#include <bits/stdc++.h>
#define rep(i,n) for(int i=0;i<(int)(n);i++)
#define rep1(i,n) for(int i=1;i<=(int)(n);i++)
#define all(c) c.begin(),c.end()
#define pb push_back
#define fs first
#define sc second
#define chmin(x,y) x=min(x,y)
#define chmax(x,y) x=max(x,y)
using namespace std;
template<class S,class T> ostream& operator<<(ostream& o,const pair<S,T> &p){
	return o<<"("<<p.fs<<","<<p.sc<<")";
}
template<class T> ostream& operator<<(ostream& o,const vector<T> &vc){
	o<<"{";
	for(const T& v:vc) o<<v<<",";
	o<<"}";
	return o;
}
using ll = long long;
template<class T> using V = vector<T>;
template<class T> using VV = vector<vector<T>>;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n-1); }

#ifdef LOCAL
#define show(x) cerr << "LINE" << __LINE__ << " : " << #x << " = " << (x) << endl
#define dump(x) cerr << "LINE" << __LINE__ << " : " << #x << " = {";  \
	for(auto v: x) cerr << v << ","; cerr << "}" << endl;
#else
#define show(x) true
#define dump(x) true
#endif

template<unsigned int mod_>
struct ModInt{
	using uint = unsigned int;
	using ll = long long;
	using ull = unsigned long long;

	constexpr static uint mod = mod_;

	uint v;
	ModInt():v(0){}
	ModInt(ll _v):v(normS(_v%mod+mod)){}
	explicit operator bool() const {return v!=0;}
	static uint normS(const uint &x){return (x<mod)?x:x-mod;}		// [0 , 2*mod-1] -> [0 , mod-1]
	static ModInt make(const uint &x){ModInt m; m.v=x; return m;}
	ModInt operator+(const ModInt& b) const { return make(normS(v+b.v));}
	ModInt operator-(const ModInt& b) const { return make(normS(v+mod-b.v));}
	ModInt operator-() const { return make(normS(mod-v)); }
	ModInt operator*(const ModInt& b) const { return make((ull)v*b.v%mod);}
	ModInt operator/(const ModInt& b) const { return *this*b.inv();}
	ModInt& operator+=(const ModInt& b){ return *this=*this+b;}
	ModInt& operator-=(const ModInt& b){ return *this=*this-b;}
	ModInt& operator*=(const ModInt& b){ return *this=*this*b;}
	ModInt& operator/=(const ModInt& b){ return *this=*this/b;}
	ModInt& operator++(int){ return *this=*this+1;}
	ModInt& operator--(int){ return *this=*this-1;}
	ll extgcd(ll a,ll b,ll &x,ll &y) const{
		ll p[]={a,1,0},q[]={b,0,1};
		while(*q){
			ll t=*p/ *q;
			rep(i,3) swap(p[i]-=t*q[i],q[i]);
		}
		if(p[0]<0) rep(i,3) p[i]=-p[i];
		x=p[1],y=p[2];
		return p[0];
	}
	ModInt inv() const {
		ll x,y;
		extgcd(v,mod,x,y);
		return make(normS(x+mod));
	}
	ModInt pow(ll p) const {
		if(p<0) return inv().pow(-p);
		ModInt a = 1;
		ModInt x = *this;
		while(p){
			if(p&1) a *= x;
			x *= x;
			p >>= 1;
		}
		return a;
	}
	bool operator==(const ModInt& b) const { return v==b.v;}
	bool operator!=(const ModInt& b) const { return v!=b.v;}
	friend istream& operator>>(istream &o,ModInt& x){
		ll tmp;
		o>>tmp;
		x=ModInt(tmp);
		return o;
	}
	friend ostream& operator<<(ostream &o,const ModInt& x){ return o<<x.v;}
};
using mint = ModInt<998244353>;
struct UnionFind{
	vector<int> par,rank;
	UnionFind(int N){
		par.assign(N,0);
		rank.assign(N,0);
		rep(i,N) par[i]=i;
	}
	int find(int x){
		if(par[x]==x) return x;
		return par[x]=find(par[x]);
	}
	bool same(int x,int y){
		return find(x)==find(y);
	}
	void unite(int x,int y){
		x=find(x),y=find(y);
		if(x==y) return;
		if(rank[x]<rank[y]) swap(x,y);

		//x becomes root
		par[y]=x;
		if(rank[x]==rank[y]) rank[x]++;
	}
};
int main(){
	cin.tie(0);
	ios::sync_with_stdio(false);		//DON'T USE scanf/printf/puts !!
	cout << fixed << setprecision(20);

	int N; cin >> N; N--;
	V<string> s(N); rep(i,N) cin >> s[i];
	VV<int> is(N,V<int>(N));
	rep(i,N) rep(j,N){
		int x=i,y=j;
		if(x>y) swap(x,y);
		if(x+y>=N){
			swap(x,y);x=N-1-x,y=N-1-y;
		}
		// cout<<x<<y<<" ";
		// if(j == N-1) cout << endl;
		if(s[i][j] == 'o') is[x][y] |= 2;
		if(s[i][j] == 'x') is[x][y] |= 1;
	}
	rep(i,N) rep(j,N) if(is[i][j] == 3){
		cout << 0 << endl;
		return 0;
	}
	int RANK = 0;
	rep(r,2){
		show(r);show("======================");
		int n = (r == 0) ? (N+3)/2 : (N+2)/2;
		show(n);
		V<int> deg0(n),deg1(n);
		UnionFind UF(n);
		int M = 0;
		rep(i,N) rep(j,N) if(i<=j && i+j<=N-1 && (i+j)%2 == r){
			int u = (j-i)/2;
			int v = (j+i)/2+1;
			show(u);show(v);
			assert(0<=u&&u<n);
			assert(0<=v&&v<n);
			show(is[i][j]);
			show("-");
			if(is[i][j] == 0){
				UF.unite(u,v);
				M++;
			}else if(is[i][j] == 1){
				deg0[u]++,deg0[v]++;
			}else{
				deg1[u]++,deg1[v]++;
			}
		}
		RANK += M;
		if(n&1){
			RANK -= n;
			map<int,V<int>> cons;
			rep(v,n) cons[UF.find(v)].pb(v);
			for(auto con: cons){
				auto vs = con.sc;
				int odd = 0;
				for(int v: vs) odd += deg1[v];
				if(odd&1){
					cout << 0 << endl;
					return 0;
				}
				RANK++;
			}
		}
		show(RANK);
	}
	cout << mint(2).pow(RANK) << endl;
}

### Prompt

Please provide a Cpp coded solution to the problem described below:

In a two-dimensional plane, there is a square frame whose vertices are at coordinates (0,0), (N,0), (0,N), and (N,N). The frame is made of mirror glass. A ray of light striking an edge of the frame (but not a vertex) will be reflected so that the angle of incidence is equal to the angle of reflection. A ray of light striking a vertex of the frame will be reflected in the direction opposite to the direction it is coming from.

We will define the path for a grid point (a point with integer coordinates) (i,j) (0<i,j<N) strictly within the frame, as follows:

* The path for (i,j) is the union of the trajectories of four rays of light emitted from (i,j) to (i-1,j-1), (i-1,j+1), (i+1,j-1), and (i+1,j+1).



<image>

Figure: an example of a path for a grid point

There is a light bulb at each grid point strictly within the frame. We will assign a state - ON or OFF - to each bulb. The state of the whole set of bulbs are called beautiful if it is possible to turn OFF all the bulbs by repeating the following operation:

* Choose a grid point strictly within the frame, and switch the states of all the bulbs on its path.



Takahashi has set the states of some of the bulbs, but not for the remaining bulbs. Find the number of ways to set the states of the remaining bulbs so that the state of the whole set of bulbs is beautiful, modulo 998244353. The state of the bulb at the grid point (i,j) is set to be ON if A_{i,j}=`o`, OFF if A_{i,j}=`x`, and unset if A_{i,j}=`?`.

Constraints

* 2 \leq N \leq 1500
* A_{ij} is `o`, `x`, or `?`.

Input

Input is given from Standard Input in the following format:


N
A_{1,1}...A_{1,N-1}
:
A_{N-1,1}...A_{N-1,N-1}


Output

Print the answer.

Examples

Input

4
o?o
???
?x?


Output

1


Input

5
o?o?
????
o?x?
????


Output

0


Input

6
?o???
????o
??x??
o????
???o?


Output

32


Input

9
????o??x
?????x??
??o?o???
?o?x????
???????x
x?o?o???
????????
x?????x?


Output

4

### Response

```cpp
#include <bits/stdc++.h>
#define rep(i,n) for(int i=0;i<(int)(n);i++)
#define rep1(i,n) for(int i=1;i<=(int)(n);i++)
#define all(c) c.begin(),c.end()
#define pb push_back
#define fs first
#define sc second
#define chmin(x,y) x=min(x,y)
#define chmax(x,y) x=max(x,y)
using namespace std;
template<class S,class T> ostream& operator<<(ostream& o,const pair<S,T> &p){
	return o<<"("<<p.fs<<","<<p.sc<<")";
}
template<class T> ostream& operator<<(ostream& o,const vector<T> &vc){
	o<<"{";
	for(const T& v:vc) o<<v<<",";
	o<<"}";
	return o;
}
using ll = long long;
template<class T> using V = vector<T>;
template<class T> using VV = vector<vector<T>>;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n-1); }

#ifdef LOCAL
#define show(x) cerr << "LINE" << __LINE__ << " : " << #x << " = " << (x) << endl
#define dump(x) cerr << "LINE" << __LINE__ << " : " << #x << " = {";  \
	for(auto v: x) cerr << v << ","; cerr << "}" << endl;
#else
#define show(x) true
#define dump(x) true
#endif

template<unsigned int mod_>
struct ModInt{
	using uint = unsigned int;
	using ll = long long;
	using ull = unsigned long long;

	constexpr static uint mod = mod_;

	uint v;
	ModInt():v(0){}
	ModInt(ll _v):v(normS(_v%mod+mod)){}
	explicit operator bool() const {return v!=0;}
	static uint normS(const uint &x){return (x<mod)?x:x-mod;}		// [0 , 2*mod-1] -> [0 , mod-1]
	static ModInt make(const uint &x){ModInt m; m.v=x; return m;}
	ModInt operator+(const ModInt& b) const { return make(normS(v+b.v));}
	ModInt operator-(const ModInt& b) const { return make(normS(v+mod-b.v));}
	ModInt operator-() const { return make(normS(mod-v)); }
	ModInt operator*(const ModInt& b) const { return make((ull)v*b.v%mod);}
	ModInt operator/(const ModInt& b) const { return *this*b.inv();}
	ModInt& operator+=(const ModInt& b){ return *this=*this+b;}
	ModInt& operator-=(const ModInt& b){ return *this=*this-b;}
	ModInt& operator*=(const ModInt& b){ return *this=*this*b;}
	ModInt& operator/=(const ModInt& b){ return *this=*this/b;}
	ModInt& operator++(int){ return *this=*this+1;}
	ModInt& operator--(int){ return *this=*this-1;}
	ll extgcd(ll a,ll b,ll &x,ll &y) const{
		ll p[]={a,1,0},q[]={b,0,1};
		while(*q){
			ll t=*p/ *q;
			rep(i,3) swap(p[i]-=t*q[i],q[i]);
		}
		if(p[0]<0) rep(i,3) p[i]=-p[i];
		x=p[1],y=p[2];
		return p[0];
	}
	ModInt inv() const {
		ll x,y;
		extgcd(v,mod,x,y);
		return make(normS(x+mod));
	}
	ModInt pow(ll p) const {
		if(p<0) return inv().pow(-p);
		ModInt a = 1;
		ModInt x = *this;
		while(p){
			if(p&1) a *= x;
			x *= x;
			p >>= 1;
		}
		return a;
	}
	bool operator==(const ModInt& b) const { return v==b.v;}
	bool operator!=(const ModInt& b) const { return v!=b.v;}
	friend istream& operator>>(istream &o,ModInt& x){
		ll tmp;
		o>>tmp;
		x=ModInt(tmp);
		return o;
	}
	friend ostream& operator<<(ostream &o,const ModInt& x){ return o<<x.v;}
};
using mint = ModInt<998244353>;
struct UnionFind{
	vector<int> par,rank;
	UnionFind(int N){
		par.assign(N,0);
		rank.assign(N,0);
		rep(i,N) par[i]=i;
	}
	int find(int x){
		if(par[x]==x) return x;
		return par[x]=find(par[x]);
	}
	bool same(int x,int y){
		return find(x)==find(y);
	}
	void unite(int x,int y){
		x=find(x),y=find(y);
		if(x==y) return;
		if(rank[x]<rank[y]) swap(x,y);

		//x becomes root
		par[y]=x;
		if(rank[x]==rank[y]) rank[x]++;
	}
};
int main(){
	cin.tie(0);
	ios::sync_with_stdio(false);		//DON'T USE scanf/printf/puts !!
	cout << fixed << setprecision(20);

	int N; cin >> N; N--;
	V<string> s(N); rep(i,N) cin >> s[i];
	VV<int> is(N,V<int>(N));
	rep(i,N) rep(j,N){
		int x=i,y=j;
		if(x>y) swap(x,y);
		if(x+y>=N){
			swap(x,y);x=N-1-x,y=N-1-y;
		}
		// cout<<x<<y<<" ";
		// if(j == N-1) cout << endl;
		if(s[i][j] == 'o') is[x][y] |= 2;
		if(s[i][j] == 'x') is[x][y] |= 1;
	}
	rep(i,N) rep(j,N) if(is[i][j] == 3){
		cout << 0 << endl;
		return 0;
	}
	int RANK = 0;
	rep(r,2){
		show(r);show("======================");
		int n = (r == 0) ? (N+3)/2 : (N+2)/2;
		show(n);
		V<int> deg0(n),deg1(n);
		UnionFind UF(n);
		int M = 0;
		rep(i,N) rep(j,N) if(i<=j && i+j<=N-1 && (i+j)%2 == r){
			int u = (j-i)/2;
			int v = (j+i)/2+1;
			show(u);show(v);
			assert(0<=u&&u<n);
			assert(0<=v&&v<n);
			show(is[i][j]);
			show("-");
			if(is[i][j] == 0){
				UF.unite(u,v);
				M++;
			}else if(is[i][j] == 1){
				deg0[u]++,deg0[v]++;
			}else{
				deg1[u]++,deg1[v]++;
			}
		}
		RANK += M;
		if(n&1){
			RANK -= n;
			map<int,V<int>> cons;
			rep(v,n) cons[UF.find(v)].pb(v);
			for(auto con: cons){
				auto vs = con.sc;
				int odd = 0;
				for(int v: vs) odd += deg1[v];
				if(odd&1){
					cout << 0 << endl;
					return 0;
				}
				RANK++;
			}
		}
		show(RANK);
	}
	cout << mint(2).pow(RANK) << endl;
}

```