	//Poly(initializer_list<mint>init):vc<mint>(all(init)){}
	template<class...Args>
	Poly(Args&&...args):vc<mint>(forward<Args>(args)...){}*/
	Poly(const vc<mint>&val):vc<mint>(val){}
	Poly(vc<mint>&&val):vc<mint>(val){}
	using vc<mint>::vector;
	using vc<mint>::operator=;
	//Poly(const Poly<mint>&rhs):vc<mint>((vc<mint>)rhs){}
	//Poly(Poly<mint>&&rhs):vc<mint>(forward<vc<mint>>(rhs)){}
	int size()const{
		return vc<mint>::size();
	}
	void ups(int s){
		if(size()<s)this->resize(s,0);
	}
	Poly low(int s)const{
		assert(s);
		Poly res(s);
		rep(i,min(s,size()))res[i]=(*this)[i];
		return res;
	}
	Poly rev()const{
		auto r=*this;
		reverse(all(r));
		return r;
	}
	Poly operator>>(int x)const{
		assert(x<size());
		Poly res(size()-x);
		rep(i,size()-x)res[i]=(*this)[i+x];
		return res;
	}
	Poly operator<<(int x)const{
		Poly res(size()+x);
		rep(i,size())res[i+x]=(*this)[i];
		return res;
	}
	mint freq(int i)const{
		return i<size()?(*this)[i]:0;
	}
	Poly operator-()const{
		Poly res=*this;
		for(auto&v:res)v=-v;
		return res;
	}
	Poly& operator+=(const Poly&r){
		ups(r.size());
		rep(i,r.size())
			(*this)[i]+=r[i];
		return *this;
	}
	template<class T>
	Poly& operator+=(T t){
		(*this)[0]+=t;
		return *this;
	}
	Poly& operator-=(const Poly&r){
		ups(r.size());
		rep(i,r.size())
			(*this)[i]-=r[i];
		return *this;
	}
	template<class T>
	Poly& operator-=(T t){
		(*this)[0]-=t;
		return *this;
	}
	template<class T>
	Poly& operator*=(T t){
		for(auto&v:*this)
			v*=t;
		return *this;
	}
	Poly& operator*=(const Poly&r){
		*this=multiply(*this,r);;
		return *this;
	}
	Poly square()const{
		return multiply(*this,*this,true);
	}
	#ifndef USE_GOOD_MOD
	Poly inv(int s)const{
		Poly r{mint(1)/(*this)[0]};
		for(int n=1;n<s;n*=2)
			r=r*2-(r.square()*low(2*n)).low(2*n);
		r.resize(s);
		return r;
	}
	#else
	//source: Section 4 of "Removing redundancy from high-precision Newton iteration"
	// 5/3
	Poly inv(int s)const{
		Poly r(s);
		r[0]=mint(1)/(*this)[0];
		for(int n=1;n<s;n*=2){
			vc<mint> f=low(2*n);
			f.resize(2*n);
			inplace_fmt(f,false);
			vc<mint> g=r.low(2*n);
			g.resize(2*n);
			inplace_fmt(g,false);
			rep(i,2*n)f[i]*=g[i];
			inplace_fmt(f,true);
			rep(i,n)f[i]=0;
			inplace_fmt(f,false);
			rep(i,2*n)f[i]*=g[i];
			inplace_fmt(f,true);
			rng(i,n,min(2*n,s))r[i]=-f[i];
		}
		return r;
	}
	#endif
	template<class T>
	Poly& operator/=(T t){
		return *this*=mint(1)/mint(t);
	}
	Poly quotient(const Poly&r,const Poly&rri)const{
		int m=r.size();
		assert(r[m-1].v);
		int n=size();
		int s=n-m+1;
		if(s<=0) return {0};
		return (rev().low(s)*rri.low(s)).low(s).rev();
	}
	Poly& operator/=(const Poly&r){
		return *this=quotient(r,r.rev().inv(max(size()-r.size(),int(0))+1));
	}
	Poly& operator%=(const Poly&r){
		*this-=*this/r*r;
		return *this=low(r.size()-1);
	}
	Poly operator+(const Poly&r)const{return Poly(*this)+=r;}
	template<class T>
	Poly operator+(T t)const{return Poly(*this)+=t;}
	template<class T>
	friend Poly operator+(T t,Poly r){
		r[0]+=t;
		return r;
	}
	Poly operator-(const Poly&r)const{return Poly(*this)-=r;}
	template<class T>
	Poly operator-(T t)const{return Poly(*this)-=t;}
	template<class T>
	friend Poly operator-(T t,Poly r){
		for(auto&v:r)v=-v;
		r[0]+=t;
		return r;
	}
	template<class T>
	Poly operator*(T t)const{return Poly(*this)*=t;}
	Poly operator*(const Poly&r)const{return Poly(*this)*=r;}
	template<class T>
	Poly operator/(T t)const{return Poly(*this)/=t;}
	Poly operator/(const Poly&r)const{return Poly(*this)/=r;}
	Poly operator%(const Poly&r)const{return Poly(*this)%=r;}
	Poly dif()const{
		assert(size());
		if(size()==1){
			return {0};
		}else{
			Poly r(size()-1);
			rep(i,r.size())
				r[i]=(*this)[i+1]*(i+1);
			return r;
		}
	}
	Poly inte(const mint invs[])const{
		Poly r(size()+1,0);
		rep(i,size())
			r[i+1]=(*this)[i]*invs[i+1];
		return r;
	}
	//VERIFY: yosupo
	//opencupXIII GP of Peterhof H
	Poly log(int s,const mint invs[])const{
		assert((*this)[0]==1);
		if(s==1)return {0};
		return (low(s).dif()*inv(s-1)).low(s-1).inte(invs);
	}
	//Petrozavodsk 2019w mintay1 G
	//yosupo judge
	//UOJ Round23 C
	Poly exp(int s,const mint invs[])const{
		assert((*this)[0]==mint(0));
		Poly f{1},g{1};
		for(int n=1;;n*=2){
			if(n>=s)break;
			g=g*2-(g.square()*f).low(n);
			//if(n>=s)break;
			Poly q=low(n).dif();
			q=q+g*(f.dif()-f*q).low(2*n-1);
			f=f+(f*(low(2*n)-q.inte(invs))).low(2*n);
		}
		return f.low(s);
	}
	//exp(x),exp(-x) のペアを返す
	//UOJ Round23 C
	pair<Poly,Poly> exp2(int s,const mint invs[])const{
		assert((*this)[0]==mint(0));
		Poly f{1},g{1};
		for(int n=1;;n*=2){
			//if(n>=s)break;
			g=g*2-(g.square()*f).low(n);
			if(n>=s)break;
			Poly q=low(n).dif();
			q=q+g*(f.dif()-f*q).low(2*n-1);
			f=f+(f*(low(2*n)-q.inte(invs))).low(2*n);
		}
		return make_pair(f.low(s),g.low(s));
	}
	#ifndef USE_GOOD_MOD
	//CF250 E
	Poly sqrt(int s)const{
		assert((*this)[0]==1);
		static const mint half=mint(1)/mint(2);
		Poly r{1};
		for(int n=1;n<s;n*=2)
			r=(r+(r.inv(n*2)*low(n*2)).low(n*2))*half;
		return r.low(s);
	}
	#else
	//11/6
	//VERIFY: yosupo
	Poly sqrt(int s)const{
		assert((*this)[0]==1);
		static const mint half=mint(1)/mint(2);
		vc<mint> f{1},g{1},z{1};
		for(int n=1;n<s;n*=2){
			rep(i,n)z[i]*=z[i];
			inplace_fmt(z,true);
			
			vc<mint> delta(2*n);
			rep(i,n)delta[n+i]=z[i]-freq(i)-freq(n+i);
			inplace_fmt(delta,false);
			
			vc<mint> gbuf(2*n);
			rep(i,n)gbuf[i]=g[i];
			inplace_fmt(gbuf,false);
			
			rep(i,2*n)delta[i]*=gbuf[i];
			inplace_fmt(delta,true);
			f.resize(2*n);
			rng(i,n,2*n)f[i]=-half*delta[i];
			
			if(2*n>=s)break;
			
			z=f;
			inplace_fmt(z,false);
			
			vc<mint> eps=gbuf;
			rep(i,2*n)eps[i]*=z[i];
			inplace_fmt(eps,true);
			
			rep(i,n)eps[i]=0;
			inplace_fmt(eps,false);
			
			rep(i,2*n)eps[i]*=gbuf[i];
			inplace_fmt(eps,true);
			g.resize(2*n);
			rng(i,n,2*n)g[i]=-eps[i];
		}
		f.resize(s);
		return f;
	}
	#endif
	pair<Poly,Poly> divide(const Poly&r,const Poly&rri)const{
		Poly a=quotient(r,rri);
		Poly b=*this-a*r;
		return make_pair(a,b.low(r.size()-1));
	}
	//Yukicoder No.215
	Poly pow_mod(int n,const Poly&r)const{
		Poly rri=r.rev().inv(r.size());
		Poly cur{1},x=*this%r;
		while(n){
			if(n%2)
				cur=(cur*x).divide(r,rri).b;
			x=(x*x).divide(r,rri).b;
			n/=2;
		}
		return cur;
	}
	int lowzero()const{
		rep(i,size())if((*this)[i]!=0)return i;
		return size();
	}
	//VERIFY: yosupo
	//UOJ Round23 C (z=0,p<0)
	//Multiuni2023-4-B
	Poly pow(int s,int p,const mint invs[])const{
		assert(s>0);
		if(p==0){
			Poly res(s,0);
			res[0]=1;
			return res;
		}
		int n=size(),z=0;
		for(;z<n&&(*this)[z]==0;z++);
		assert(z==0||p>0);
		if(z*p>=s)return Poly(s,0);
		mint c=(*this)[z],cinv=c.inv();
		mint d=c.pow(p);
		int t=s-z*p;
		Poly x(t);
		rng(i,z,min(z+t,n))x[i-z]=(*this)[i]*cinv;
		x=x.log(t,invs);
		rep(i,t)x[i]*=p;
		x=x.exp(t,invs);
		rep(i,t)x[i]*=d;
		Poly y(s);
		rep(i,t)y[z*p+i]=x[i];
		return y;
	}
	//verified in compositional inverse
	Poly pow_const1(mint p,const mint invs[])const{
		assert((*this)[0]==1);
		Poly x=log(size(),invs);
		rep(i,size())x[i]*=p;
		return x.exp(size(),invs);
	}
	mint eval(mint x)const{
		mint r=0,w=1;
		for(auto v:*this){
			r+=w*v;
			w*=x;
		}
		return r;
	}
};
template<class T>
void print_single(const Poly<T>&v,int suc=1){
	rep(i,v.size())
		print_single(v[i],i==int(v.size())-1?3:2);
	if(suc==1){
		if(dbg)cout<<endl;
		else cout<<"\n";
	}
	if(suc==2)
		cout<<" ";
}

//CF641 F2
//f*x^(-a)
template<class mint>
struct Laurent{
	Poly<mint> f;
	int a;
	Laurent(const Poly<mint>&num,const Poly<mint>&den,int s){
		a=den.lowzero();
		assert(a<si(den));
		f=(num*(den>>a).inv(s)).low(s);
	}
	Laurent(const Poly<mint>&ff,int aa):f(ff),a(aa){}
	Laurent dif()const{
		return Laurent(f*(-a)+(f.dif()<<1),a+1);
	}
	mint&operator[](int i){
		assert(inc(0,i+a,si(f)-1));
		return f[i+a];
	}
};

template<class mint>
ll m2l(mint a){
	return a.v<mint::mod/2?a.v:ll(a.v)-ll(mint::mod);
}

template<class mint>
void showpoly(const Poly<mint>&a){
	vi tmp(si(a));
	rep(i,si(a)){
		tmp[i]=m2l(a[i]);
	}
	cerr<<tmp<<endl;
}

//source: Tellegen’s Principle into Practice

//Yukicoder No.1145
//top には，(x-a[i]) の積が入っている(a がサイズ s に拡張されていることに注意)
//なので例えば (1-a[i]x) の積が欲しい場合は，
//	Poly<mint> f=s.top;
//	Poly<mint> g(n+1);
//	rep(i,n+1)g[i]=f[si(f)-1-i];
template<class mint>
struct subproduct_tree{
	using poly=Poly<mint>;
	int raws,s,h;
	mint*buf;
	vc<mint*>f;
	vi len;
	poly top;
	void inner_product(const int n,const mint*a,const mint*b,mint*c){
		rep(i,n)c[i]=a[i]*b[i];
	}
	//first n elements are fft-ed
	//last n elements are raw data mod x^n-1
	//the coefficient of x^n is v
	//convert the whole array into size-2n fft-ed array
	void doubling_fmt(const int n,mint*a,const mint v){
		a[n]-=v*2;
		half_fmt(n,a+n);
	}
	subproduct_tree(const vc<mint>&a){
		raws=si(a);
		s=1;while(s<si(a))s*=2;
		h=__lg(s)+1;
		buf=new mint[s*h*2];
		f.resize(s*2);
		len.resize(s*2);
		{
			mint*head=buf;
			rng(i,1,s*2){
				len[i]=s>>__lg(i);
				f[i]=head;
				head+=len[i]*2;
			}
		}
		rep(i,s){
			mint w=i<si(a)?a[i]:0;
			f[s+i][0]=-w+1;
			f[s+i][1]=-w-1;
		}
		if(s==1)f[1][1]=f[1][0];
		gnr(i,1,s){
			inner_product(len[i],f[i*2],f[i*2+1],f[i]);
			copy(f[i],f[i]+len[i],f[i]+len[i]);
			inplace_fmt(len[i],f[i]+len[i],true);
			if(i>1)doubling_fmt(len[i],f[i],1);
		}
		top.resize(s+1);
		rep(i,s)top[i]=f[1][s+i];
		top[0]-=1;
		top[s]=1;
	}
	~subproduct_tree(){
		delete[] buf;
	}
	//VERIFY: yosupo
	vc<mint> multieval(const poly&b){
		mint*buf2=new mint[s*2];
		vc<mint*> c(s*2);
		{
			mint*head=buf2;
			rng(i,1,s*2){
				if((i&(i-1))==0&&__lg(i)%2==0)head=buf2;
				c[i]=head;
				head+=len[i];
			}
		}
		{
			poly tmp=top.rev().inv(si(b)).rev()*b;
			rep(i,s)c[1][i]=i<si(b)?tmp[si(b)-1+i]:0;
		}
		vc<mint> tmp(s);
		rng(i,1,s){
			inplace_fmt(len[i],c[i],false);
			rep(k,2){
				tmp.resize(len[i]);
				rep(j,len[i])tmp[j]=f[i*2+(k^1)][j]*c[i][j];
				inplace_fmt(tmp,true);
				rep(j,len[i]/2)c[i*2+k][j]=tmp[len[i]/2+j];
			}
		}
		vc<mint> ans(raws);
		rep(i,raws)ans[i]=c[s+i][0];
		delete[] buf2;
		return ans;
	}
	poly interpolate(const vc<mint>&val){
		mint*buf2=new mint[s*2*2];
		vc<mint*> c(s*2);
		{
			mint*head=buf2;
			rng(i,1,s*2){
				if((i&(i-1))==0&&__lg(i)%2==0)head=buf2;
				c[i]=head;
				head+=len[i]*2;
			}
		}
		{
			vc<mint> z=multieval(poly(top.bg+(s-si(val)),top.ed).dif());
			rep(i,s){
				mint w=i<si(val)?val[i]/z[i]:0;
				c[s+i][0]=c[s+i][1]=w;
			}
		}
		gnr(i,1,s){
			rep(j,len[i])
				c[i][j]=c[i*2][j]*f[i*2+1][j]+c[i*2+1][j]*f[i*2][j];
			copy(c[i],c[i]+len[i],c[i]+len[i]);
			inplace_fmt(len[i],c[i]+len[i],true);
			if(i>1)doubling_fmt(len[i],c[i],0);
		}
		poly res(c[1]+s+(s-si(val)),c[1]+s*2);
		delete[] buf2;
		return res;
	}
	//res[i]=prod_{j!=i} (a[i]-a[j])
	//Yukicoder 2556
	vc<mint> product_of_difference(){
		return multieval(poly(top.bg+(s-raws),top.ed).dif());
	}
};

template<class mint>
vc<mint> multieval(const Poly<mint>&f,const vc<mint>&x){
	return subproduct_tree<mint>(x).multieval(f);
}

template<class mint>
Poly<mint> interpolate(const vc<mint>&x,const vc<mint>&y){
	assert(si(x)==si(y));
	if(si(x)==1)return {y[0]};
	subproduct_tree<mint> tree(x);
	return tree.interpolate(y);
}

//a==-1 だけ verify されてる
//LOJ 6703
//product of ax+b
//ARC155F
template<class mint>
Poly<mint> product_linear(const vc<pair<mint,mint>>&rw){
	mint w=1;
	vc<mint> buf;
	for(auto ab:rw){
		if(ab.a==0){
			w*=ab.b;
		}else{
			w*=ab.a;
			buf.pb(-ab.b/ab.a);
		}
	}
	subproduct_tree<mint> tree(buf);
	const auto&f=tree.top;
	int n=si(buf)+1;
	vc<mint> res(n);
	rep(i,n)res[i]=f[si(f)-n+i]*w;
	res.resize(si(rw)+1);
	return res;
}

#ifndef DYNAMIC_MOD
extern constexpr modinfo base{998244353,3};
//extern constexpr modinfo base{1000000007,0};
//extern constexpr modinfo base{2147483579,1689685080};//2^31 未満の最大の安全素数
//modinfo base{1,0};
#ifdef USE_GOOD_MOD
static_assert(base.mod==998244353);
#endif
#else
modinfo base(1,0);
extern constexpr modinfo base107(1000000007,0);
using mint107=modular<base107>;
#endif
using mint=modular<base>;

mint parity(int i){
	return i%2==0?1:-1;
}

#ifdef LOCAL
const int vmax=10010;
#else
const int vmax=(1<<21)+10;
#endif
mint fact[vmax],finv[vmax],invs[vmax];
void initfact(){
	const int s=min<int>(vmax,base.mod);
	fact[0]=1;
	rng(i,1,s){
		fact[i]=fact[i-1]*i;
	}
	finv[s-1]=fact[s-1].inv();
	for(int i=s-2;i>=0;i--){
		finv[i]=finv[i+1]*(i+1);
	}
	for(int i=s-1;i>=1;i--){
		invs[i]=finv[i]*fact[i-1];
	}
}
mint choose(int n,int k){
	return inc(0,k,n)?fact[n]*finv[n-k]*finv[k]:0;
}
mint binom(int a,int b){
	return 0<=a&&0<=b?fact[a+b]*finv[a]*finv[b]:0;
}
mint catalan(int n){
	return binom(n,n)-(n-1>=0?binom(n-1,n+1):0);
}
//対角線を超えず (x,y) に至る方法の数
mint catalan(int x,int y){
	assert(y<=x);
	return binom(x,y)-binom(x+1,y-1);
}
//y=x+c を超えず (x,y) に至る方法の数
mint catalan(int x,int y,int c){
	assert(y<=x+c);
	return binom(x,y)-binom(x+c+1,y-c-1);
}

/*
const int vmax=610;
mint fact[vmax+1],binbuf[vmax+1][vmax+1];
mint choose(int n,int k){
	return 0<=k&&k<=n?binbuf[n-k][k]:0;
}
mint binom(int a,int b){
	return 0<=a&&0<=b?binbuf[a][b]:0;
}
void initfact(int n){
	fact[0]=1;
	rep(i,n)fact[i+1]=fact[i]*(i+1);
	rep(i,n+1)rep(j,n+1){
		if(i==0&&j==0){
			binbuf[i][j]=1;
		}else{
			binbuf[i][j]=0;
			if(i)binbuf[i][j]+=binbuf[i-1][j];
			if(j)binbuf[i][j]+=binbuf[i][j-1];
		}
	}
}
*/

mint p2[vmax],p2inv[vmax];
void initp2(){
	p2[0]=1;
	rep(i,vmax-1)p2[i+1]=p2[i]*2;
	p2inv[vmax-1]=p2[vmax-1].inv();
	per(i,vmax-1)p2inv[i]=p2inv[i+1]*2;
}

int m2i(mint a){
	uint v=a.v;
	return v<mint::mod/2?v:int(v)-int(mint::mod);
}

pi m2f(mint x){
	pi res(inf,inf);
	auto upd=[&](int a,int b){
		if(abs(res.a)+res.b>abs(a)+b){
			res=pi(a,b);
		}
	};
	rng(b,1,1000){
		mint y=x*b;
		upd(y.v,b);
		upd(-int((-y).v),b);
	}
	return res;
}

//FFT の結果をキャッシュして持っておく
//rw に生データ，buf に fft 後のデータ
//どちらかが空でもよい
//si(buf) は 0 or si(rw) 以上の 2 冪
//Codechef 2021 January Lunchtime EXPGROUP
//yosupo product of polynomial sequence
//Yukicoder 2166
//メモリ食いまくり
//ABC278H
//Osijek Day6 A
//Yukicoder 2575 (a<b<c の畳込みがほしいときはここへ)
struct F{
	int n;
	vc<mint> rw,buf;
	F():n(0){}
	F(const vc<mint>&given):rw(given){
		n=si(rw);
		assert(n>0);
	}
	F(initializer_list<mint> init):rw(all(init)){
		n=si(rw);
		assert(n>0);
	}
	int size()const{return n;}
	bool empty()const{return n==0;}
	void assume_have(){
		if(rw.empty()){
			int s=minp2(n);
			assert(si(buf)>=s);
			rw.resize(s);
			rep(i,s)rw[i]=buf[i].v;
			inplace_fmt(rw,true);
			rw.resize(n);
		}
		assert(si(rw)==n);
	}
	vc<mint> getrw(){
		if(n==0)return {};
		assume_have();
		return rw;
	}
	void prepare(int len){
		if(si(buf)<len)assume_have();
		if(buf.empty()){
			int s=minp2(n);
			buf.resize(s);
			rep(i,n)buf[i]=rw[i];
			inplace_fmt(buf,false);
		}
		while(si(buf)<len){
			int s=si(buf);
			buf.resize(s*2);
			rep(i,n)buf[s+i]=rw[i];
			half_fmt(s,buf.data()+s);
		}
	}
	void copy_from(F&a){
		n=a.n;
		rw=a.rw;
		if(si(a.buf)){
			int s=minp2(n);
			buf.resize(s);
			rep(i,s)buf[i]=a.buf[i];
		}else buf.clear();
	}
	void init_from_sum(F&a,F&b){
		if(a.empty())return copy_from(b);
		if(b.empty())return copy_from(a);
		n=max(a.n,b.n);
		if(si(a.rw)&&si(b.rw)){
			rw.resize(n);
			rep(i,n){
				rw[i]=0;
				if(i<a.n)rw[i]+=a.rw[i];
				if(i<b.n)rw[i]+=b.rw[i];
			}
		}else rw.clear();
		int s=minp2(n);
		if(si(a.buf)>=s&&si(b.buf)>=s){
			buf.resize(s);
			rep(i,s)buf[i]=mint(a.buf[i].v)+mint(b.buf[i].v);
		}else buf.clear();
		if(rw.empty()&&buf.empty()){
			/*
			a.prepare(n);
			b.prepare(n);
			buf.resize(s);
			rep(i,s)buf[i]=mint(a.buf[i].v)+mint(b.buf[i].v);
			*/
			//こっちのほうが微小にメモリが少なくなる?
			a.assume_have();
			b.assume_have();
			rw.assign(n,0);
			rep(i,a.n)rw[i]+=a.rw[i];
			rep(i,b.n)rw[i]+=b.rw[i];
		}
	}
	//Osijek Day6 A (n=0,middle=true)
	void init_from_product(F&a,F&b,bool middle=false){
		if(a.n==0||b.n==0){
			n=0;
			rw.clear();
			buf.clear();
			return;
		}
		assert(a.n>0);
		assert(b.n>0);
		if(middle){
			assert(a.n>=b.n);
			n=a.n;
		}else{
			n=a.n+b.n-1;
		}
		rw.clear();
		int s=minp2(n);
		a.prepare(n);
		b.prepare(n);
		buf.resize(s);
		rep(i,s)buf[i]=a.buf[i]*b.buf[i];
	}
	F operator*(F&b){
		F res;
		res.init_from_product(*this,b);
		return res;
	}
	F operator*(F&&b){
		F res;
		res.init_from_product(*this,b);
		return res;
	}
	F operator+(F&b){
		F res;
		res.init_from_sum(*this,b);
		return res;
	}
	F operator+(F&&b){
		F res;
		res.init_from_sum(*this,b);
		return res;
	}
	F& operator*=(F&b){
		return *this=(*this)*b;
	}
	F& operator+=(F&b){
		return *this=(*this)+b;
	}
	F& operator+=(F&&b){
		return *this=(*this)+b;
	}
	void freememory(){
		n=0;
		vc<mint>().swap(rw);
		vc<mint>().swap(buf);
	}
	//middle product の情報をつくる
	//a,b,c が x の関数だとする
	//ans+=[x^deg(a)] abc だとしよう
	//ab の係数上位 deg(c)+1 項を d とおけば
	//ans+=[x^deg(d)] dc とできるので，この d を求めよう
	//単にサイズを広げない prod をするだけ
	//deg(a)>=deg(b)+deg(c) でないと破綻する
	F middle(F&b){
		F res;
		res.init_from_product(*this,b,true);
		return res;
	}
	//middle product の情報が buf に入っているとする
	//サイズ s に縮める
	//[n-s,n) が重要で，それ以外のパートにはゴミが入っているかもしれない
	//Osijek Day6 A
	void middle_shrink(int s){
		if(empty())return;
		assert(s<=n);
		if(s<n){
			assume_have();
			rep(i,s)rw[i]=rw[n-s+i];
			rw.resize(n=s);
			buf.clear();
		}
	}
	//転置原理+分割統治と組み合わせるといい感じ
	//ありがちなパターン
	//for i=0,1,...: ans[i]=[x^(n-1)]v; v=m_i*v
	//m_i が（高々）一次なら，dfs(l,r) では v[k].middle_shrink(r-l)
	//m_i の要素の次数はバラバラでも大丈夫
	//空の prod で空が返るようにしてあるので m_i に空を入れてもいい
	//行列とかで戻していくとき，middle 側は空or全部同じ次数という形にしておく
};



struct Query{
	int kind;
	mint a,b;
};
//kind 0: ax+b
//kind 1: eval at a
vc<mint> prefix_multieval(vc<Query> qs){
	/*vc<mint> res;
	rep(i,si(qs)){
		if(qs[i].kind==1){
			mint v=1;
			rep(j,i){
				if(qs[j].kind==0){
					v*=qs[j].a*qs[i].a+qs[j].b;
				}
			}
			res.pb(v);
		}
	}
	return res;*/
	
	int s=si(qs);
	int d=0;
	for(auto [kind,a,b]:qs)if(kind==0)d++;
	assert(d<s);
	
	using P=pair<F,F>;
	vc<P> buf(2*s);
	auto getid=[&](int l,int r){
		if(r-l==1)return l+r;
		else return (l+r)/2*2;
	};
	{
		auto dfs=[&](auto self,int l,int r)->void{
			auto&[a,b]=buf[getid(l,r)];
			if(r-l==1){
				if(qs[l].kind==1){
					a={1,-qs[l].a};
					b={1,-qs[l].a};
				}else{
					a={qs[l].a,qs[l].b};
					b={0,1};
				}
			}else{
				int m=(l+r)/2;
				self(self,l,m);
				self(self,m,r);
				auto&[al,bl]=buf[getid(l,m)];
				auto&[ar,br]=buf[getid(m,r)];
				a=al*ar;
				b=bl*br;
				
				ar.freememory();
				bl.freememory();
			}
		};
		dfs(dfs,0,s);
	}
	F ini;
	{
		vc<pair<mint,mint>> dens;
		for(auto [kind,a,b]:qs)if(kind==1)dens.eb(-a,1);
		Poly<mint> den=product_linear(dens);
		den=den.inv(d+1);
		assert(d+1<=s);
		den.resize(s);
		rotate(den.bg,den.bg+d+1,den.ed);
		ini=den;
	}
	vc<mint> ans;
	{
		auto dfs=[&](auto self,int l,int r,F&z)->void{
			z.middle_shrink(r-l);
			{
				auto&[a,b]=buf[getid(l,r)];
				a.freememory();
				b.freememory();
			}
			if(r-l==1){
				mint val=z.getrw()[0];
				if(qs[l].kind==1)ans.pb(val);
				z.freememory();
			}else{
				int m=(l+r)/2;
				auto&[al,bl]=buf[getid(l,m)];
				auto&[ar,br]=buf[getid(m,r)];
				
				F zl=z.middle(br),zr=z.middle(al);
				z.freememory();
				self(self,l,m,zl);
				self(self,m,r,zr);
			}
		};
		dfs(dfs,0,s,ini);
	}
	return ans;
}

mint sub(int n,vc<pi> lr,int sh){
	if(n==0)return 0;
	vi fix(n+1);
	for(auto [l,r]:lr){
		if(l!=-1&&r!=-1){
			fix[l]++;
			fix[r]--;
		}
	}
	rep(i,n)fix[i+1]+=fix[i];
	int bothfree=0;
	for(auto [l,r]:lr){
		if(l==-1&&r==-1){
			bothfree++;
		}
	}
	
	vc<mint> rcnt(n+1,1),lcnt(n+1,1);
	for(auto [l,r]:lr){
		if(l==-1&&r!=-1){
			rcnt[r]*=(r+sh);
		}
		if(l!=-1&&r==-1){
			lcnt[l]*=(n-l+sh);
		}
	}
	rep(i,n)rcnt[i+1]*=rcnt[i];
	per(i,n)lcnt[i]*=lcnt[i+1];
	
	vc<mint> res(n);
	rep(i,n){
		res[i]=mint(2).pow(fix[i])*mint((n+sh+1)*(n+sh)/2+(i+1)*(n-i)).pow(bothfree);
		res[i]*=rcnt[i];
		res[i]*=lcnt[i+1];
	}
	
	{//about left
		vvc<pair<mint,mint>> buf(n+1);
		for(auto [l,r]:lr){
			if(l!=-1&&r==-1){
				buf[l].eb(-1,n+sh-l+n);
			}
		}
		vc<Query> qs;
		rep(i,n){
			for(auto [a,b]:buf[i]){
				qs.pb({0,a,b});
			}
			qs.pb({1,i,0});
		}
		auto z=prefix_multieval(qs);
		
		rep(i,n)res[i]*=z[i];
	}

	{//about right
		vvc<pair<mint,mint>> buf(n+1);
		for(auto [l,r]:lr){
			if(l==-1&&r!=-1){
				buf[r].eb(1,r+sh+1);
			}
		}
		vc<Query> qs;
		per(i,n){
			for(auto [a,b]:buf[i+1]){
				qs.pb({0,a,b});
			}
			qs.pb({1,i,0});
		}
		auto z=prefix_multieval(qs);
		rein(z);
		rep(i,n)res[i]*=z[i];
	}
	
	return SUM(res);
}

void slv(){
	INT(n,m);
	VPI(lr,n);
	mint tot=1;
	for(auto [l,r]:lr){
		if(l==-1){
			if(r==-1){
				tot*=m*(m+1)/2;
			}else{
				tot*=r;
			}
		}else{
			if(r==-1){
				tot*=m+1-l;
			}else{
				tot*=1;
			}
		}
	}
	
	for(auto&[l,r]:lr){
		if(l!=-1)l--;
	}
	mint ans=sub(m,lr,0);
	m--;
	for(auto&[l,r]:lr){
		if(r!=-1)r--;
	}
	ans-=sub(m,lr,1);
	
	ans-=tot;
	
	print(ans);
}

signed main(signed argc,char*argv[]){
	if(argc>1&&strcmp(argv[1],"D")==0)dbg=true;
	
	cin.tie(0);
	ios::sync_with_stdio(0);
	cout<<fixed<<setprecision(20);
	
	if(dbg){
		while(1){
			if(current_run_id%run_batch_size==0){
				cerr<<"Current Run "<<current_run_id<<endl;
			}
			slv();
			current_run_id++;
		}
	}else{
		//int t;cin>>t;rep(_,t)
		slv();
	}
}